Theorem stoweidlem20 38913

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 10912	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ)
4	3	leidd 10473	. . . 4 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
5	4	ancli 572	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝑀 ≤ 𝑀))
6		eleq1 2676	. . . . 5 ⊢ (𝑛 = 𝑀 → (𝑛 ∈ ℕ ↔ 𝑀 ∈ ℕ))
7		breq1 4586	. . . . . . 7 ⊢ (𝑛 = 𝑀 → (𝑛 ≤ 𝑀 ↔ 𝑀 ≤ 𝑀))
8	7	anbi2d 736	. . . . . 6 ⊢ (𝑛 = 𝑀 → ((𝜑 ∧ 𝑛 ≤ 𝑀) ↔ (𝜑 ∧ 𝑀 ≤ 𝑀)))
9		oveq2 6557	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (1...𝑛) = (1...𝑀))
10	9	sumeq1d 14279	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))
11	10	mpteq2dv 4673	. . . . . . 7 ⊢ (𝑛 = 𝑀 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
12	11	eleq1d 2672	. . . . . 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 333	. . . . 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 333	. . . 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 4586	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ 𝑀 ↔ 1 ≤ 𝑀))
16	15	anbi2d 736	. . . . . 6 ⊢ (𝑥 = 1 → ((𝜑 ∧ 𝑥 ≤ 𝑀) ↔ (𝜑 ∧ 1 ≤ 𝑀)))
17		oveq2 6557	. . . . . . . . 9 ⊢ (𝑥 = 1 → (1...𝑥) = (1...1))
18	17	sumeq1d 14279	. . . . . . . 8 ⊢ (𝑥 = 1 → Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡))
19	18	mpteq2dv 4673	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)))
20	19	eleq1d 2672	. . . . . 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 333	. . . . 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 4586	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑀 ↔ 𝑦 ≤ 𝑀))
23	22	anbi2d 736	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ≤ 𝑀) ↔ (𝜑 ∧ 𝑦 ≤ 𝑀)))
24		oveq2 6557	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
25	24	sumeq1d 14279	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))
26	25	mpteq2dv 4673	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)))
27	26	eleq1d 2672	. . . . . 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 333	. . . . 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 4586	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 ≤ 𝑀 ↔ (𝑦 + 1) ≤ 𝑀))
30	29	anbi2d 736	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝜑 ∧ 𝑥 ≤ 𝑀) ↔ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)))
31		oveq2 6557	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
32	31	sumeq1d 14279	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡))
33	32	mpteq2dv 4673	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺‘𝑖)‘𝑡)))
34	33	eleq1d 2672	. . . . . 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 333	. . . . 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 4586	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑥 ≤ 𝑀 ↔ 𝑛 ≤ 𝑀))
37	36	anbi2d 736	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝜑 ∧ 𝑥 ≤ 𝑀) ↔ (𝜑 ∧ 𝑛 ≤ 𝑀)))
38		oveq2 6557	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (1...𝑥) = (1...𝑛))
39	38	sumeq1d 14279	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡))
40	39	mpteq2dv 4673	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡)))
41	40	eleq1d 2672	. . . . . 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 333	. . . . 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 11284	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
45		stoweidlem20.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝐴)
46		nnuz 11599	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ≥‘1)
47	2, 46	syl6eleq 2698	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1))
48		eluzfz1 12219	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑀))
49	47, 48	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ (1...𝑀))
50	45, 49	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺‘1) ∈ 𝐴)
51	50	ancli 572	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝜑 ∧ (𝐺‘1) ∈ 𝐴))
52		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝐺‘1) → (𝑓 ∈ 𝐴 ↔ (𝐺‘1) ∈ 𝐴))
53	52	anbi2d 736	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐺‘1) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝐺‘1) ∈ 𝐴)))
54		feq1 5939	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐺‘1) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘1):𝑇⟶ℝ))
55	53, 54	imbi12d 333	. . . . . . . . . . . . . 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 3239	. . . . . . . . . . . . 13 ⊢ ((𝐺‘1) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘1) ∈ 𝐴) → (𝐺‘1):𝑇⟶ℝ))
58	50, 51, 57	sylc 63	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘1):𝑇⟶ℝ)
59	58	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘1)‘𝑡) ∈ ℝ)
60	59	recnd 9947	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘1)‘𝑡) ∈ ℂ)
61		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (𝐺‘1))
62	61	fveq1d 6105	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → ((𝐺‘𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
63	62	fsum1 14320	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ ((𝐺‘1)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
64	44, 60, 63	sylancr 694	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
65	43, 64	mpteq2da 4671	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘1)‘𝑡)))
66	58	feqmptd 6159	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘1) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘1)‘𝑡)))
67	65, 66	eqtr4d 2647	. . . . . . 7 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)) = (𝐺‘1))
68	67, 50	eqeltrd 2688	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)
69	68	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)
70		simprl 790	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ ∧ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → 𝜑)
71		simpll 786	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ ∧ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → 𝑦 ∈ ℕ)
72		simprr 792	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ ∧ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑦 + 1) ≤ 𝑀)
