Theorem stoweidlem20 43103

Description: If a set A of real functions from a common domain T is closed under the sum of two functions, then it is closed under the sum of a finite number of functions, indexed by G. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem20.1	⊢ Ⅎ𝑡𝜑
stoweidlem20.2	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))
stoweidlem20.3	⊢ (𝜑 → 𝑀 ∈ ℕ)
stoweidlem20.4	⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝐴)
stoweidlem20.5	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem20.6	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)

Assertion

Ref	Expression
stoweidlem20	⊢ (𝜑 → 𝐹 ∈ 𝐴)

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

Allowed substitution hints:   𝜑(𝑡)   𝐴(𝑡,𝑖)   𝐹(𝑡,𝑓,𝑔,𝑖)   𝑀(𝑓,𝑔)

Proof of Theorem stoweidlem20

Dummy variables 𝑦 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem20.2	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))
2		stoweidlem20.3	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3	2	nnred 11731	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ)
4	3	leidd 11284	. . . 4 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
5	4	ancli 552	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝑀 ≤ 𝑀))
6		eleq1 2820	. . . . 5 ⊢ (𝑛 = 𝑀 → (𝑛 ∈ ℕ ↔ 𝑀 ∈ ℕ))
7		breq1 5033	. . . . . . 7 ⊢ (𝑛 = 𝑀 → (𝑛 ≤ 𝑀 ↔ 𝑀 ≤ 𝑀))
8	7	anbi2d 632	. . . . . 6 ⊢ (𝑛 = 𝑀 → ((𝜑 ∧ 𝑛 ≤ 𝑀) ↔ (𝜑 ∧ 𝑀 ≤ 𝑀)))
9		oveq2 7178	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (1...𝑛) = (1...𝑀))
10	9	sumeq1d 15151	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))
11	10	mpteq2dv 5126	. . . . . . 7 ⊢ (𝑛 = 𝑀 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
12	11	eleq1d 2817	. . . . . 6 ⊢ (𝑛 = 𝑀 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴))
13	8, 12	imbi12d 348	. . . . 5 ⊢ (𝑛 = 𝑀 → (((𝜑 ∧ 𝑛 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑀 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)))
14	6, 13	imbi12d 348	. . . 4 ⊢ (𝑛 = 𝑀 → ((𝑛 ∈ ℕ → ((𝜑 ∧ 𝑛 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ↔ (𝑀 ∈ ℕ → ((𝜑 ∧ 𝑀 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴))))
15		breq1 5033	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ 𝑀 ↔ 1 ≤ 𝑀))
16	15	anbi2d 632	. . . . . 6 ⊢ (𝑥 = 1 → ((𝜑 ∧ 𝑥 ≤ 𝑀) ↔ (𝜑 ∧ 1 ≤ 𝑀)))
17		oveq2 7178	. . . . . . . . 9 ⊢ (𝑥 = 1 → (1...𝑥) = (1...1))
18	17	sumeq1d 15151	. . . . . . . 8 ⊢ (𝑥 = 1 → Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡))
19	18	mpteq2dv 5126	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)))
20	19	eleq1d 2817	. . . . . 6 ⊢ (𝑥 = 1 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴))
21	16, 20	imbi12d 348	. . . . 5 ⊢ (𝑥 = 1 → (((𝜑 ∧ 𝑥 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑 ∧ 1 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)))
22		breq1 5033	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑀 ↔ 𝑦 ≤ 𝑀))
23	22	anbi2d 632	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ≤ 𝑀) ↔ (𝜑 ∧ 𝑦 ≤ 𝑀)))
24		oveq2 7178	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
25	24	sumeq1d 15151	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))
26	25	mpteq2dv 5126	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)))
27	26	eleq1d 2817	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴))
28	23, 27	imbi12d 348	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)))
29		breq1 5033	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 ≤ 𝑀 ↔ (𝑦 + 1) ≤ 𝑀))
30	29	anbi2d 632	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝜑 ∧ 𝑥 ≤ 𝑀) ↔ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)))
31		oveq2 7178	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
32	31	sumeq1d 15151	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡))
33	32	mpteq2dv 5126	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡)))
34	33	eleq1d 2817	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡)) ∈ 𝐴))
35	30, 34	imbi12d 348	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (((𝜑 ∧ 𝑥 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑦 + 1) ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)))
36		breq1 5033	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑥 ≤ 𝑀 ↔ 𝑛 ≤ 𝑀))
37	36	anbi2d 632	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝜑 ∧ 𝑥 ≤ 𝑀) ↔ (𝜑 ∧ 𝑛 ≤ 𝑀)))
38		oveq2 7178	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (1...𝑥) = (1...𝑛))
39	38	sumeq1d 15151	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡))
40	39	mpteq2dv 5126	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡)))
41	40	eleq1d 2817	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴))
42	37, 41	imbi12d 348	. . . . 5 ⊢ (𝑥 = 𝑛 → (((𝜑 ∧ 𝑥 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑛 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)))
43		stoweidlem20.1	. . . . . . . . 9 ⊢ Ⅎ𝑡𝜑
44		1z 12093	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
45		stoweidlem20.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝐴)
