Theorem rngcifuestrc 20691

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 2764	. . . 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 20673	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
6		incom 4163	. . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
7	5, 6	eqtrdi 2815	. . . 4 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))
8		eqid 2764	. . . . 5 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
9	3, 4, 2, 8	rngchomfval 20674	. . . 4 ⊢ (𝜑 → (Hom ‘𝑅) = ( RngHom ↾ (𝐵 × 𝐵)))
10	1, 2, 7, 9	rnghmsubcsetc 20685	. . 3 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
11		eqid 2764	. . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))
12		eqid 2764	. . 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 20670	. . . . . . 7 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
15	14	fveq2d 6873	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
16	4, 15	eqtrid 2811	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
17	16	reseq2d 5967	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
18	13, 17	eqtrd 2799	. . 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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
21	9	oveqdr 7426	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(Hom ‘𝑅)𝑦) = (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑦))
22		ovres 7564	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHom 𝑦))
23	22	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHom 𝑦))
24	21, 23	eqtr2d 2800	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHom 𝑦) = (𝑥(Hom ‘𝑅)𝑦))
25	24	reseq2d 5967	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHom 𝑦)) = ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))
26	16, 20, 25	mpoeq123dva 7472	. . . 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 2799	. . 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 17978	. 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 7416	. . 3 ⊢ (𝜑 → (𝑅 Func 𝐸) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈)))
32	31	breqd 5113	. 2 ⊢ (𝜑 → (𝐹(𝑅 Func 𝐸)𝐺 ↔ 𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺))
33	28, 32	mpbird 259	1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝐸)𝐺)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144   ∩ cin 3905   class class class wbr 5102   I cid 5543   × cxp 5647   ↾ cres 5651  ‘cfv 6523  (class class class)co 7398   ∈ cmpo 7400  Basecbs 17247  Hom chom 17299   ↾_cat cresc 17843   Func cfunc 17889  ExtStrCatcestrc 18156  Rngcrng 20200   RngHom crnghm 20485  RngCatcrngc 20668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-map 8812  df-pm 8813  df-ixp 8882  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-z 12571  df-dec 12691  df-uz 12842  df-fz 13515  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17248  df-ress 17269  df-plusg 17301  df-hom 17312  df-cco 17313  df-0g 17472  df-cat 17702  df-cid 17703  df-homf 17704  df-ssc 17845  df-resc 17846  df-subc 17847  df-func 17893  df-idfu 17894  df-full 17941  df-fth 17942  df-estrc 18157  df-mgm 18676  df-mgmhm 18728  df-sgrp 18755  df-mnd 18771  df-mhm 18819  df-grp 18980  df-ghm 19256  df-abl 19825  df-mgp 20189  df-rng 20201  df-rnghm 20487  df-rngc 20669

This theorem is referenced by:  funcrngcsetcALT  20693