Theorem rngcifuestrc 20559

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 2729	. . . 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 20541	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
6		incom 4168	. . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
7	5, 6	eqtrdi 2780	. . . 4 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))
8		eqid 2729	. . . . 5 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
9	3, 4, 2, 8	rngchomfval 20542	. . . 4 ⊢ (𝜑 → (Hom ‘𝑅) = ( RngHom ↾ (𝐵 × 𝐵)))
10	1, 2, 7, 9	rnghmsubcsetc 20553	. . 3 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
11		eqid 2729	. . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))
12		eqid 2729	. . 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 20538	. . . . . . 7 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
15	14	fveq2d 6844	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
16	4, 15	eqtrid 2776	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
17	16	reseq2d 5939	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
18	13, 17	eqtrd 2764	. . 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 7397	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(Hom ‘𝑅)𝑦) = (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑦))
22		ovres 7535	. . . . . . . 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 2765	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHom 𝑦) = (𝑥(Hom ‘𝑅)𝑦))
25	24	reseq2d 5939	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHom 𝑦)) = ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))
26	16, 20, 25	mpoeq123dva 7443	. . . 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 2764	. . 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 17885	. 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 7387	. . 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 257	1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝐸)𝐺)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3910   class class class wbr 5102   I cid 5525   × cxp 5629   ↾ cres 5633  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  Basecbs 17155  Hom chom 17207   ↾_cat cresc 17750   Func cfunc 17796  ExtStrCatcestrc 18063  Rngcrng 20072   RngHom crnghm 20354  RngCatcrngc 20536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-hom 17220  df-cco 17221  df-0g 17380  df-cat 17609  df-cid 17610  df-homf 17611  df-ssc 17752  df-resc 17753  df-subc 17754  df-func 17800  df-idfu 17801  df-full 17848  df-fth 17849  df-estrc 18064  df-mgm 18549  df-mgmhm 18601  df-sgrp 18628  df-mnd 18644  df-mhm 18692  df-grp 18850  df-ghm 19127  df-abl 19697  df-mgp 20061  df-rng 20073  df-rnghm 20356  df-rngc 20537

This theorem is referenced by:  funcrngcsetcALT  20561