Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rhmsubcrngc Structured version   Visualization version   GIF version

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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