Theorem comffval 17040

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 17039	. . 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 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑥 = ⟨𝑋, 𝑌⟩)
8	7	fveq2d 6667	. . . . 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 7712	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
12	9, 10, 11	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	12	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
14	8, 13	eqtrd 2793	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑥) = 𝑌)
15		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
16	14, 15	oveq12d 7174	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑥)𝐻𝑧) = (𝑌𝐻𝑍))
17	7	fveq2d 6667	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
18		df-ov 7159	. . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
19	17, 18	eqtr4di 2811	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝑋𝐻𝑌))
20	7, 15	oveq12d 7174	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑥 · 𝑧) = (⟨𝑋, 𝑌⟩ · 𝑍))
21	20	oveqd 7173	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔(𝑥 · 𝑧)𝑓) = (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓))
22	16, 19, 21	mpoeq123dv 7229	. 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 5566	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
24		comffval.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25		ovex 7189	. . . 4 ⊢ (𝑌𝐻𝑍) ∈ V
26		ovex 7189	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
27	25, 26	mpoex 7788	. . 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	ovmpod 7303	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4531   × cxp 5526  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  2^nd c2nd 7698  Basecbs 16554  Hom chom 16647  compcco 16648  comp_fccomf 17009

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-comf 17013

This theorem is referenced by:  comfval  17041  comffval2  17043  comffn  17046