Theorem rngcresringcat 45947

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 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
5		eqidd 2737	. . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)))
6	1, 2, 3, 4, 5	dfrngc2 45889	. . 3 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
7		inex1g 5263	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
8	2, 7	syl 17	. . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
9		rnghmfn 45807	. . . . 5 ⊢ RngHomo Fn (Rng × Rng)
10		fnfun 6585	. . . . 5 ⊢ ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
11	9, 10	mp1i 13	. . . 4 ⊢ (𝜑 → Fun RngHomo )
12		sqxpexg 7667	. . . . 5 ⊢ ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
13	8, 12	syl 17	. . . 4 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
14		resfunexg 7147	. . . 4 ⊢ ((Fun RngHomo ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
15	11, 13, 14	syl2anc 584	. . 3 ⊢ (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
16		fvexd 6840	. . 3 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
17		rhmsubcrngc.h	. . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
18		rhmfn 45835	. . . . . 6 ⊢ RingHom Fn (Ring × Ring)
19		fnfun 6585	. . . . . 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 4148	. . . . . . . 8 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
23	21, 22	eqtrdi 2792	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
24		inex1g 5263	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
25	2, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V)
26	23, 25	eqeltrd 2837	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
27		sqxpexg 7667	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
28	26, 27	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V)
29		resfunexg 7147	. . . . 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 2837	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
32		ringrng 45796	. . . . . . 7 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng)
33	32	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
34	33	ssrdv 3938	. . . . 5 ⊢ (𝜑 → Ring ⊆ Rng)
35	34	ssrind 4182	. . . 4 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
36		incom 4148	. . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
37	36	a1i 11	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
38	35, 21, 37	3sstr4d 3979	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝑈 ∩ Rng))
39	6, 8, 15, 16, 31, 38	estrres 17953	. 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 6840	. . . 4 ⊢ (𝜑 → (RngCat‘𝑈) ∈ V)
42	1, 41	eqeltrid 2841	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
43	23, 17	rhmresfn 45926	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
44	40, 42, 26, 43	rescval2 17637	. 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 45935	. 2 ⊢ (𝜑 → (RingCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
47	39, 44, 46	3eqtr4d 2786	1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  Vcvv 3441   ∩ cin 3897  {ctp 4577  ⟨cop 4579   × cxp 5618   ↾ cres 5622  Fun wfun 6473   Fn wfn 6474  ‘cfv 6479  (class class class)co 7337   sSet csts 16961  ndxcnx 16991  Basecbs 17009   ↾_s cress 17038  Hom chom 17070  compcco 17071   ↾_cat cresc 17617  ExtStrCatcestrc 17935  Ringcrg 19878   RingHom crh 20051  Rngcrng 45791   RngHomo crngh 45802  RngCatcrngc 45874  RingCatcringc 45920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-er 8569  df-map 8688  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-2 12137  df-3 12138  df-4 12139  df-5 12140  df-6 12141  df-7 12142  df-8 12143  df-9 12144  df-n0 12335  df-z 12421  df-dec 12539  df-uz 12684  df-fz 13341  df-struct 16945  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-hom 17083  df-cco 17084  df-0g 17249  df-resc 17620  df-estrc 17936  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-mhm 18527  df-grp 18676  df-minusg 18677  df-ghm 18928  df-cmn 19483  df-abl 19484  df-mgp 19816  df-ur 19833  df-ring 19880  df-rnghom 20054  df-rng0 45792  df-rnghomo 45804  df-rngc 45876  df-ringc 45922

This theorem is referenced by: (None)