Theorem stoweidlem8 43549

Description: Lemma for stoweid 43604: two class variables replace two setvar variables, for the sum of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem8.1	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem8.2	⊢ Ⅎ𝑡𝐹
stoweidlem8.3	⊢ Ⅎ𝑡𝐺

Assertion

Ref	Expression
stoweidlem8	⊢ ((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 stoweidlem8

Step	Hyp	Ref	Expression
1		simp3 1137	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → 𝐺 ∈ 𝐴)
2		eleq1 2826	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
3	2	3anbi3d 1441	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴)))
4		stoweidlem8.3	. . . . . . 7 ⊢ Ⅎ𝑡𝐺
5	4	nfeq2 2924	. . . . . 6 ⊢ Ⅎ𝑡 𝑔 = 𝐺
6		fveq1 6773	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑡) = (𝐺‘𝑡))
7	6	oveq2d 7291	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝐹‘𝑡) + (𝑔‘𝑡)) = ((𝐹‘𝑡) + (𝐺‘𝑡)))
8	7	adantr 481	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + (𝑔‘𝑡)) = ((𝐹‘𝑡) + (𝐺‘𝑡)))
9	5, 8	mpteq2da 5172	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐺‘𝑡))))
10	9	eleq1d 2823	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐺‘𝑡))) ∈ 𝐴))
11	3, 10	imbi12d 345	. . 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)))
12		simp2 1136	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → 𝐹 ∈ 𝐴)
13		eleq1 2826	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴))
14	13	3anbi2d 1440	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴)))
15		stoweidlem8.2	. . . . . . . . 9 ⊢ Ⅎ𝑡𝐹
16	15	nfeq2 2924	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑓 = 𝐹
17		fveq1 6773	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑡) = (𝐹‘𝑡))
18	17	oveq1d 7290	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑡) + (𝑔‘𝑡)) = ((𝐹‘𝑡) + (𝑔‘𝑡)))
19	18	adantr 481	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) + (𝑔‘𝑡)) = ((𝐹‘𝑡) + (𝑔‘𝑡)))
20	16, 19	mpteq2da 5172	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝑔‘𝑡))))
21	20	eleq1d 2823	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴))
22	14, 21	imbi12d 345	. . . . 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)))
23		stoweidlem8.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
24	22, 23	vtoclg 3505	. . . 4 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴))
25	12, 24	mpcom 38	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
26	11, 25	vtoclg 3505	. 2 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐺‘𝑡))) ∈ 𝐴))
27	1, 26	mpcom 38	1 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐺‘𝑡))) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275   + caddc 10874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  stoweidlem20  43561  stoweidlem21  43562  stoweidlem22  43563  stoweidlem23  43564