Theorem upciclem2 49642

Description: Lemma for upciclem3 49643 and upeu2 49647. (Contributed by Zhi Wang, 19-Sep-2025.)

Hypotheses

Ref	Expression
upcic.b	⊢ 𝐵 = (Base‘𝐷)
upcic.c	⊢ 𝐶 = (Base‘𝐸)
upcic.h	⊢ 𝐻 = (Hom ‘𝐷)
upcic.j	⊢ 𝐽 = (Hom ‘𝐸)
upcic.o	⊢ 𝑂 = (comp‘𝐸)
upcic.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
upcic.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
upcic.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
upciclem2.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
upciclem2.w	⊢ (𝜑 → 𝑊 ∈ 𝐶)
upciclem2.m	⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋)))
upciclem2.od	⊢ · = (comp‘𝐷)
upciclem2.k	⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
upciclem2.l	⊢ (𝜑 → 𝐿 ∈ (𝑌𝐻𝑍))
upciclem2.nm	⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))

Assertion

Ref	Expression
upciclem2	⊢ (𝜑 → (((𝑋𝐺𝑍)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑍)𝐾))(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑍))𝑀) = (((𝑌𝐺𝑍)‘𝐿)(⟨𝑊, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))𝑁))

Proof of Theorem upciclem2

Step	Hyp	Ref	Expression
1		upcic.c	. . 3 ⊢ 𝐶 = (Base‘𝐸)
2		upcic.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐸)
3		upcic.o	. . 3 ⊢ 𝑂 = (comp‘𝐸)
4		upcic.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
5	4	funcrcl3 49555	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
6		upciclem2.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐶)
7		upcic.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
8	7, 1, 4	funcf1 17833	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
9		upcic.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10	8, 9	ffvelcdmd 7037	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐶)
11		upcic.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
12	8, 11	ffvelcdmd 7037	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ 𝐶)
13		upciclem2.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋)))
14		upcic.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐷)
15	7, 14, 2, 4, 9, 11	funcf2 17835	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
16		upciclem2.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
17	15, 16	ffvelcdmd 7037	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐾) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
18		upciclem2.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
19	8, 18	ffvelcdmd 7037	. . 3 ⊢ (𝜑 → (𝐹‘𝑍) ∈ 𝐶)
20	7, 14, 2, 4, 11, 18	funcf2 17835	. . . 4 ⊢ (𝜑 → (𝑌𝐺𝑍):(𝑌𝐻𝑍)⟶((𝐹‘𝑌)𝐽(𝐹‘𝑍)))
21		upciclem2.l	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (𝑌𝐻𝑍))
22	20, 21	ffvelcdmd 7037	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑍)‘𝐿) ∈ ((𝐹‘𝑌)𝐽(𝐹‘𝑍)))
23	1, 2, 3, 5, 6, 10, 12, 13, 17, 19, 22	catass 17652	. 2 ⊢ (𝜑 → ((((𝑌𝐺𝑍)‘𝐿)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐾))(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑍))𝑀) = (((𝑌𝐺𝑍)‘𝐿)(⟨𝑊, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))(((𝑋𝐺𝑌)‘𝐾)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
24		upciclem2.od	. . . 4 ⊢ · = (comp‘𝐷)
25	7, 14, 24, 3, 4, 9, 11, 18, 16, 21	funcco 17838	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑍)𝐾)) = (((𝑌𝐺𝑍)‘𝐿)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐾)))
26	25	oveq1d 7382	. 2 ⊢ (𝜑 → (((𝑋𝐺𝑍)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑍)𝐾))(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑍))𝑀) = ((((𝑌𝐺𝑍)‘𝐿)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐾))(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑍))𝑀))
27		upciclem2.nm	. . 3 ⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
28	27	oveq2d 7383	. 2 ⊢ (𝜑 → (((𝑌𝐺𝑍)‘𝐿)(⟨𝑊, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))𝑁) = (((𝑌𝐺𝑍)‘𝐿)(⟨𝑊, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))(((𝑋𝐺𝑌)‘𝐾)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
29	23, 26, 28	3eqtr4d 2781	1 ⊢ (𝜑 → (((𝑋𝐺𝑍)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑍)𝐾))(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑍))𝑀) = (((𝑌𝐺𝑍)‘𝐿)(⟨𝑊, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))𝑁))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231  compcco 17232   Func cfunc 17821

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  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-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ixp 8846  df-cat 17634  df-func 17825

This theorem is referenced by:  upciclem3  49643  upeu2  49647