Theorem rngcifuestrc 46885

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 2732	. . . . 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 46853	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
6		incom 4201	. . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
7	5, 6	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))
8		eqid 2732	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
9	3, 4, 2, 8	rngchomfval 46854	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) = ( RngHomo ↾ (𝐵 × 𝐵)))
10	1, 2, 7, 9	rnghmsubcsetc 46865	. . . 4 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
11	10	idi 1	. . 3 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
12		eqid 2732	. . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))
13		eqid 2732	. . 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 46850	. . . . . . 7 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
16	15	fveq2d 6895	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
17	4, 16	eqtrid 2784	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
18	17	reseq2d 5981	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
19	14, 18	eqtrd 2772	. . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
22	9	oveqdr 7436	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(Hom ‘𝑅)𝑦) = (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦))
23		ovres 7572	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦))
24	23	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦))
25	22, 24	eqtr2d 2773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHomo 𝑦) = (𝑥(Hom ‘𝑅)𝑦))
26	25	reseq2d 5981	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))
27	17, 21, 26	mpoeq123dva 7482	. . . 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 2772	. . 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 46631	. 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 7426	. . 3 ⊢ (𝜑 → (𝑅 Func 𝐸) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈)))
33	32	breqd 5159	. 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 396   = wceq 1541   ∈ wcel 2106   ∩ cin 3947   class class class wbr 5148   I cid 5573   × cxp 5674   ↾ cres 5678  ‘cfv 6543  (class class class)co 7408   ∈ cmpo 7410  Basecbs 17143  Hom chom 17207   ↾_cat cresc 17754  Subcatcsubc 17755   Func cfunc 17803  ExtStrCatcestrc 18072  Rngcrng 46638   RngHomo crngh 46673  RngCatcrngc 46845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-pm 8822  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-uz 12822  df-fz 13484  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-hom 17220  df-cco 17221  df-0g 17386  df-cat 17611  df-cid 17612  df-homf 17613  df-ssc 17756  df-resc 17757  df-subc 17758  df-func 17807  df-idfu 17808  df-full 17854  df-fth 17855  df-estrc 18073  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-mhm 18670  df-grp 18821  df-ghm 19089  df-abl 19650  df-mgp 19987  df-mgmhm 46539  df-rng 46639  df-rnghomo 46675  df-rngc 46847

This theorem is referenced by:  funcrngcsetcALT  46887