Theorem rngcifuestrc 45443

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	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rngcifuestrc.f	⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵))
rngcifuestrc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))

Assertion

Ref	Expression
rngcifuestrc	⊢ (𝜑 → 𝐹(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

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . 5 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2		rngcifuestrc.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		rngcifuestrc.r	. . . . . . 7 ⊢ 𝑅 = (RngCat‘𝑈)
4		rngcifuestrc.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4, 2	rngcbas 45411	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
6		incom 4131	. . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
7	5, 6	eqtrdi 2795	. . . . 5 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))
8		eqid 2738	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
9	3, 4, 2, 8	rngchomfval 45412	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) = ( RngHomo ↾ (𝐵 × 𝐵)))
10	1, 2, 7, 9	rnghmsubcsetc 45423	. . . 4 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
11	10	idi 1	. . 3 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
12		eqid 2738	. . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))
13		eqid 2738	. . 3 ⊢ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
14		rngcifuestrc.f	. . . 4 ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵))
15	3, 2, 5, 9	rngcval 45408	. . . . . . 7 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
16	15	fveq2d 6760	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
17	4, 16	syl5eq 2791	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
18	17	reseq2d 5880	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
19	14, 18	eqtrd 2778	. . 3 ⊢ (𝜑 → 𝐹 = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
20		rngcifuestrc.g	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
21	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
22	9	oveqdr 7283	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(Hom ‘𝑅)𝑦) = (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦))
23		ovres 7416	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦))
24	23	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦))
25	22, 24	eqtr2d 2779	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHomo 𝑦) = (𝑥(Hom ‘𝑅)𝑦))
26	25	reseq2d 5880	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))
27	17, 21, 26	mpoeq123dva 7327	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦))))
28	20, 27	eqtrd 2778	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦))))
29	11, 12, 13, 19, 28	inclfusubc 45313	. 2 ⊢ (𝜑 → 𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺)
30		rngcifuestrc.e	. . . . 5 ⊢ 𝐸 = (ExtStrCat‘𝑈)
31	30	a1i 11	. . . 4 ⊢ (𝜑 → 𝐸 = (ExtStrCat‘𝑈))
32	15, 31	oveq12d 7273	. . 3 ⊢ (𝜑 → (𝑅 Func 𝐸) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈)))
33	32	breqd 5081	. 2 ⊢ (𝜑 → (𝐹(𝑅 Func 𝐸)𝐺 ↔ 𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺))
34	29, 33	mpbird 256	1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝐸)𝐺)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∩ cin 3882   class class class wbr 5070   I cid 5479   × cxp 5578   ↾ cres 5582  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  Basecbs 16840  Hom chom 16899   ↾_cat cresc 17437  Subcatcsubc 17438   Func cfunc 17485  ExtStrCatcestrc 17754  Rngcrng 45320   RngHomo crngh 45331  RngCatcrngc 45403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-hom 16912  df-cco 16913  df-0g 17069  df-cat 17294  df-cid 17295  df-homf 17296  df-ssc 17439  df-resc 17440  df-subc 17441  df-func 17489  df-idfu 17490  df-full 17536  df-fth 17537  df-estrc 17755  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-grp 18495  df-ghm 18747  df-abl 19304  df-mgp 19636  df-mgmhm 45221  df-rng0 45321  df-rnghomo 45333  df-rngc 45405

This theorem is referenced by:  funcrngcsetcALT  45445