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Theorem rhmsubc 46941
Description: According to df-subc 17755, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17786 and subcss2 17789). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.)
Hypotheses
Ref Expression
rngcrescrhm.u (𝜑𝑈𝑉)
rngcrescrhm.c 𝐶 = (RngCat‘𝑈)
rngcrescrhm.r (𝜑𝑅 = (Ring ∩ 𝑈))
rngcrescrhm.h 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
Assertion
Ref Expression
rhmsubc (𝜑𝐻 ∈ (Subcat‘(RngCat‘𝑈)))

Proof of Theorem rhmsubc
Dummy variables 𝑥 𝑦 𝑧 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngcrescrhm.u . . . 4 (𝜑𝑈𝑉)
2 rngcrescrhm.r . . . 4 (𝜑𝑅 = (Ring ∩ 𝑈))
3 eqidd 2733 . . . 4 (𝜑 → (Rng ∩ 𝑈) = (Rng ∩ 𝑈))
41, 2, 3rhmsscrnghm 46877 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
5 rngcrescrhm.h . . . 4 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
65a1i 11 . . 3 (𝜑𝐻 = ( RingHom ↾ (𝑅 × 𝑅)))
7 rngcrescrhm.c . . . . . . 7 𝐶 = (RngCat‘𝑈)
87a1i 11 . . . . . 6 (𝜑𝐶 = (RngCat‘𝑈))
98eqcomd 2738 . . . . 5 (𝜑 → (RngCat‘𝑈) = 𝐶)
109fveq2d 6892 . . . 4 (𝜑 → (Homf ‘(RngCat‘𝑈)) = (Homf𝐶))
11 eqid 2732 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
127, 11, 1rngchomfeqhom 46820 . . . 4 (𝜑 → (Homf𝐶) = (Hom ‘𝐶))
13 eqid 2732 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
147, 11, 1, 13rngchomfval 46817 . . . . 5 (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))))
157, 11, 1rngcbas 46816 . . . . . . . 8 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
16 incom 4200 . . . . . . . 8 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
1715, 16eqtrdi 2788 . . . . . . 7 (𝜑 → (Base‘𝐶) = (Rng ∩ 𝑈))
1817sqxpeqd 5707 . . . . . 6 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))
1918reseq2d 5979 . . . . 5 (𝜑 → ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
2014, 19eqtrd 2772 . . . 4 (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
2110, 12, 203eqtrd 2776 . . 3 (𝜑 → (Homf ‘(RngCat‘𝑈)) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
224, 6, 213brtr4d 5179 . 2 (𝜑𝐻cat (Homf ‘(RngCat‘𝑈)))
231, 7, 2, 5rhmsubclem3 46939 . . . 4 ((𝜑𝑥𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
241, 7, 2, 5rhmsubclem4 46940 . . . . . 6 ((((𝜑𝑥𝑅) ∧ (𝑦𝑅𝑧𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2524ralrimivva 3200 . . . . 5 (((𝜑𝑥𝑅) ∧ (𝑦𝑅𝑧𝑅)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2625ralrimivva 3200 . . . 4 ((𝜑𝑥𝑅) → ∀𝑦𝑅𝑧𝑅𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2723, 26jca 512 . . 3 ((𝜑𝑥𝑅) → (((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝑅𝑧𝑅𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
2827ralrimiva 3146 . 2 (𝜑 → ∀𝑥𝑅 (((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝑅𝑧𝑅𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
29 eqid 2732 . . 3 (Homf ‘(RngCat‘𝑈)) = (Homf ‘(RngCat‘𝑈))
30 eqid 2732 . . 3 (Id‘(RngCat‘𝑈)) = (Id‘(RngCat‘𝑈))
31 eqid 2732 . . 3 (comp‘(RngCat‘𝑈)) = (comp‘(RngCat‘𝑈))
32 eqid 2732 . . . . 5 (RngCat‘𝑈) = (RngCat‘𝑈)
3332rngccat 46829 . . . 4 (𝑈𝑉 → (RngCat‘𝑈) ∈ Cat)
341, 33syl 17 . . 3 (𝜑 → (RngCat‘𝑈) ∈ Cat)
351, 7, 2, 5rhmsubclem1 46937 . . 3 (𝜑𝐻 Fn (𝑅 × 𝑅))
3629, 30, 31, 34, 35issubc2 17782 . 2 (𝜑 → (𝐻 ∈ (Subcat‘(RngCat‘𝑈)) ↔ (𝐻cat (Homf ‘(RngCat‘𝑈)) ∧ ∀𝑥𝑅 (((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝑅𝑧𝑅𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)))))
3722, 28, 36mpbir2and 711 1 (𝜑𝐻 ∈ (Subcat‘(RngCat‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  cin 3946  cop 4633   class class class wbr 5147   × cxp 5673  cres 5677  cfv 6540  (class class class)co 7405  Basecbs 17140  Hom chom 17204  compcco 17205  Catccat 17604  Idccid 17605  Homf chomf 17606  cat cssc 17750  Subcatcsubc 17752  Ringcrg 20049   RingHom crh 20240  Rngcrng 46634   RngHomo crngh 46668  RngCatcrngc 46808
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-hom 17217  df-cco 17218  df-0g 17383  df-cat 17608  df-cid 17609  df-homf 17610  df-ssc 17753  df-resc 17754  df-subc 17755  df-estrc 18070  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-grp 18818  df-minusg 18819  df-ghm 19084  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-rnghom 20243  df-mgmhm 46535  df-rng 46635  df-rnghomo 46670  df-rngc 46810
This theorem is referenced by:  rhmsubccat  46942
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