73		simp1 1054	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝜑)
74		nnre 10904	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
75	74	3ad2ant2 1076	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ∈ ℝ)
76		1red 9934	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℝ)
77	75, 76	readdcld 9948	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℝ)
78	2	3ad2ant1 1075	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℕ)
79	78	nnred 10912	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℝ)
80	75	lep1d 10834	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ≤ (𝑦 + 1))
81		simp3 1056	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ≤ 𝑀)
82	75, 77, 79, 80, 81	letrd 10073	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ≤ 𝑀)
83	73, 82	jca 553	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝜑 ∧ 𝑦 ≤ 𝑀))
84	70, 71, 72, 83	syl3anc 1318	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ ∧ ((𝜑 ∧ 𝑦 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝜑 ∧ 𝑦 ≤ 𝑀))
85		simplr 788	. . . . . . . 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 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑦 ∈ ℕ
88		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑦 + 1) ≤ 𝑀
89	43, 87, 88	nf3an 1819	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀)
90		simpl2 1058	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → 𝑦 ∈ ℕ)
91	90, 46	syl6eleq 2698	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → 𝑦 ∈ (ℤ≥‘1))
92		simpll1 1093	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝜑)
93		1zzd 11285	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ∈ ℤ)
94	2	nnzd 11357	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℤ)
95	94	3ad2ant1 1075	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℤ)
96	95	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑀 ∈ ℤ)
97		elfzelz 12213	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℤ)
98	97	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ ℤ)
99		elfzle1 12215	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...(𝑦 + 1)) → 1 ≤ 𝑖)
100	99	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ≤ 𝑖)
101	97	zred 11358	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℝ)
102	101	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ ℝ)
103	77	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → (𝑦 + 1) ∈ ℝ)
104	79	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑀 ∈ ℝ)
105		elfzle2 12216	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ≤ (𝑦 + 1))
106	105	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ≤ (𝑦 + 1))
107		simpll3 1095	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → (𝑦 + 1) ≤ 𝑀)
108	102, 103, 104, 106, 107	letrd 10073	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ≤ 𝑀)
109		elfz4 12206	. . . . . . . . . . . . . 14 ⊢ (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)) → 𝑖 ∈ (1...𝑀))
110	93, 96, 98, 100, 108, 109	syl32anc 1326	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ (1...𝑀))
111		simplr 788	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑡 ∈ 𝑇)
112	45	ffvelrnda 6267	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘𝑖) ∈ 𝐴)
113	112	3adant3 1074	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑖) ∈ 𝐴)
114		simp1 1054	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → 𝜑)
115	114, 113	jca 553	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴))
116		eleq1 2676	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝐺‘𝑖) → (𝑓 ∈ 𝐴 ↔ (𝐺‘𝑖) ∈ 𝐴))
117	116	anbi2d 736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝐺‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴)))
118		feq1 5939	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝐺‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘𝑖):𝑇⟶ℝ))
119	117, 118	imbi12d 333	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝐺‘𝑖) → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴) → (𝐺‘𝑖):𝑇⟶ℝ)))
120	119, 56	vtoclg 3239	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴) → (𝐺‘𝑖):𝑇⟶ℝ))
121	113, 115, 120	sylc 63	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑖):𝑇⟶ℝ)
122		simp3 1056	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
123	121, 122	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
124	123	recnd 9947	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑖)‘𝑡) ∈ ℂ)
125	92, 110, 111, 124	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → ((𝐺‘𝑖)‘𝑡) ∈ ℂ)
126		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑦 + 1) → (𝐺‘𝑖) = (𝐺‘(𝑦 + 1)))
127	126	fveq1d 6105	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑦 + 1) → ((𝐺‘𝑖)‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
128	91, 125, 127	fsump1 14329	. . . . . . . . . . 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)))
129		simpr 476	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
130		fzfid 12634	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → (1...𝑦) ∈ Fin)
131		simpll1 1093	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝜑)
132		1zzd 11285	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ∈ ℤ)
133	95	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℤ)
134		elfzelz 12213	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℤ)
135	134	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℤ)
136		elfzle1 12215	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑦) → 1 ≤ 𝑖)
137	136	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ≤ 𝑖)
138	134	zred 11358	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℝ)
139	138	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℝ)
140	77	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ∈ ℝ)
141	79	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℝ)
142	75	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑦 ∈ ℝ)
143		elfzle2 12216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1...𝑦) → 𝑖 ≤ 𝑦)