46		nnuz 12363	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ≥‘1)
47	2, 46	eleqtrdi 2843	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1))
48		eluzfz1 13005	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑀))
49	47, 48	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ (1...𝑀))
50	45, 49	ffvelrnd 6862	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺‘1) ∈ 𝐴)
51	50	ancli 552	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝜑 ∧ (𝐺‘1) ∈ 𝐴))
52		eleq1 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝐺‘1) → (𝑓 ∈ 𝐴 ↔ (𝐺‘1) ∈ 𝐴))
53	52	anbi2d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐺‘1) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝐺‘1) ∈ 𝐴)))
54		feq1 6485	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐺‘1) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘1):𝑇⟶ℝ))
55	53, 54	imbi12d 348	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝐺‘1) → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘1) ∈ 𝐴) → (𝐺‘1):𝑇⟶ℝ)))
56		stoweidlem20.6	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
57	55, 56	vtoclg 3470	. . . . . . . . . . . . 13 ⊢ ((𝐺‘1) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘1) ∈ 𝐴) → (𝐺‘1):𝑇⟶ℝ))
58	50, 51, 57	sylc 65	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘1):𝑇⟶ℝ)
59	58	ffvelrnda 6861	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘1)‘𝑡) ∈ ℝ)
60	59	recnd 10747	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘1)‘𝑡) ∈ ℂ)
61		fveq2 6674	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (𝐺‘1))
62	61	fveq1d 6676	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → ((𝐺‘𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
63	62	fsum1 15195	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝐺‘1)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
64	44, 60, 63	sylancr 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
65	43, 64	mpteq2da 5124	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘1)‘𝑡)))
66	58	feqmptd 6737	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘1) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘1)‘𝑡)))
67	65, 66	eqtr4d 2776	. . . . . . 7 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)) = (𝐺‘1))
68	67, 50	eqeltrd 2833	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)
69	68	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)
70		simprl 771	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ ∧ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → 𝜑)
71		simpll 767	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ ∧ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → 𝑦 ∈ ℕ)
72		simprr 773	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ ∧ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑦 + 1) ≤ 𝑀)
73		simp1 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝜑)
74		nnre 11723	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
75	74	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ∈ ℝ)
76		1red 10720	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℝ)
77	75, 76	readdcld 10748	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℝ)
78	2	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℕ)
79	78	nnred 11731	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℝ)
80	75	lep1d 11649	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ≤ (𝑦 + 1))
81		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ≤ 𝑀)
82	75, 77, 79, 80, 81	letrd 10875	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ≤ 𝑀)
83	73, 82	jca 515	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝜑 ∧ 𝑦 ≤ 𝑀))
84	70, 71, 72, 83	syl3anc 1372	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ ∧ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝜑 ∧ 𝑦 ≤ 𝑀))
85		simplr 769	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ ∧ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴))
86	84, 85	mpd 15	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ ∧ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)
87		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑦 ∈ ℕ
88		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑦 + 1) ≤ 𝑀
89	43, 87, 88	nf3an 1908	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀)
90		simpl2 1193	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → 𝑦 ∈ ℕ)
91	90, 46	eleqtrdi 2843	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → 𝑦 ∈ (ℤ≥‘1))
92		simpll1 1213	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝜑)
93		1zzd 12094	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ∈ ℤ)
94	2	nnzd 12167	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℤ)
95	94	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℤ)
96	95	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑀 ∈ ℤ)
97		elfzelz 12998	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℤ)
98	97	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ ℤ)
