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Theorem rngcifuestrc 42525
Description: The "inclusion functor" from the category of non-unital rings into the category of extensible structures. (Contributed by AV, 30-Mar-2020.)
Hypotheses
Ref Expression
rngcifuestrc.r 𝑅 = (RngCat‘𝑈)
rngcifuestrc.e 𝐸 = (ExtStrCat‘𝑈)
rngcifuestrc.b 𝐵 = (Base‘𝑅)
rngcifuestrc.u (𝜑𝑈𝑉)
rngcifuestrc.f (𝜑𝐹 = ( I ↾ 𝐵))
rngcifuestrc.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
Assertion
Ref Expression
rngcifuestrc (𝜑𝐹(𝑅 Func 𝐸)𝐺)
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rngcifuestrc
StepHypRef Expression
1 eqid 2771 . . . . 5 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2 rngcifuestrc.u . . . . 5 (𝜑𝑈𝑉)
3 rngcifuestrc.r . . . . . . 7 𝑅 = (RngCat‘𝑈)
4 rngcifuestrc.b . . . . . . 7 𝐵 = (Base‘𝑅)
53, 4, 2rngcbas 42493 . . . . . 6 (𝜑𝐵 = (𝑈 ∩ Rng))
6 incom 3956 . . . . . 6 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
75, 6syl6eq 2821 . . . . 5 (𝜑𝐵 = (Rng ∩ 𝑈))
8 eqid 2771 . . . . . 6 (Hom ‘𝑅) = (Hom ‘𝑅)
93, 4, 2, 8rngchomfval 42494 . . . . 5 (𝜑 → (Hom ‘𝑅) = ( RngHomo ↾ (𝐵 × 𝐵)))
101, 2, 7, 9rnghmsubcsetc 42505 . . . 4 (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
1110idi 2 . . 3 (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
12 eqid 2771 . . 3 ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))
13 eqid 2771 . . 3 (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
14 rngcifuestrc.f . . . 4 (𝜑𝐹 = ( I ↾ 𝐵))
153, 2, 5, 9rngcval 42490 . . . . . . 7 (𝜑𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
1615fveq2d 6336 . . . . . 6 (𝜑 → (Base‘𝑅) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
174, 16syl5eq 2817 . . . . 5 (𝜑𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
1817reseq2d 5534 . . . 4 (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
1914, 18eqtrd 2805 . . 3 (𝜑𝐹 = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
20 rngcifuestrc.g . . . 4 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
2117adantr 466 . . . . 5 ((𝜑𝑥𝐵) → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
229oveqdr 6819 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(Hom ‘𝑅)𝑦) = (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦))
23 ovres 6947 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦))
2423adantl 467 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦))
2522, 24eqtr2d 2806 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 RngHomo 𝑦) = (𝑥(Hom ‘𝑅)𝑦))
2625reseq2d 5534 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))
2717, 21, 26mpt2eq123dva 6863 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦))))
2820, 27eqtrd 2805 . . 3 (𝜑𝐺 = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦))))
2911, 12, 13, 19, 28inclfusubc 42395 . 2 (𝜑𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺)
30 rngcifuestrc.e . . . . 5 𝐸 = (ExtStrCat‘𝑈)
3130a1i 11 . . . 4 (𝜑𝐸 = (ExtStrCat‘𝑈))
3215, 31oveq12d 6811 . . 3 (𝜑 → (𝑅 Func 𝐸) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈)))
3332breqd 4797 . 2 (𝜑 → (𝐹(𝑅 Func 𝐸)𝐺𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺))
3429, 33mpbird 247 1 (𝜑𝐹(𝑅 Func 𝐸)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cin 3722   class class class wbr 4786   I cid 5156   × cxp 5247  cres 5251  cfv 6031  (class class class)co 6793  cmpt2 6795  Basecbs 16064  Hom chom 16160  cat cresc 16675  Subcatcsubc 16676   Func cfunc 16721  ExtStrCatcestrc 16969  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 835  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-func 16725  df-idfu 16726  df-full 16771  df-fth 16772  df-estrc 16970  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-grp 17633  df-ghm 17866  df-abl 18403  df-mgp 18698  df-mgmhm 42307  df-rng0 42403  df-rnghomo 42415  df-rngc 42487
This theorem is referenced by:  funcrngcsetcALT  42527
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