Theorem comffval 16744

Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
comfffval.o	⊢ 𝑂 = (compf‘𝐶)
comfffval.b	⊢ 𝐵 = (Base‘𝐶)
comfffval.h	⊢ 𝐻 = (Hom ‘𝐶)
comfffval.x	⊢ · = (comp‘𝐶)
comffval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
comffval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
comffval.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
comffval	⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))

Distinct variable groups:   𝑓,𝑔,𝐶   𝜑,𝑓,𝑔   · ,𝑓,𝑔   𝑓,𝑋,𝑔   𝑓,𝑌,𝑔   𝑓,𝑍,𝑔   𝑓,𝐻,𝑔

Allowed substitution hints:   𝐵(𝑓,𝑔)   𝑂(𝑓,𝑔)

Proof of Theorem comffval

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		comfffval.o	. . . 4 ⊢ 𝑂 = (compf‘𝐶)
2		comfffval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		comfffval.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
4		comfffval.x	. . . 4 ⊢ · = (comp‘𝐶)
5	1, 2, 3, 4	comfffval 16743	. . 3 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓)))
6	5	a1i 11	. 2 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓))))
7		simprl 761	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑥 = ⟨𝑋, 𝑌⟩)
8	7	fveq2d 6450	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
9		comffval.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		comffval.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
11		op2ndg 7458	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
12	9, 10, 11	syl2anc 579	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	12	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
14	8, 13	eqtrd 2813	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑥) = 𝑌)
15		simprr 763	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
16	14, 15	oveq12d 6940	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑥)𝐻𝑧) = (𝑌𝐻𝑍))
17	7	fveq2d 6450	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
18		df-ov 6925	. . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
19	17, 18	syl6eqr 2831	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝑋𝐻𝑌))
20	7, 15	oveq12d 6940	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑥 · 𝑧) = (⟨𝑋, 𝑌⟩ · 𝑍))
21	20	oveqd 6939	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔(𝑥 · 𝑧)𝑓) = (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓))
22	16, 19, 21	mpt2eq123dv 6994	. 2 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓)) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))
23	9, 10	opelxpd 5393	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
24		comffval.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25		ovex 6954	. . . 4 ⊢ (𝑌𝐻𝑍) ∈ V
26		ovex 6954	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
27	25, 26	mpt2ex 7527	. . 3 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)) ∈ V
28	27	a1i 11	. 2 ⊢ (𝜑 → (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)) ∈ V)
29	6, 22, 23, 24, 28	ovmpt2d 7065	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106  Vcvv 3397  ⟨cop 4403   × cxp 5353  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  2^nd c2nd 7444  Basecbs 16255  Hom chom 16349  compcco 16350  comp_fccomf 16713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-comf 16717

This theorem is referenced by:  comfval  16745  comffval2  16747  comffn  16750