Theorem rngcresringcat 20595

Description: The restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings. (Contributed by AV, 16-Mar-2020.)

Hypotheses

Ref	Expression
rhmsubcrngc.c	⊢ 𝐶 = (RngCat‘𝑈)
rhmsubcrngc.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rhmsubcrngc.b	⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))
rhmsubcrngc.h	⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))

Assertion

Ref	Expression
rngcresringcat	⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈))

Proof of Theorem rngcresringcat

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmsubcrngc.c	. . . 4 ⊢ 𝐶 = (RngCat‘𝑈)
2		rhmsubcrngc.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		eqidd 2728	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4		eqidd 2728	. . . 4 ⊢ (𝜑 → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
5		eqidd 2728	. . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)))
6	1, 2, 3, 4, 5	dfrngc2 20554	. . 3 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
7		inex1g 5313	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
8	2, 7	syl 17	. . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
9		rnghmfn 20371	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
10		fnfun 6648	. . . . 5 ⊢ ( RngHom Fn (Rng × Rng) → Fun RngHom )
11	9, 10	mp1i 13	. . . 4 ⊢ (𝜑 → Fun RngHom )
12		sqxpexg 7751	. . . . 5 ⊢ ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
13	8, 12	syl 17	. . . 4 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
14		resfunexg 7221	. . . 4 ⊢ ((Fun RngHom ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
15	11, 13, 14	syl2anc 583	. . 3 ⊢ (𝜑 → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
16		fvexd 6906	. . 3 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
17		rhmsubcrngc.h	. . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
18		rhmfn 20431	. . . . . 6 ⊢ RingHom Fn (Ring × Ring)
19		fnfun 6648	. . . . . 6 ⊢ ( RingHom Fn (Ring × Ring) → Fun RingHom )
20	18, 19	mp1i 13	. . . . 5 ⊢ (𝜑 → Fun RingHom )
21		rhmsubcrngc.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))
22		incom 4197	. . . . . . . 8 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
23	21, 22	eqtrdi 2783	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
24		inex1g 5313	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
25	2, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V)
26	23, 25	eqeltrd 2828	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
27		sqxpexg 7751	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
28	26, 27	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V)
29		resfunexg 7221	. . . . 5 ⊢ ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
30	20, 28, 29	syl2anc 583	. . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
31	17, 30	eqeltrd 2828	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
32		ringrng 20214	. . . . . . 7 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng)
33	32	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
34	33	ssrdv 3984	. . . . 5 ⊢ (𝜑 → Ring ⊆ Rng)
35	34	ssrind 4231	. . . 4 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
36		incom 4197	. . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
37	36	a1i 11	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
38	35, 21, 37	3sstr4d 4025	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝑈 ∩ Rng))
39	6, 8, 15, 16, 31, 38	estrres 18123	. 2 ⊢ (𝜑 → ((𝐶 ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
40		eqid 2727	. . 3 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻)
41		fvexd 6906	. . . 4 ⊢ (𝜑 → (RngCat‘𝑈) ∈ V)
42	1, 41	eqeltrid 2832	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
43	23, 17	rhmresfn 20574	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
44	40, 42, 26, 43	rescval2 17804	. 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
45		eqid 2727	. . 3 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈)
46	45, 2, 23, 17, 5	dfringc2 20583	. 2 ⊢ (𝜑 → (RingCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
47	39, 44, 46	3eqtr4d 2777	1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3469   ∩ cin 3943  {ctp 4628  ⟨cop 4630   × cxp 5670   ↾ cres 5674  Fun wfun 6536   Fn wfn 6537  ‘cfv 6542  (class class class)co 7414   sSet csts 17125  ndxcnx 17155  Basecbs 17173   ↾_s cress 17202  Hom chom 17237  compcco 17238   ↾_cat cresc 17784  ExtStrCatcestrc 18105  Rngcrng 20085  Ringcrg 20166   RngHom crnghm 20366   RingHom crh 20401  RngCatcrngc 20542  RingCatcringc 20571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12497  df-z 12583  df-dec 12702  df-uz 12847  df-fz 13511  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17174  df-ress 17203  df-plusg 17239  df-hom 17250  df-cco 17251  df-0g 17416  df-resc 17787  df-estrc 18106  df-mgm 18593  df-sgrp 18672  df-mnd 18688  df-mhm 18733  df-grp 18886  df-minusg 18887  df-ghm 19161  df-cmn 19730  df-abl 19731  df-mgp 20068  df-rng 20086  df-ur 20115  df-ring 20168  df-rnghm 20368  df-rhm 20404  df-rngc 20543  df-ringc 20572

This theorem is referenced by: (None)