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Theorem rngcresringcat 44647
 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 (𝜑𝑈𝑉)
rhmsubcrngc.b (𝜑𝐵 = (Ring ∩ 𝑈))
rhmsubcrngc.h (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
Assertion
Ref Expression
rngcresringcat (𝜑 → (𝐶cat 𝐻) = (RingCat‘𝑈))

Proof of Theorem rngcresringcat
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 rhmsubcrngc.c . . . 4 𝐶 = (RngCat‘𝑈)
2 rhmsubcrngc.u . . . 4 (𝜑𝑈𝑉)
3 eqidd 2802 . . . 4 (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4 eqidd 2802 . . . 4 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
5 eqidd 2802 . . . 4 (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)))
61, 2, 3, 4, 5dfrngc2 44589 . . 3 (𝜑𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
7 inex1g 5190 . . . 4 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
82, 7syl 17 . . 3 (𝜑 → (𝑈 ∩ Rng) ∈ V)
9 rnghmfn 44507 . . . . 5 RngHomo Fn (Rng × Rng)
10 fnfun 6427 . . . . 5 ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
119, 10mp1i 13 . . . 4 (𝜑 → Fun RngHomo )
12 sqxpexg 7461 . . . . 5 ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
138, 12syl 17 . . . 4 (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
14 resfunexg 6959 . . . 4 ((Fun RngHomo ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
1511, 13, 14syl2anc 587 . . 3 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
16 fvexd 6664 . . 3 (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
17 rhmsubcrngc.h . . . 4 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
18 rhmfn 44535 . . . . . 6 RingHom Fn (Ring × Ring)
19 fnfun 6427 . . . . . 6 ( RingHom Fn (Ring × Ring) → Fun RingHom )
2018, 19mp1i 13 . . . . 5 (𝜑 → Fun RingHom )
21 rhmsubcrngc.b . . . . . . . 8 (𝜑𝐵 = (Ring ∩ 𝑈))
22 incom 4131 . . . . . . . 8 (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
2321, 22eqtrdi 2852 . . . . . . 7 (𝜑𝐵 = (𝑈 ∩ Ring))
24 inex1g 5190 . . . . . . . 8 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
252, 24syl 17 . . . . . . 7 (𝜑 → (𝑈 ∩ Ring) ∈ V)
2623, 25eqeltrd 2893 . . . . . 6 (𝜑𝐵 ∈ V)
27 sqxpexg 7461 . . . . . 6 (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
2826, 27syl 17 . . . . 5 (𝜑 → (𝐵 × 𝐵) ∈ V)
29 resfunexg 6959 . . . . 5 ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3020, 28, 29syl2anc 587 . . . 4 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3117, 30eqeltrd 2893 . . 3 (𝜑𝐻 ∈ V)
32 ringrng 44496 . . . . . . 7 (𝑟 ∈ Ring → 𝑟 ∈ Rng)
3332a1i 11 . . . . . 6 (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
3433ssrdv 3924 . . . . 5 (𝜑 → Ring ⊆ Rng)
3534ssrind 4165 . . . 4 (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
36 incom 4131 . . . . 5 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
3736a1i 11 . . . 4 (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
3835, 21, 373sstr4d 3965 . . 3 (𝜑𝐵 ⊆ (𝑈 ∩ Rng))
396, 8, 15, 16, 31, 38estrres 17385 . 2 (𝜑 → ((𝐶s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
40 eqid 2801 . . 3 (𝐶cat 𝐻) = (𝐶cat 𝐻)
41 fvexd 6664 . . . 4 (𝜑 → (RngCat‘𝑈) ∈ V)
421, 41eqeltrid 2897 . . 3 (𝜑𝐶 ∈ V)
4323, 17rhmresfn 44626 . . 3 (𝜑𝐻 Fn (𝐵 × 𝐵))
4440, 42, 26, 43rescval2 17094 . 2 (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
45 eqid 2801 . . 3 (RingCat‘𝑈) = (RingCat‘𝑈)
4645, 2, 23, 17, 5dfringc2 44635 . 2 (𝜑 → (RingCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
4739, 44, 463eqtr4d 2846 1 (𝜑 → (𝐶cat 𝐻) = (RingCat‘𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  Vcvv 3444   ∩ cin 3883  {ctp 4532  ⟨cop 4534   × cxp 5521   ↾ cres 5525  Fun wfun 6322   Fn wfn 6323  ‘cfv 6328  (class class class)co 7139  ndxcnx 16476   sSet csts 16477  Basecbs 16479   ↾s cress 16480  Hom chom 16572  compcco 16573   ↾cat cresc 17074  ExtStrCatcestrc 17368  Ringcrg 19294   RingHom crh 19464  Rngcrng 44491   RngHomo crngh 44502  RngCatcrngc 44574  RingCatcringc 44620 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-hom 16585  df-cco 16586  df-0g 16711  df-resc 17077  df-estrc 17369  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-mhm 17952  df-grp 18102  df-minusg 18103  df-ghm 18352  df-cmn 18904  df-abl 18905  df-mgp 19237  df-ur 19249  df-ring 19296  df-rnghom 19467  df-rng0 44492  df-rnghomo 44504  df-rngc 44576  df-ringc 44622 This theorem is referenced by: (None)
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