Theorem fucoaltvallem 50697

Description: Lemma for fucoaltval 50698. (Contributed by Zhi Wang, 14-Sep-2026.) (New usage is discouraged.)

Hypotheses

Ref	Expression
fucoaltfunc.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fucoaltval.l	⊢ 𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)
fucoaltval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
fucoaltval.e	⊢ (𝜑 → 𝐸 ∈ Cat)
fucoaltval.k	⊢ (𝜑 → 𝐾 ∈ (𝑄 Func 𝑄))
fucoaltval.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐸))
fucoaltval.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
fucoaltval.a	⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺))
fucoaltval.i	⊢ 𝐼 = (idfunc‘𝐶)
fucoaltval.j	⊢ 𝐽 = (Id‘(𝐶 FuncCat 𝐶))
fucoaltvallem.t	⊢ 𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))

Assertion

Ref	Expression
fucoaltvallem	⊢ (𝜑 → (𝐴(⟨𝐹, 𝐼⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, 𝐼⟩)(𝐽‘𝐼)) = ((𝐹(2nd ‘𝐾)𝐺)‘𝐴))

Proof of Theorem fucoaltvallem

Step	Hyp	Ref	Expression
1		df-ov 7446	. 2 ⊢ (𝐴(⟨𝐹, 𝐼⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, 𝐼⟩)(𝐽‘𝐼)) = ((⟨𝐹, 𝐼⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, 𝐼⟩)‘⟨𝐴, (𝐽‘𝐼)⟩)
2		fucoaltvallem.t	. . . . 5 ⊢ 𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))
3	2	xpcfucbas 50579	. . . 4 ⊢ ((𝐶 Func 𝐸) × (𝐶 Func 𝐶)) = (Base‘𝑇)
4		fucoaltval.l	. . . . 5 ⊢ 𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)
5		fucoaltfunc.q	. . . . . 6 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
6		fucoaltval.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		fucoaltval.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
8	2, 5, 6, 6, 7	fucofunca 50687	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐶⟩ ∘F 𝐸) ∈ (𝑇 Func 𝑄))
9	4, 8	eqeltrid 2868	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (𝑇 Func 𝑄))
10		fucoaltval.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝑄 Func 𝑄))
11		fucoaltval.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐸))
12		fucoaltval.i	. . . . . . 7 ⊢ 𝐼 = (idfunc‘𝐶)
13	12	idfucl 18085	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
14	6, 13	syl 18	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝐶 Func 𝐶))
15	11, 14	opelxpd 5694	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐼⟩ ∈ ((𝐶 Func 𝐸) × (𝐶 Func 𝐶)))
16		fucoaltval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
17	16, 14	opelxpd 5694	. . . 4 ⊢ (𝜑 → ⟨𝐺, 𝐼⟩ ∈ ((𝐶 Func 𝐸) × (𝐶 Func 𝐶)))
18		eqid 2763	. . . 4 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
19		fucoaltval.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺))
20		eqid 2763	. . . . . . . 8 ⊢ (𝐶 FuncCat 𝐶) = (𝐶 FuncCat 𝐶)
21	20	fucbas 18167	. . . . . . 7 ⊢ (𝐶 Func 𝐶) = (Base‘(𝐶 FuncCat 𝐶))
22		eqid 2763	. . . . . . . 8 ⊢ (𝐶 Nat 𝐶) = (𝐶 Nat 𝐶)
23	20, 22	fuchom 18168	. . . . . . 7 ⊢ (𝐶 Nat 𝐶) = (Hom ‘(𝐶 FuncCat 𝐶))
24		fucoaltval.j	. . . . . . 7 ⊢ 𝐽 = (Id‘(𝐶 FuncCat 𝐶))
25	20, 6, 6	fuccat 18177	. . . . . . 7 ⊢ (𝜑 → (𝐶 FuncCat 𝐶) ∈ Cat)
26	21, 23, 24, 25, 14	catidcl 17885	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝐼) ∈ (𝐼(𝐶 Nat 𝐶)𝐼))
