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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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