144	143	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ 𝑦)
145		letrp1 10744	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑖 ≤ 𝑦) → 𝑖 ≤ (𝑦 + 1))
146	139, 142, 144, 145	syl3anc 1318	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ (𝑦 + 1))
147		simpl3 1059	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ≤ 𝑀)
148	139, 140, 141, 146, 147	letrd 10073	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ 𝑀)
149	148	adantlr 747	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ 𝑀)
150	132, 133, 135, 137, 149, 109	syl32anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ (1...𝑀))
151		simplr 788	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑡 ∈ 𝑇)
152	131, 150, 151, 123	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑦)) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ)
153	130, 152	fsumrecl 14312	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡) ∈ ℝ)
154		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))
155	154	fvmpt2 6200	. . . . . . . . . . . . 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))
156	129, 153, 155	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))
157	156	oveq1d 6564	. . . . . . . . . . 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)))
158	128, 157	eqtr4d 2647	. . . . . . . . . 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)))
159	89, 158	mpteq2da 4671	. . . . . . . . 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))))
160	159	adantr 480	. . . . . . . 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))))
161		1zzd 11285	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℤ)
162		peano2nn 10909	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
163	162	nnzd 11357	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℤ)
164	163	3ad2ant2 1076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℤ)
165	162	nnge1d 10940	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ → 1 ≤ (𝑦 + 1))
166	165	3ad2ant2 1076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ≤ (𝑦 + 1))
167		elfz4 12206	. . . . . . . . . . . . . . . . 17 ⊢ (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ) ∧ (1 ≤ (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑦 + 1) ∈ (1...𝑀))
168	161, 95, 164, 166, 81, 167	syl32anc 1326	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ (1...𝑀))
169	45	ffvelrnda 6267	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
170	73, 168, 169	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
171		eleq1 2676	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓 ∈ 𝐴 ↔ (𝐺‘(𝑦 + 1)) ∈ 𝐴))
172	171	anbi2d 736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝐺‘(𝑦 + 1)) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴)))
173		feq1 5939	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
174	172, 173	imbi12d 333	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝐺‘(𝑦 + 1)) → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)))
175	174, 56	vtoclg 3239	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘(𝑦 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
176	175	anabsi7 856	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
177	73, 170, 176	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
178	177	ffvelrnda 6267	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ)
179		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
180	179	fvmpt2 6200	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
181	129, 178, 180	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
182	181	oveq2d 6565	. . . . . . . . . . 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)))
183	89, 182	mpteq2da 4671	. . . . . . . . . 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))))
184	183	adantr 480	. . . . . . . . 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))))
185		simpl1 1057	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → 𝜑)
186		simpr 476	. . . . . . . . . 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)
187	168	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → (𝑦 + 1) ∈ (1...𝑀))
188	176	feqmptd 6159	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
189	169, 188	syldan 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
190	189, 169	eqeltrrd 2689	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴)
191	185, 187, 190	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴)
192		stoweidlem20.5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
193		nfmpt1 4675	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺‘𝑖)‘𝑡))
194		nfmpt1 4675	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
195	192, 193, 194	stoweidlem8 38901	. . . . . . . . . 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)
196	185, 186, 191, 195	syl3anc 1318	. . . . . . . . 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)
197	184, 196	eqeltrrd 2689	. . . . . . . 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)
198	160, 197	eqeltrd 2688	. . . . . . 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)
199	70, 71, 72, 86, 198	syl31anc 1321	. . . . . 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)
200	199	exp31 628	. . . . 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)))
201	21, 28, 35, 42, 69, 200	nnind 10915	. . . 4 ⊢ (𝑛 ∈ ℕ → ((𝜑 ∧ 𝑛 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴))
202	14, 201	vtoclg 3239	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑀 ∈ ℕ → ((𝜑 ∧ 𝑀 ≤ 𝑀) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)))
203	2, 2, 5, 202	syl3c 64	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) ∈ 𝐴)
204	1, 203	syl5eqel 2692	1 ⊢ (𝜑 → 𝐹 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  1c1 9816   + caddc 9818   ≤ cle 9954  ℕcn 10897  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  Σcsu 14264

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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265

This theorem is referenced by:  stoweidlem32  38925