Theorem rngcresringcat 20634

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 2737	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4		eqidd 2737	. . . 4 ⊢ (𝜑 → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
5		eqidd 2737	. . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)))
6	1, 2, 3, 4, 5	dfrngc2 20593	. . 3 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
7		inex1g 5294	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
8	2, 7	syl 17	. . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
9		rnghmfn 20404	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
10		fnfun 6643	. . . . 5 ⊢ ( RngHom Fn (Rng × Rng) → Fun RngHom )
11	9, 10	mp1i 13	. . . 4 ⊢ (𝜑 → Fun RngHom )
12		sqxpexg 7754	. . . . 5 ⊢ ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
13	8, 12	syl 17	. . . 4 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
14		resfunexg 7212	. . . 4 ⊢ ((Fun RngHom ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
15	11, 13, 14	syl2anc 584	. . 3 ⊢ (𝜑 → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
16		fvexd 6896	. . 3 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
17		rhmsubcrngc.h	. . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
18		rhmfn 20464	. . . . . 6 ⊢ RingHom Fn (Ring × Ring)
19		fnfun 6643	. . . . . 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 4189	. . . . . . . 8 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
23	21, 22	eqtrdi 2787	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
24		inex1g 5294	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
25	2, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V)
26	23, 25	eqeltrd 2835	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
27		sqxpexg 7754	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
28	26, 27	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V)
29		resfunexg 7212	. . . . 5 ⊢ ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
30	20, 28, 29	syl2anc 584	. . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
31	17, 30	eqeltrd 2835	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
32		ringrng 20250	. . . . . . 7 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng)
33	32	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
34	33	ssrdv 3969	. . . . 5 ⊢ (𝜑 → Ring ⊆ Rng)
35	34	ssrind 4224	. . . 4 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
36		incom 4189	. . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
37	36	a1i 11	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
38	35, 21, 37	3sstr4d 4019	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝑈 ∩ Rng))
39	6, 8, 15, 16, 31, 38	estrres 18156	. 2 ⊢ (𝜑 → ((𝐶 ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
40		eqid 2736	. . 3 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻)
41		fvexd 6896	. . . 4 ⊢ (𝜑 → (RngCat‘𝑈) ∈ V)
42	1, 41	eqeltrid 2839	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
43	23, 17	rhmresfn 20613	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
44	40, 42, 26, 43	rescval2 17846	. 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
45		eqid 2736	. . 3 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈)
46	45, 2, 23, 17, 5	dfringc2 20622	. 2 ⊢ (𝜑 → (RingCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
47	39, 44, 46	3eqtr4d 2781	1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∩ cin 3930  {ctp 4610  ⟨cop 4612   × cxp 5657   ↾ cres 5661  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   sSet csts 17187  ndxcnx 17217  Basecbs 17233   ↾_s cress 17256  Hom chom 17287  compcco 17288   ↾_cat cresc 17826  ExtStrCatcestrc 18139  Rngcrng 20117  Ringcrg 20198   RngHom crnghm 20399   RingHom crh 20434  RngCatcrngc 20581  RingCatcringc 20610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-hom 17300  df-cco 17301  df-0g 17460  df-resc 17829  df-estrc 18140  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-mhm 18766  df-grp 18924  df-minusg 18925  df-ghm 19201  df-cmn 19768  df-abl 19769  df-mgp 20106  df-rng 20118  df-ur 20147  df-ring 20200  df-rnghm 20401  df-rhm 20437  df-rngc 20582  df-ringc 20611

This theorem is referenced by: (None)