Theorem comffval 17757

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 17756	. . 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 6924	. . . . 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 8043	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
12	9, 10, 11	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
14	8, 13	eqtrd 2780	. . . 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 7466	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑥)𝐻𝑧) = (𝑌𝐻𝑍))
17	7	fveq2d 6924	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
18		df-ov 7451	. . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
19	17, 18	eqtr4di 2798	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝑋𝐻𝑌))
20	7, 15	oveq12d 7466	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑥 · 𝑧) = (⟨𝑋, 𝑌⟩ · 𝑍))
21	20	oveqd 7465	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔(𝑥 · 𝑧)𝑓) = (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓))
22	16, 19, 21	mpoeq123dv 7525	. 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 5739	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
24		comffval.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25		ovex 7481	. . . 4 ⊢ (𝑌𝐻𝑍) ∈ V
26		ovex 7481	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
27	25, 26	mpoex 8120	. . 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 7602	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ⟨cop 4654   × cxp 5698  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  2^nd c2nd 8029  Basecbs 17258  Hom chom 17322  compcco 17323  comp_fccomf 17725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-comf 17729

This theorem is referenced by:  comfval  17758  comffval2  17760  comffn  17763