Theorem stoweidlem6 38899

Description: Lemma for stoweid 38956: 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 1056	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → 𝐺 ∈ 𝐴)
2		eleq1 2676	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
3	2	3anbi3d 1397	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴)))
4		stoweidlem6.2	. . . . . 6 ⊢ Ⅎ𝑡 𝑔 = 𝐺
5		fveq1 6102	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑡) = (𝐺‘𝑡))
6	5	oveq2d 6565	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝐹‘𝑡) · (𝑔‘𝑡)) = ((𝐹‘𝑡) · (𝐺‘𝑡)))
7	6	adantr 480	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) · (𝑔‘𝑡)) = ((𝐹‘𝑡) · (𝐺‘𝑡)))
8	4, 7	mpteq2da 4671	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐺‘𝑡))))
9	8	eleq1d 2672	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴))
10	3, 9	imbi12d 333	. . 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 1055	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → 𝐹 ∈ 𝐴)
12		eleq1 2676	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴))
13	12	3anbi2d 1396	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴)))
14		stoweidlem6.1	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑓 = 𝐹
15		fveq1 6102	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑡) = (𝐹‘𝑡))
16	15	oveq1d 6564	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑡) · (𝑔‘𝑡)) = ((𝐹‘𝑡) · (𝑔‘𝑡)))
17	16	adantr 480	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) = ((𝐹‘𝑡) · (𝑔‘𝑡)))
18	14, 17	mpteq2da 4671	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))))
19	18	eleq1d 2672	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
20	13, 19	imbi12d 333	. . . . 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 3239	. . . 4 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
23	11, 22	mpcom 37	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
24	10, 23	vtoclg 3239	. 2 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴))
25	1, 24	mpcom 37	1 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   · cmul 9820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-iota 5768  df-fv 5812  df-ov 6552

This theorem is referenced by:  stoweidlem19  38912  stoweidlem22  38915  stoweidlem32  38925  stoweidlem36  38929