99		elfzle1 13001	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...(𝑦 + 1)) → 1 ≤ 𝑖)
100	99	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ≤ 𝑖)
101	97	zred 12168	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℝ)
102	101	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ ℝ)
103	77	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → (𝑦 + 1) ∈ ℝ)
104	79	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑀 ∈ ℝ)
105		elfzle2 13002	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ≤ (𝑦 + 1))
106	105	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ≤ (𝑦 + 1))
107		simpll3 1215	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → (𝑦 + 1) ≤ 𝑀)
108	102, 103, 104, 106, 107	letrd 10875	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ≤ 𝑀)
109	93, 96, 98, 100, 108	elfzd 12989	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ (1...𝑀))
110		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑡 ∈ 𝑇)
111	45	ffvelrnda 6861	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘𝑖) ∈ 𝐴)
112	111	3adant3 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑖) ∈ 𝐴)
113		simp1 1137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → 𝜑)
114	113, 112	jca 515	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴))
115		eleq1 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝐺‘𝑖) → (𝑓 ∈ 𝐴 ↔ (𝐺‘𝑖) ∈ 𝐴))
116	115	anbi2d 632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝐺‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴)))
117		feq1 6485	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝐺‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘𝑖):𝑇⟶ℝ))
118	116, 117	imbi12d 348	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝐺‘𝑖) → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴) → (𝐺‘𝑖):𝑇⟶ℝ)))
119	118, 56	vtoclg 3470	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴) → (𝐺‘𝑖):𝑇⟶ℝ))
120	112, 114, 119	sylc 65	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑖):𝑇⟶ℝ)
121		simp3 1139	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
122	120, 121	ffvelrnd 6862	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
123	122	recnd 10747	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑖)‘𝑡) ∈ ℂ)
124	92, 109, 110, 123	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → ((𝐺‘𝑖)‘𝑡) ∈ ℂ)
125		fveq2 6674	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑦 + 1) → (𝐺‘𝑖) = (𝐺‘(𝑦 + 1)))
126	125	fveq1d 6676	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑦 + 1) → ((𝐺‘𝑖)‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
127	91, 124, 126	fsump1 15204	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡) = (Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
128		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
129		fzfid 13432	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → (1...𝑦) ∈ Fin)
130		simpll1 1213	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝜑)
131		1zzd 12094	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ∈ ℤ)
132	95	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℤ)
133		elfzelz 12998	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℤ)
134	133	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℤ)
135		elfzle1 13001	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑦) → 1 ≤ 𝑖)
136	135	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ≤ 𝑖)
137	133	zred 12168	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℝ)
138	137	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℝ)
139	77	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ∈ ℝ)
140	79	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℝ)
141	75	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑦 ∈ ℝ)
142		elfzle2 13002	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1...𝑦) → 𝑖 ≤ 𝑦)
143	142	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ 𝑦)
144		letrp1 11562	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑖 ≤ 𝑦) → 𝑖 ≤ (𝑦 + 1))
145	138, 141, 143, 144	syl3anc 1372	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ (𝑦 + 1))
146		simpl3 1194	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ≤ 𝑀)
147	138, 139, 140, 145, 146	letrd 10875	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ 𝑀)
148	147	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ 𝑀)
149	131, 132, 134, 136, 148	elfzd 12989	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ (1...𝑀))
150		simplr 769	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑡 ∈ 𝑇)
151	130, 149, 150, 122	syl3anc 1372	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
152	129, 151	fsumrecl 15184	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡) ∈ ℝ)
153		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))
154	153	fvmpt2 6786	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝑇 ∧ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))
155	128, 152, 154	syl2anc 587	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))