27	19, 26	opelxpd 5694	. . . . 5 ⊢ (𝜑 → ⟨𝐴, (𝐽‘𝐼)⟩ ∈ ((𝐹(𝐶 Nat 𝐸)𝐺) × (𝐼(𝐶 Nat 𝐶)𝐼)))
28	2, 11, 14, 16, 14, 18	xpcfuchom2 50582	. . . . 5 ⊢ (𝜑 → (⟨𝐹, 𝐼⟩(Hom ‘𝑇)⟨𝐺, 𝐼⟩) = ((𝐹(𝐶 Nat 𝐸)𝐺) × (𝐼(𝐶 Nat 𝐶)𝐼)))
29	27, 28	eleqtrrd 2867	. . . 4 ⊢ (𝜑 → ⟨𝐴, (𝐽‘𝐼)⟩ ∈ (⟨𝐹, 𝐼⟩(Hom ‘𝑇)⟨𝐺, 𝐼⟩))
30	3, 9, 10, 15, 17, 18, 29	cofu2 18090	. . 3 ⊢ (𝜑 → ((⟨𝐹, 𝐼⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, 𝐼⟩)‘⟨𝐴, (𝐽‘𝐼)⟩) = ((((1st ‘𝐿)‘⟨𝐹, 𝐼⟩)(2nd ‘𝐾)((1st ‘𝐿)‘⟨𝐺, 𝐼⟩))‘((⟨𝐹, 𝐼⟩(2nd ‘𝐿)⟨𝐺, 𝐼⟩)‘⟨𝐴, (𝐽‘𝐼)⟩)))
31		df-ov 7446	. . . . . 6 ⊢ (𝐹(1st ‘𝐿)𝐼) = ((1st ‘𝐿)‘⟨𝐹, 𝐼⟩)
32	4	eqcomi 2772	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐶⟩ ∘F 𝐸) = 𝐿
33	32	fveq2i 6911	. . . . . . . . 9 ⊢ (1st ‘(⟨𝐶, 𝐶⟩ ∘F 𝐸)) = (1st ‘𝐿)
34	33	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐶⟩ ∘F 𝐸)) = (1st ‘𝐿))
35	34, 14, 11	fuco11b 50664	. . . . . . 7 ⊢ (𝜑 → (𝐹(1st ‘𝐿)𝐼) = (𝐹 ∘func 𝐼))
36	11, 12	cofurid 18095	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘func 𝐼) = 𝐹)
37	35, 36	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → (𝐹(1st ‘𝐿)𝐼) = 𝐹)
38	31, 37	eqtr3id 2812	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐿)‘⟨𝐹, 𝐼⟩) = 𝐹)
39		df-ov 7446	. . . . . 6 ⊢ (𝐺(1st ‘𝐿)𝐼) = ((1st ‘𝐿)‘⟨𝐺, 𝐼⟩)
40	34, 14, 16	fuco11b 50664	. . . . . . 7 ⊢ (𝜑 → (𝐺(1st ‘𝐿)𝐼) = (𝐺 ∘func 𝐼))
41	16, 12	cofurid 18095	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∘func 𝐼) = 𝐺)
42	40, 41	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → (𝐺(1st ‘𝐿)𝐼) = 𝐺)
43	39, 42	eqtr3id 2812	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐿)‘⟨𝐺, 𝐼⟩) = 𝐺)
44	38, 43	oveq12d 7461	. . . 4 ⊢ (𝜑 → (((1st ‘𝐿)‘⟨𝐹, 𝐼⟩)(2nd ‘𝐾)((1st ‘𝐿)‘⟨𝐺, 𝐼⟩)) = (𝐹(2nd ‘𝐾)𝐺))
45		df-ov 7446	. . . . 5 ⊢ (𝐴(⟨𝐹, 𝐼⟩(2nd ‘𝐿)⟨𝐺, 𝐼⟩)(𝐽‘𝐼)) = ((⟨𝐹, 𝐼⟩(2nd ‘𝐿)⟨𝐺, 𝐼⟩)‘⟨𝐴, (𝐽‘𝐼)⟩)
46	32	fveq2i 6911	. . . . . . 7 ⊢ (2nd ‘(⟨𝐶, 𝐶⟩ ∘F 𝐸)) = (2nd ‘𝐿)
47	46	a1i 11	. . . . . 6 ⊢ (𝜑 → (2nd ‘(⟨𝐶, 𝐶⟩ ∘F 𝐸)) = (2nd ‘𝐿))
48	47, 20, 24, 12, 6, 19	fucoridid 50691	. . . . 5 ⊢ (𝜑 → (𝐴(⟨𝐹, 𝐼⟩(2nd ‘𝐿)⟨𝐺, 𝐼⟩)(𝐽‘𝐼)) = 𝐴)
49	45, 48	eqtr3id 2812	. . . 4 ⊢ (𝜑 → ((⟨𝐹, 𝐼⟩(2nd ‘𝐿)⟨𝐺, 𝐼⟩)‘⟨𝐴, (𝐽‘𝐼)⟩) = 𝐴)
50	44, 49	fveq12d 6915	. . 3 ⊢ (𝜑 → ((((1st ‘𝐿)‘⟨𝐹, 𝐼⟩)(2nd ‘𝐾)((1st ‘𝐿)‘⟨𝐺, 𝐼⟩))‘((⟨𝐹, 𝐼⟩(2nd ‘𝐿)⟨𝐺, 𝐼⟩)‘⟨𝐴, (𝐽‘𝐼)⟩)) = ((𝐹(2nd ‘𝐾)𝐺)‘𝐴))
51	30, 50	eqtrd 2798	. 2 ⊢ (𝜑 → ((⟨𝐹, 𝐼⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, 𝐼⟩)‘⟨𝐴, (𝐽‘𝐼)⟩) = ((𝐹(2nd ‘𝐾)𝐺)‘𝐴))
52	1, 51	eqtrid 2810	1 ⊢ (𝜑 → (𝐴(⟨𝐹, 𝐼⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, 𝐼⟩)(𝐽‘𝐼)) = ((𝐹(2nd ‘𝐾)𝐺)‘𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ⟨cop 4838   × cxp 5653  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024  Hom chom 17468  Catccat 17867  Idccid 17868   Func cfunc 18058  id_funccidfu 18059   ∘_func ccofu 18060   Nat cnat 18148   FuncCat cfuc 18149   ×_c cxpc 18371   ∘_F cfuco 50643

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-map 8877  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-struct 17354  df-slot 17389  df-ndx 17401  df-base 17417  df-hom 17481  df-cco 17482  df-cat 17871  df-cid 17872  df-func 18062  df-idfu 18063  df-cofu 18064  df-nat 18150  df-fuc 18151  df-xpc 18375  df-fuco 50644

This theorem is used by:  fucoaltval  50698