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Theorem rngcifuestrc 41789
 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 2610 . . . . 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 41757 . . . . . 6 (𝜑𝐵 = (𝑈 ∩ Rng))
6 incom 3767 . . . . . 6 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
75, 6syl6eq 2660 . . . . 5 (𝜑𝐵 = (Rng ∩ 𝑈))
8 eqid 2610 . . . . . 6 (Hom ‘𝑅) = (Hom ‘𝑅)
93, 4, 2, 8rngchomfval 41758 . . . . 5 (𝜑 → (Hom ‘𝑅) = ( RngHomo ↾ (𝐵 × 𝐵)))
101, 2, 7, 9rnghmsubcsetc 41769 . . . 4 (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
1110idi 2 . . 3 (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
12 eqid 2610 . . 3 ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))
13 eqid 2610 . . 3 (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
14 rngcifuestrc.f . . . 4 (𝜑𝐹 = ( I ↾ 𝐵))
153, 2, 5, 9rngcval 41754 . . . . . . 7 (𝜑𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
1615fveq2d 6107 . . . . . 6 (𝜑 → (Base‘𝑅) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
174, 16syl5eq 2656 . . . . 5 (𝜑𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
1817reseq2d 5317 . . . 4 (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
1914, 18eqtrd 2644 . . 3 (𝜑𝐹 = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
20 rngcifuestrc.g . . . 4 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
2117adantr 480 . . . . 5 ((𝜑𝑥𝐵) → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
229oveqdr 6573 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(Hom ‘𝑅)𝑦) = (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦))
23 ovres 6698 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦))
2423adantl 481 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦))
2522, 24eqtr2d 2645 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 RngHomo 𝑦) = (𝑥(Hom ‘𝑅)𝑦))
2625reseq2d 5317 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))
2717, 21, 26mpt2eq123dva 6614 . . . 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 2644 . . 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 41657 . 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 6567 . . 3 (𝜑 → (𝑅 Func 𝐸) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈)))
3332breqd 4594 . 2 (𝜑 → (𝐹(𝑅 Func 𝐸)𝐺𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺))
3429, 33mpbird 246 1 (𝜑𝐹(𝑅 Func 𝐸)𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   class class class wbr 4583   I cid 4948   × cxp 5036   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  Hom chom 15779   ↾cat cresc 16291  Subcatcsubc 16292   Func cfunc 16337  ExtStrCatcestrc 16585  Rngcrng 41664   RngHomo crngh 41675  RngCatcrngc 41749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-hom 15793  df-cco 15794  df-0g 15925  df-cat 16152  df-cid 16153  df-homf 16154  df-ssc 16293  df-resc 16294  df-subc 16295  df-func 16341  df-idfu 16342  df-full 16387  df-fth 16388  df-estrc 16586  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-ghm 17481  df-abl 18019  df-mgp 18313  df-mgmhm 41569  df-rng0 41665  df-rnghomo 41677  df-rngc 41751 This theorem is referenced by:  funcrngcsetcALT  41791
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