156	155	oveq1d 7185	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)) = (Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
157	127, 156	eqtr4d 2776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡) = (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
158	89, 157	mpteq2da 5124	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡))))
159	158	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡))))
160		1zzd 12094	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℤ)
161		peano2nn 11728	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
162	161	nnzd 12167	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℤ)
163	162	3ad2ant2 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℤ)
164	161	nnge1d 11764	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ → 1 ≤ (𝑦 + 1))
165	164	3ad2ant2 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ≤ (𝑦 + 1))
166	160, 95, 163, 165, 81	elfzd 12989	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ (1...𝑀))
167	45	ffvelrnda 6861	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
168	73, 166, 167	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
169		eleq1 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓 ∈ 𝐴 ↔ (𝐺‘(𝑦 + 1)) ∈ 𝐴))
170	169	anbi2d 632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝐺‘(𝑦 + 1)) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴)))
171		feq1 6485	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
172	170, 171	imbi12d 348	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝐺‘(𝑦 + 1)) → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)))
173	172, 56	vtoclg 3470	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘(𝑦 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
174	173	anabsi7 671	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
175	73, 168, 174	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
176	175	ffvelrnda 6861	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ)
177		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
178	177	fvmpt2 6786	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
179	128, 176, 178	syl2anc 587	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
180	179	oveq2d 7186	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
181	89, 180	mpteq2da 5124	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡))))
182	181	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡))))
183		simpl1 1192	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → 𝜑)
184		simpr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)
185	166	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → (𝑦 + 1) ∈ (1...𝑀))
186	174	feqmptd 6737	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
187	167, 186	syldan 594	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
188	187, 167	eqeltrrd 2834	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴)
189	183, 185, 188	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴)
190		stoweidlem20.5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
191		nfmpt1 5128	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))
192		nfmpt1 5128	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
193	190, 191, 192	stoweidlem8 43091	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡))) ∈ 𝐴)
194	183, 184, 189, 193	syl3anc 1372	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡))) ∈ 𝐴)
195	182, 194	eqeltrrd 2834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡))) ∈ 𝐴)
196	159, 195	eqeltrd 2833	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)
197	70, 71, 72, 86, 196	syl31anc 1374	. . . . . 6 ⊢ (((𝑦 ∈ ℕ ∧ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)
198	197	exp31 423	. . . . 5 ⊢ (𝑦 ∈ ℕ → (((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → ((𝜑 ∧ (𝑦 + 1) ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)))
199	21, 28, 35, 42, 69, 198	nnind 11734	. . . 4 ⊢ (𝑛 ∈ ℕ → ((𝜑 ∧ 𝑛 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴))
200	14, 199	vtoclg 3470	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑀 ∈ ℕ → ((𝜑 ∧ 𝑀 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)))
201	2, 2, 5, 200	syl3c 66	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)
202	1, 201	eqeltrid 2837	1 ⊢ (𝜑 → 𝐹 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1088   = wceq 1542  Ⅎwnf 1790   ∈ wcel 2114   class class class wbr 5030   ↦ cmpt 5110  ⟶wf 6335  ‘cfv 6339  (class class class)co 7170  ℂcc 10613  ℝcr 10614  1c1 10616   + caddc 10618   ≤ cle 10754  ℕcn 11716  ℤcz 12062  ℤ_≥cuz 12324  ...cfz 12981  Σcsu 15135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-sup 8979  df-oi 9047  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-n0 11977  df-z 12063  df-uz 12325  df-rp 12473  df-fz 12982  df-fzo 13125  df-seq 13461  df-exp 13522  df-hash 13783  df-cj 14548  df-re 14549  df-im 14550  df-sqrt 14684  df-abs 14685  df-clim 14935  df-sum 15136

This theorem is referenced by:  stoweidlem32  43115