Theorem upciclem2 50093

Description: Lemma for upciclem3 50094 and upeu2 50098. (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 50006	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
6		upciclem2.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐶)
7		upcic.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
8	7, 1, 4	funcf1 17955	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
9		upcic.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10	8, 9	ffvelcdmd 7078	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐶)
11		upcic.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
12	8, 11	ffvelcdmd 7078	. . 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 17957	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
16		upciclem2.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
17	15, 16	ffvelcdmd 7078	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐾) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
18		upciclem2.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
19	8, 18	ffvelcdmd 7078	. . 3 ⊢ (𝜑 → (𝐹‘𝑍) ∈ 𝐶)
20	7, 14, 2, 4, 11, 18	funcf2 17957	. . . 4 ⊢ (𝜑 → (𝑌𝐺𝑍):(𝑌𝐻𝑍)⟶((𝐹‘𝑌)𝐽(𝐹‘𝑍)))
21		upciclem2.l	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (𝑌𝐻𝑍))
22	20, 21	ffvelcdmd 7078	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑍)‘𝐿) ∈ ((𝐹‘𝑌)𝐽(𝐹‘𝑍)))
23	1, 2, 3, 5, 6, 10, 12, 13, 17, 19, 22	catass 17774	. 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 17960	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑍)𝐾)) = (((𝑌𝐺𝑍)‘𝐿)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐾)))
26	25	oveq1d 7428	. 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 7429	. 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 2805	1 ⊢ (𝜑 → (((𝑋𝐺𝑍)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑍)𝐾))(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑍))𝑀) = (((𝑌𝐺𝑍)‘𝐿)(⟨𝑊, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))𝑁))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ⟨cop 4590   class class class wbr 5103  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  Hom chom 17353  compcco 17354   Func cfunc 17943

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  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-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-map 8828  df-ixp 8905  df-cat 17756  df-func 17947

This theorem is used by:  upciclem3  50094  upeu2  50098