Theorem rngcresringcat 45476

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 2739	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4		eqidd 2739	. . . 4 ⊢ (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
5		eqidd 2739	. . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)))
6	1, 2, 3, 4, 5	dfrngc2 45418	. . 3 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
7		inex1g 5238	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
8	2, 7	syl 17	. . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
9		rnghmfn 45336	. . . . 5 ⊢ RngHomo Fn (Rng × Rng)
10		fnfun 6517	. . . . 5 ⊢ ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
11	9, 10	mp1i 13	. . . 4 ⊢ (𝜑 → Fun RngHomo )
12		sqxpexg 7583	. . . . 5 ⊢ ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
13	8, 12	syl 17	. . . 4 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
14		resfunexg 7073	. . . 4 ⊢ ((Fun RngHomo ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
15	11, 13, 14	syl2anc 583	. . 3 ⊢ (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
16		fvexd 6771	. . 3 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
17		rhmsubcrngc.h	. . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
18		rhmfn 45364	. . . . . 6 ⊢ RingHom Fn (Ring × Ring)
19		fnfun 6517	. . . . . 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 4131	. . . . . . . 8 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
23	21, 22	eqtrdi 2795	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
24		inex1g 5238	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
25	2, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V)
26	23, 25	eqeltrd 2839	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
27		sqxpexg 7583	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
28	26, 27	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V)
29		resfunexg 7073	. . . . 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 2839	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
32		ringrng 45325	. . . . . . 7 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng)
33	32	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
34	33	ssrdv 3923	. . . . 5 ⊢ (𝜑 → Ring ⊆ Rng)
35	34	ssrind 4166	. . . 4 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
36		incom 4131	. . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
37	36	a1i 11	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
38	35, 21, 37	3sstr4d 3964	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝑈 ∩ Rng))
39	6, 8, 15, 16, 31, 38	estrres 17772	. 2 ⊢ (𝜑 → ((𝐶 ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
40		eqid 2738	. . 3 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻)
41		fvexd 6771	. . . 4 ⊢ (𝜑 → (RngCat‘𝑈) ∈ V)
42	1, 41	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
43	23, 17	rhmresfn 45455	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
44	40, 42, 26, 43	rescval2 17457	. 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
45		eqid 2738	. . 3 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈)
46	45, 2, 23, 17, 5	dfringc2 45464	. 2 ⊢ (𝜑 → (RingCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
47	39, 44, 46	3eqtr4d 2788	1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882  {ctp 4562  ⟨cop 4564   × cxp 5578   ↾ cres 5582  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255   sSet csts 16792  ndxcnx 16822  Basecbs 16840   ↾_s cress 16867  Hom chom 16899  compcco 16900   ↾_cat cresc 17437  ExtStrCatcestrc 17754  Ringcrg 19698   RingHom crh 19871  Rngcrng 45320   RngHomo crngh 45331  RngCatcrngc 45403  RingCatcringc 45449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-hom 16912  df-cco 16913  df-0g 17069  df-resc 17440  df-estrc 17755  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-grp 18495  df-minusg 18496  df-ghm 18747  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-rnghom 19874  df-rng0 45321  df-rnghomo 45333  df-rngc 45405  df-ringc 45451

This theorem is referenced by: (None)