Theorem fucoaltvallem 50583

Description: Lemma for fucoaltval 50584. (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 7423	. 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 50465	. . . 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 50573	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐶⟩ ∘F 𝐸) ∈ (𝑇 Func 𝑄))
9	4, 8	eqeltrid 2864	. . . 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 18058	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
14	6, 13	syl 18	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝐶 Func 𝐶))
15	11, 14	opelxpd 5689	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐼⟩ ∈ ((𝐶 Func 𝐸) × (𝐶 Func 𝐶)))
16		fucoaltval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
17	16, 14	opelxpd 5689	. . . 4 ⊢ (𝜑 → ⟨𝐺, 𝐼⟩ ∈ ((𝐶 Func 𝐸) × (𝐶 Func 𝐶)))
18		eqid 2760	. . . 4 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
19		fucoaltval.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺))
20		eqid 2760	. . . . . . . 8 ⊢ (𝐶 FuncCat 𝐶) = (𝐶 FuncCat 𝐶)
21	20	fucbas 18140	. . . . . . 7 ⊢ (𝐶 Func 𝐶) = (Base‘(𝐶 FuncCat 𝐶))
22		eqid 2760	. . . . . . . 8 ⊢ (𝐶 Nat 𝐶) = (𝐶 Nat 𝐶)
23	20, 22	fuchom 18141	. . . . . . 7 ⊢ (𝐶 Nat 𝐶) = (Hom ‘(𝐶 FuncCat 𝐶))
24		fucoaltval.j	. . . . . . 7 ⊢ 𝐽 = (Id‘(𝐶 FuncCat 𝐶))
25	20, 6, 6	fuccat 18150	. . . . . . 7 ⊢ (𝜑 → (𝐶 FuncCat 𝐶) ∈ Cat)
26	21, 23, 24, 25, 14	catidcl 17858	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝐼) ∈ (𝐼(𝐶 Nat 𝐶)𝐼))
27	19, 26	opelxpd 5689	. . . . 5 ⊢ (𝜑 → ⟨𝐴, (𝐽‘𝐼)⟩ ∈ ((𝐹(𝐶 Nat 𝐸)𝐺) × (𝐼(𝐶 Nat 𝐶)𝐼)))
28	2, 11, 14, 16, 14, 18	xpcfuchom2 50468	. . . . 5 ⊢ (𝜑 → (⟨𝐹, 𝐼⟩(Hom ‘𝑇)⟨𝐺, 𝐼⟩) = ((𝐹(𝐶 Nat 𝐸)𝐺) × (𝐼(𝐶 Nat 𝐶)𝐼)))
29	27, 28	eleqtrrd 2863	. . . 4 ⊢ (𝜑 → ⟨𝐴, (𝐽‘𝐼)⟩ ∈ (⟨𝐹, 𝐼⟩(Hom ‘𝑇)⟨𝐺, 𝐼⟩))
30	3, 9, 10, 15, 17, 18, 29	cofu2 18063	. . 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 7423	. . . . . 6 ⊢ (𝐹(1st ‘𝐿)𝐼) = ((1st ‘𝐿)‘⟨𝐹, 𝐼⟩)
32	4	eqcomi 2769	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐶⟩ ∘F 𝐸) = 𝐿
33	32	fveq2i 6888	. . . . . . . . 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 50550	. . . . . . 7 ⊢ (𝜑 → (𝐹(1st ‘𝐿)𝐼) = (𝐹 ∘func 𝐼))
36	11, 12	cofurid 18068	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘func 𝐼) = 𝐹)
37	35, 36	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → (𝐹(1st ‘𝐿)𝐼) = 𝐹)
38	31, 37	eqtr3id 2809	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐿)‘⟨𝐹, 𝐼⟩) = 𝐹)
39		df-ov 7423	. . . . . 6 ⊢ (𝐺(1st ‘𝐿)𝐼) = ((1st ‘𝐿)‘⟨𝐺, 𝐼⟩)
