Theorem rngcifuestrc 20570

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 RngHom 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 2734	. . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2		rngcifuestrc.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		rngcifuestrc.r	. . . . . 6 ⊢ 𝑅 = (RngCat‘𝑈)
4		rngcifuestrc.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4, 2	rngcbas 20552	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
6		incom 4159	. . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
7	5, 6	eqtrdi 2785	. . . 4 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))
8		eqid 2734	. . . . 5 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
9	3, 4, 2, 8	rngchomfval 20553	. . . 4 ⊢ (𝜑 → (Hom ‘𝑅) = ( RngHom ↾ (𝐵 × 𝐵)))
10	1, 2, 7, 9	rnghmsubcsetc 20564	. . 3 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
11		eqid 2734	. . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))
12		eqid 2734	. . 3 ⊢ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
13		rngcifuestrc.f	. . . 4 ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵))
14	3, 2, 5, 9	rngcval 20549	. . . . . . 7 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
15	14	fveq2d 6836	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
16	4, 15	eqtrid 2781	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
17	16	reseq2d 5936	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
18	13, 17	eqtrd 2769	. . 3 ⊢ (𝜑 → 𝐹 = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
19		rngcifuestrc.g	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))))
20	16	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
21	9	oveqdr 7384	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(Hom ‘𝑅)𝑦) = (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑦))
22		ovres 7522	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHom 𝑦))
23	22	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHom 𝑦))
24	21, 23	eqtr2d 2770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHom 𝑦) = (𝑥(Hom ‘𝑅)𝑦))
25	24	reseq2d 5936	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHom 𝑦)) = ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))
26	16, 20, 25	mpoeq123dva 7430	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))) = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦))))
27	19, 26	eqtrd 2769	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦))))
28	10, 11, 12, 18, 27	inclfusubc 17865	. 2 ⊢ (𝜑 → 𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺)
29		rngcifuestrc.e	. . . . 5 ⊢ 𝐸 = (ExtStrCat‘𝑈)
30	29	a1i 11	. . . 4 ⊢ (𝜑 → 𝐸 = (ExtStrCat‘𝑈))
31	14, 30	oveq12d 7374	. . 3 ⊢ (𝜑 → (𝑅 Func 𝐸) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈)))
32	31	breqd 5107	. 2 ⊢ (𝜑 → (𝐹(𝑅 Func 𝐸)𝐺 ↔ 𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺))
33	28, 32	mpbird 257	1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝐸)𝐺)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∩ cin 3898   class class class wbr 5096   I cid 5516   × cxp 5620   ↾ cres 5624  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  Basecbs 17134  Hom chom 17186   ↾_cat cresc 17730   Func cfunc 17776  ExtStrCatcestrc 18043  Rngcrng 20085   RngHom crnghm 20368  RngCatcrngc 20547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8763  df-pm 8764  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-z 12487  df-dec 12606  df-uz 12750  df-fz 13422  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-hom 17199  df-cco 17200  df-0g 17359  df-cat 17589  df-cid 17590  df-homf 17591  df-ssc 17732  df-resc 17733  df-subc 17734  df-func 17780  df-idfu 17781  df-full 17828  df-fth 17829  df-estrc 18044  df-mgm 18563  df-mgmhm 18615  df-sgrp 18642  df-mnd 18658  df-mhm 18706  df-grp 18864  df-ghm 19140  df-abl 19710  df-mgp 20074  df-rng 20086  df-rnghm 20370  df-rngc 20548

This theorem is referenced by:  funcrngcsetcALT  20572