Theorem stoweidlem6 43014

Description: Lemma for stoweid 43071: two class variables replace two setvar variables, for multiplication of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem6.1	⊢ Ⅎ𝑡 𝑓 = 𝐹
stoweidlem6.2	⊢ Ⅎ𝑡 𝑔 = 𝐺
stoweidlem6.3	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)

Assertion

Ref	Expression
stoweidlem6	⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)

Distinct variable groups:   𝑓,𝑔,𝑡   𝐴,𝑓,𝑔   𝑓,𝐹,𝑔   𝑇,𝑓,𝑔   𝜑,𝑓,𝑔   𝑔,𝐺

Allowed substitution hints:   𝜑(𝑡)   𝐴(𝑡)   𝑇(𝑡)   𝐹(𝑡)   𝐺(𝑡,𝑓)

Proof of Theorem stoweidlem6

Step	Hyp	Ref	Expression
1		simp3 1135	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → 𝐺 ∈ 𝐴)
2		eleq1 2839	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
3	2	3anbi3d 1439	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴)))
4		stoweidlem6.2	. . . . . 6 ⊢ Ⅎ𝑡 𝑔 = 𝐺
5		fveq1 6657	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑡) = (𝐺‘𝑡))
6	5	oveq2d 7166	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝐹‘𝑡) · (𝑔‘𝑡)) = ((𝐹‘𝑡) · (𝐺‘𝑡)))
7	6	adantr 484	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) · (𝑔‘𝑡)) = ((𝐹‘𝑡) · (𝐺‘𝑡)))
8	4, 7	mpteq2da 5126	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐺‘𝑡))))
9	8	eleq1d 2836	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴))
10	3, 9	imbi12d 348	. . 3 ⊢ (𝑔 = 𝐺 → (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)))
11		simp2 1134	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → 𝐹 ∈ 𝐴)
12		eleq1 2839	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴))
13	12	3anbi2d 1438	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴)))
14		stoweidlem6.1	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑓 = 𝐹
15		fveq1 6657	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑡) = (𝐹‘𝑡))
16	15	oveq1d 7165	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑡) · (𝑔‘𝑡)) = ((𝐹‘𝑡) · (𝑔‘𝑡)))
17	16	adantr 484	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) = ((𝐹‘𝑡) · (𝑔‘𝑡)))
18	14, 17	mpteq2da 5126	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))))
19	18	eleq1d 2836	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
20	13, 19	imbi12d 348	. . . . 5 ⊢ (𝑓 = 𝐹 → (((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)))
21		stoweidlem6.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
22	20, 21	vtoclg 3485	. . . 4 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
23	11, 22	mpcom 38	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
24	10, 23	vtoclg 3485	. 2 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴))
25	1, 24	mpcom 38	1 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111   ↦ cmpt 5112  ‘cfv 6335  (class class class)co 7150   · cmul 10580

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-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-iota 6294  df-fv 6343  df-ov 7153

This theorem is referenced by:  stoweidlem19  43027  stoweidlem22  43030  stoweidlem32  43040  stoweidlem36  43044