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Theorem rnghmsubcsetc 41238
Description: The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory of the category of extensible structures. (Contributed by AV, 9-Mar-2020.)
Hypotheses
Ref Expression
rnghmsubcsetc.c 𝐶 = (ExtStrCat‘𝑈)
rnghmsubcsetc.u (𝜑𝑈𝑉)
rnghmsubcsetc.b (𝜑𝐵 = (Rng ∩ 𝑈))
rnghmsubcsetc.h (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
Assertion
Ref Expression
rnghmsubcsetc (𝜑𝐻 ∈ (Subcat‘𝐶))

Proof of Theorem rnghmsubcsetc
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghmsubcsetc.u . . . 4 (𝜑𝑈𝑉)
2 rnghmsubcsetc.b . . . 4 (𝜑𝐵 = (Rng ∩ 𝑈))
31, 2rnghmsscmap 41235 . . 3 (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
4 rnghmsubcsetc.h . . 3 (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
5 rnghmsubcsetc.c . . . . 5 𝐶 = (ExtStrCat‘𝑈)
6 eqid 2626 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
75, 1, 6estrchomfeqhom 16692 . . . 4 (𝜑 → (Homf𝐶) = (Hom ‘𝐶))
85, 1, 6estrchomfval 16682 . . . 4 (𝜑 → (Hom ‘𝐶) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
97, 8eqtrd 2660 . . 3 (𝜑 → (Homf𝐶) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
103, 4, 93brtr4d 4650 . 2 (𝜑𝐻cat (Homf𝐶))
115, 1, 2, 4rnghmsubcsetclem1 41236 . . . 4 ((𝜑𝑥𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
125, 1, 2, 4rnghmsubcsetclem2 41237 . . . 4 ((𝜑𝑥𝐵) → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))
1311, 12jca 554 . . 3 ((𝜑𝑥𝐵) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
1413ralrimiva 2965 . 2 (𝜑 → ∀𝑥𝐵 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
15 eqid 2626 . . 3 (Homf𝐶) = (Homf𝐶)
16 eqid 2626 . . 3 (Id‘𝐶) = (Id‘𝐶)
17 eqid 2626 . . 3 (comp‘𝐶) = (comp‘𝐶)
185estrccat 16689 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
191, 18syl 17 . . 3 (𝜑𝐶 ∈ Cat)
20 incom 3788 . . . . 5 (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
212, 20syl6eq 2676 . . . 4 (𝜑𝐵 = (𝑈 ∩ Rng))
2221, 4rnghmresfn 41224 . . 3 (𝜑𝐻 Fn (𝐵 × 𝐵))
2315, 16, 17, 19, 22issubc2 16412 . 2 (𝜑 → (𝐻 ∈ (Subcat‘𝐶) ↔ (𝐻cat (Homf𝐶) ∧ ∀𝑥𝐵 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)))))
2410, 14, 23mpbir2and 956 1 (𝜑𝐻 ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  cin 3559  cop 4159   class class class wbr 4618   × cxp 5077  cres 5081  cfv 5850  (class class class)co 6605  cmpt2 6607  𝑚 cmap 7803  Basecbs 15776  Hom chom 15868  compcco 15869  Catccat 16241  Idccid 16242  Homf chomf 16243  cat cssc 16383  Subcatcsubc 16385  ExtStrCatcestrc 16678  Rngcrng 41135   RngHomo crngh 41146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-fz 12266  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-hom 15882  df-cco 15883  df-0g 16018  df-cat 16245  df-cid 16246  df-homf 16247  df-ssc 16386  df-resc 16387  df-subc 16388  df-estrc 16679  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-grp 17341  df-ghm 17574  df-abl 18112  df-mgp 18406  df-mgmhm 41040  df-rng0 41136  df-rnghomo 41148  df-rngc 41220
This theorem is referenced by:  rngccat  41239  rngcid  41240  rngcifuestrc  41258  funcrngcsetc  41259
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