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Theorem rhmsubcrngc 44646
Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of non-unital rings. (Contributed by AV, 12-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
rhmsubcrngc (𝜑𝐻 ∈ (Subcat‘𝐶))

Proof of Theorem rhmsubcrngc
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rhmsubcrngc.u . . . . 5 (𝜑𝑈𝑉)
2 rhmsubcrngc.b . . . . 5 (𝜑𝐵 = (Ring ∩ 𝑈))
3 eqid 2801 . . . . . . 7 (RngCat‘𝑈) = (RngCat‘𝑈)
4 eqid 2801 . . . . . . 7 (Base‘(RngCat‘𝑈)) = (Base‘(RngCat‘𝑈))
53, 4, 1rngcbas 44582 . . . . . 6 (𝜑 → (Base‘(RngCat‘𝑈)) = (𝑈 ∩ Rng))
6 incom 4131 . . . . . 6 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
75, 6eqtrdi 2852 . . . . 5 (𝜑 → (Base‘(RngCat‘𝑈)) = (Rng ∩ 𝑈))
81, 2, 7rhmsscrnghm 44643 . . . 4 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ⊆cat ( RngHomo ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈)))))
9 rhmsubcrngc.c . . . . . . . 8 𝐶 = (RngCat‘𝑈)
109a1i 11 . . . . . . 7 (𝜑𝐶 = (RngCat‘𝑈))
1110fveq2d 6653 . . . . . 6 (𝜑 → (Base‘𝐶) = (Base‘(RngCat‘𝑈)))
1211sqxpeqd 5555 . . . . 5 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))))
1312reseq2d 5822 . . . 4 (𝜑 → ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHomo ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈)))))
148, 13breqtrrd 5061 . . 3 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ⊆cat ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))))
15 rhmsubcrngc.h . . 3 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
16 eqid 2801 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
179, 16, 1rngchomfeqhom 44586 . . . 4 (𝜑 → (Homf𝐶) = (Hom ‘𝐶))
18 eqid 2801 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
199, 16, 1, 18rngchomfval 44583 . . . 4 (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))))
2017, 19eqtrd 2836 . . 3 (𝜑 → (Homf𝐶) = ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))))
2114, 15, 203brtr4d 5065 . 2 (𝜑𝐻cat (Homf𝐶))
229, 1, 2, 15rhmsubcrngclem1 44644 . . . 4 ((𝜑𝑥𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
239, 1, 2, 15rhmsubcrngclem2 44645 . . . 4 ((𝜑𝑥𝐵) → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2422, 23jca 515 . . 3 ((𝜑𝑥𝐵) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
2524ralrimiva 3152 . 2 (𝜑 → ∀𝑥𝐵 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
26 eqid 2801 . . 3 (Homf𝐶) = (Homf𝐶)
27 eqid 2801 . . 3 (Id‘𝐶) = (Id‘𝐶)
28 eqid 2801 . . 3 (comp‘𝐶) = (comp‘𝐶)
299rngccat 44595 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
301, 29syl 17 . . 3 (𝜑𝐶 ∈ Cat)
31 incom 4131 . . . . 5 (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
322, 31eqtrdi 2852 . . . 4 (𝜑𝐵 = (𝑈 ∩ Ring))
3332, 15rhmresfn 44626 . . 3 (𝜑𝐻 Fn (𝐵 × 𝐵))
3426, 27, 28, 30, 33issubc2 17102 . 2 (𝜑 → (𝐻 ∈ (Subcat‘𝐶) ↔ (𝐻cat (Homf𝐶) ∧ ∀𝑥𝐵 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)))))
3521, 25, 34mpbir2and 712 1 (𝜑𝐻 ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  cin 3883  cop 4534   class class class wbr 5033   × cxp 5521  cres 5525  cfv 6328  (class class class)co 7139  Basecbs 16479  Hom chom 16572  compcco 16573  Catccat 16931  Idccid 16932  Homf chomf 16933  cat cssc 17073  Subcatcsubc 17075  Ringcrg 19294   RingHom crh 19464  Rngcrng 44491   RngHomo crngh 44502  RngCatcrngc 44574
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-pm 8396  df-ixp 8449  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-cat 16935  df-cid 16936  df-homf 16937  df-ssc 17076  df-resc 17077  df-subc 17078  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-mgmhm 44392  df-rng0 44492  df-rnghomo 44504  df-rngc 44576  df-ringc 44622
This theorem is referenced by: (None)
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