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Theorem rhmsubc 42618
 Description: According to df-subc 16679, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16707 and subcss2 16710). 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 2772 . . . 4 (𝜑 → (Rng ∩ 𝑈) = (Rng ∩ 𝑈))
41, 2, 3rhmsscrnghm 42554 . . 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 2777 . . . . 5 (𝜑 → (RngCat‘𝑈) = 𝐶)
109fveq2d 6336 . . . 4 (𝜑 → (Homf ‘(RngCat‘𝑈)) = (Homf𝐶))
11 eqid 2771 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
127, 11, 1rngchomfeqhom 42497 . . . 4 (𝜑 → (Homf𝐶) = (Hom ‘𝐶))
13 eqid 2771 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
147, 11, 1, 13rngchomfval 42494 . . . . 5 (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))))
157, 11, 1rngcbas 42493 . . . . . . . 8 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
16 incom 3956 . . . . . . . 8 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
1715, 16syl6eq 2821 . . . . . . 7 (𝜑 → (Base‘𝐶) = (Rng ∩ 𝑈))
1817sqxpeqd 5281 . . . . . 6 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))
1918reseq2d 5534 . . . . 5 (𝜑 → ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
2014, 19eqtrd 2805 . . . 4 (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
2110, 12, 203eqtrd 2809 . . 3 (𝜑 → (Homf ‘(RngCat‘𝑈)) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
224, 6, 213brtr4d 4818 . 2 (𝜑𝐻cat (Homf ‘(RngCat‘𝑈)))
231, 7, 2, 5rhmsubclem3 42616 . . . 4 ((𝜑𝑥𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
241, 7, 2, 5rhmsubclem4 42617 . . . . . 6 ((((𝜑𝑥𝑅) ∧ (𝑦𝑅𝑧𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2524ralrimivva 3120 . . . . 5 (((𝜑𝑥𝑅) ∧ (𝑦𝑅𝑧𝑅)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2625ralrimivva 3120 . . . 4 ((𝜑𝑥𝑅) → ∀𝑦𝑅𝑧𝑅𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2723, 26jca 501 . . 3 ((𝜑𝑥𝑅) → (((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝑅𝑧𝑅𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
2827ralrimiva 3115 . 2 (𝜑 → ∀𝑥𝑅 (((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝑅𝑧𝑅𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
29 eqid 2771 . . 3 (Homf ‘(RngCat‘𝑈)) = (Homf ‘(RngCat‘𝑈))
30 eqid 2771 . . 3 (Id‘(RngCat‘𝑈)) = (Id‘(RngCat‘𝑈))
31 eqid 2771 . . 3 (comp‘(RngCat‘𝑈)) = (comp‘(RngCat‘𝑈))
32 eqid 2771 . . . . 5 (RngCat‘𝑈) = (RngCat‘𝑈)
3332rngccat 42506 . . . 4 (𝑈𝑉 → (RngCat‘𝑈) ∈ Cat)
341, 33syl 17 . . 3 (𝜑 → (RngCat‘𝑈) ∈ Cat)
351, 7, 2, 5rhmsubclem1 42614 . . 3 (𝜑𝐻 Fn (𝑅 × 𝑅))
3629, 30, 31, 34, 35issubc2 16703 . 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 692 1 (𝜑𝐻 ∈ (Subcat‘(RngCat‘𝑈)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061   ∩ cin 3722  ⟨cop 4322   class class class wbr 4786   × cxp 5247   ↾ cres 5251  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532  Idccid 16533  Homf chomf 16534   ⊆cat cssc 16674  Subcatcsubc 16676  Ringcrg 18755   RingHom crh 18922  Rngcrng 42402   RngHomo crngh 42413  RngCatcrngc 42485 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-fz 12534  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-hom 16174  df-cco 16175  df-0g 16310  df-cat 16536  df-cid 16537  df-homf 16538  df-ssc 16677  df-resc 16678  df-subc 16679  df-estrc 16970  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-grp 17633  df-minusg 17634  df-ghm 17866  df-cmn 18402  df-abl 18403  df-mgp 18698  df-ur 18710  df-ring 18757  df-rnghom 18925  df-mgmhm 42307  df-rng0 42403  df-rnghomo 42415  df-rngc 42487 This theorem is referenced by:  rhmsubccat  42619
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