40	34, 14, 16	fuco11b 50550	. . . . . . 7 ⊢ (𝜑 → (𝐺(1st ‘𝐿)𝐼) = (𝐺 ∘func 𝐼))
41	16, 12	cofurid 18068	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∘func 𝐼) = 𝐺)
42	40, 41	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → (𝐺(1st ‘𝐿)𝐼) = 𝐺)
43	39, 42	eqtr3id 2809	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐿)‘⟨𝐺, 𝐼⟩) = 𝐺)
44	38, 43	oveq12d 7438	. . . 4 ⊢ (𝜑 → (((1st ‘𝐿)‘⟨𝐹, 𝐼⟩)(2nd ‘𝐾)((1st ‘𝐿)‘⟨𝐺, 𝐼⟩)) = (𝐹(2nd ‘𝐾)𝐺))
45		df-ov 7423	. . . . 5 ⊢ (𝐴(⟨𝐹, 𝐼⟩(2nd ‘𝐿)⟨𝐺, 𝐼⟩)(𝐽‘𝐼)) = ((⟨𝐹, 𝐼⟩(2nd ‘𝐿)⟨𝐺, 𝐼⟩)‘⟨𝐴, (𝐽‘𝐼)⟩)
46	32	fveq2i 6888	. . . . . . 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 50577	. . . . 5 ⊢ (𝜑 → (𝐴(⟨𝐹, 𝐼⟩(2nd ‘𝐿)⟨𝐺, 𝐼⟩)(𝐽‘𝐼)) = 𝐴)
49	45, 48	eqtr3id 2809	. . . 4 ⊢ (𝜑 → ((⟨𝐹, 𝐼⟩(2nd ‘𝐿)⟨𝐺, 𝐼⟩)‘⟨𝐴, (𝐽‘𝐼)⟩) = 𝐴)
50	44, 49	fveq12d 6892	. . 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 2795	. 2 ⊢ (𝜑 → ((⟨𝐹, 𝐼⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, 𝐼⟩)‘⟨𝐴, (𝐽‘𝐼)⟩) = ((𝐹(2nd ‘𝐾)𝐺)‘𝐴))
52	1, 51	eqtrid 2807	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 1569   ∈ wcel 2144  ⟨cop 4589   × cxp 5648  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Hom chom 17441  Catccat 17840  Idccid 17841   Func cfunc 18031  id_funccidfu 18032   ∘_func ccofu 18033   Nat cnat 18121   FuncCat cfuc 18122   ×_c cxpc 18344   ∘_F cfuco 50529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11258  ax-resscn 11259  ax-1cn 11260  ax-icn 11261  ax-addcl 11262  ax-addrcl 11263  ax-mulcl 11264  ax-mulrcl 11265  ax-mulcom 11266  ax-addass 11267  ax-mulass 11268  ax-distr 11269  ax-i2m1 11270  ax-1ne0 11271  ax-1rid 11272  ax-rnegex 11273  ax-rrecex 11274  ax-cnre 11275  ax-pre-lttri 11276  ax-pre-lttrn 11277  ax-pre-ltadd 11278  ax-pre-mulgt0 11279

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8301  df-wrecs 8332  df-recs 8381  df-rdg 8420  df-1o 8478  df-er 8719  df-map 8851  df-ixp 8928  df-en 8976  df-dom 8977  df-sdom 8978  df-fin 8979  df-pnf 11347  df-mnf 11348  df-xr 11349  df-ltxr 11350  df-le 11351  df-sub 11545  df-neg 11546  df-nn 12338  df-2 12407  df-3 12408  df-4 12409  df-5 12410  df-6 12411  df-7 12412  df-8 12413  df-9 12414  df-n0 12609  df-z 12696  df-dec 12817  df-uz 12968  df-fz 13642  df-struct 17327  df-slot 17362  df-ndx 17374  df-base 17390  df-hom 17454  df-cco 17455  df-cat 17844  df-cid 17845  df-func 18035  df-idfu 18036  df-cofu 18037  df-nat 18123  df-fuc 18124  df-xpc 18348  df-fuco 50530

This theorem is used by:  fucoaltval  50584