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

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
  Copyright terms: Public domain W3C validator