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Theorem stoweidlem20 40975
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 𝑡𝜑
stoweidlem20.2 𝐹 = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
stoweidlem20.3 (𝜑𝑀 ∈ ℕ)
stoweidlem20.4 (𝜑𝐺:(1...𝑀)⟶𝐴)
stoweidlem20.5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stoweidlem20.6 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
Assertion
Ref Expression
stoweidlem20 (𝜑𝐹𝐴)
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.
StepHypRef Expression
1 stoweidlem20.2 . 2 𝐹 = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
2 stoweidlem20.3 . . 3 (𝜑𝑀 ∈ ℕ)
32nnred 11330 . . . . 5 (𝜑𝑀 ∈ ℝ)
43leidd 10887 . . . 4 (𝜑𝑀𝑀)
54ancli 545 . . 3 (𝜑 → (𝜑𝑀𝑀))
6 eleq1 2867 . . . . 5 (𝑛 = 𝑀 → (𝑛 ∈ ℕ ↔ 𝑀 ∈ ℕ))
7 breq1 4847 . . . . . . 7 (𝑛 = 𝑀 → (𝑛𝑀𝑀𝑀))
87anbi2d 623 . . . . . 6 (𝑛 = 𝑀 → ((𝜑𝑛𝑀) ↔ (𝜑𝑀𝑀)))
9 oveq2 6887 . . . . . . . . 9 (𝑛 = 𝑀 → (1...𝑛) = (1...𝑀))
109sumeq1d 14771 . . . . . . . 8 (𝑛 = 𝑀 → Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
1110mpteq2dv 4939 . . . . . . 7 (𝑛 = 𝑀 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1211eleq1d 2864 . . . . . 6 (𝑛 = 𝑀 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
138, 12imbi12d 336 . . . . 5 (𝑛 = 𝑀 → (((𝜑𝑛𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑𝑀𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
146, 13imbi12d 336 . . . 4 (𝑛 = 𝑀 → ((𝑛 ∈ ℕ → ((𝜑𝑛𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ↔ (𝑀 ∈ ℕ → ((𝜑𝑀𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴))))
15 breq1 4847 . . . . . . 7 (𝑥 = 1 → (𝑥𝑀 ↔ 1 ≤ 𝑀))
1615anbi2d 623 . . . . . 6 (𝑥 = 1 → ((𝜑𝑥𝑀) ↔ (𝜑 ∧ 1 ≤ 𝑀)))
17 oveq2 6887 . . . . . . . . 9 (𝑥 = 1 → (1...𝑥) = (1...1))
1817sumeq1d 14771 . . . . . . . 8 (𝑥 = 1 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡))
1918mpteq2dv 4939 . . . . . . 7 (𝑥 = 1 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)))
2019eleq1d 2864 . . . . . 6 (𝑥 = 1 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
2116, 20imbi12d 336 . . . . 5 (𝑥 = 1 → (((𝜑𝑥𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑 ∧ 1 ≤ 𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
22 breq1 4847 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑀𝑦𝑀))
2322anbi2d 623 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥𝑀) ↔ (𝜑𝑦𝑀)))
24 oveq2 6887 . . . . . . . . 9 (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
2524sumeq1d 14771 . . . . . . . 8 (𝑥 = 𝑦 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
2625mpteq2dv 4939 . . . . . . 7 (𝑥 = 𝑦 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)))
2726eleq1d 2864 . . . . . 6 (𝑥 = 𝑦 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
2823, 27imbi12d 336 . . . . 5 (𝑥 = 𝑦 → (((𝜑𝑥𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
29 breq1 4847 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑥𝑀 ↔ (𝑦 + 1) ≤ 𝑀))
3029anbi2d 623 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝜑𝑥𝑀) ↔ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)))
31 oveq2 6887 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
3231sumeq1d 14771 . . . . . . . 8 (𝑥 = (𝑦 + 1) → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡))
3332mpteq2dv 4939 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)))
3433eleq1d 2864 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴))
3530, 34imbi12d 336 . . . . 5 (𝑥 = (𝑦 + 1) → (((𝜑𝑥𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑦 + 1) ≤ 𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
36 breq1 4847 . . . . . . 7 (𝑥 = 𝑛 → (𝑥𝑀𝑛𝑀))
3736anbi2d 623 . . . . . 6 (𝑥 = 𝑛 → ((𝜑𝑥𝑀) ↔ (𝜑𝑛𝑀)))
38 oveq2 6887 . . . . . . . . 9 (𝑥 = 𝑛 → (1...𝑥) = (1...𝑛))
3938sumeq1d 14771 . . . . . . . 8 (𝑥 = 𝑛 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡))
4039mpteq2dv 4939 . . . . . . 7 (𝑥 = 𝑛 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)))
4140eleq1d 2864 . . . . . 6 (𝑥 = 𝑛 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
4237, 41imbi12d 336 . . . . 5 (𝑥 = 𝑛 → (((𝜑𝑥𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑𝑛𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
43 stoweidlem20.1 . . . . . . . . 9 𝑡𝜑
44 1z 11696 . . . . . . . . . 10 1 ∈ ℤ
45 stoweidlem20.4 . . . . . . . . . . . . . 14 (𝜑𝐺:(1...𝑀)⟶𝐴)
46 nnuz 11966 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
472, 46syl6eleq 2889 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (ℤ‘1))
48 eluzfz1 12601 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ‘1) → 1 ∈ (1...𝑀))
4947, 48syl 17 . . . . . . . . . . . . . 14 (𝜑 → 1 ∈ (1...𝑀))
5045, 49ffvelrnd 6587 . . . . . . . . . . . . 13 (𝜑 → (𝐺‘1) ∈ 𝐴)
5150ancli 545 . . . . . . . . . . . . 13 (𝜑 → (𝜑 ∧ (𝐺‘1) ∈ 𝐴))
52 eleq1 2867 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐺‘1) → (𝑓𝐴 ↔ (𝐺‘1) ∈ 𝐴))
5352anbi2d 623 . . . . . . . . . . . . . . 15 (𝑓 = (𝐺‘1) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺‘1) ∈ 𝐴)))
54 feq1 6238 . . . . . . . . . . . . . . 15 (𝑓 = (𝐺‘1) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘1):𝑇⟶ℝ))
5553, 54imbi12d 336 . . . . . . . . . . . . . 14 (𝑓 = (𝐺‘1) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘1) ∈ 𝐴) → (𝐺‘1):𝑇⟶ℝ)))
56 stoweidlem20.6 . . . . . . . . . . . . . 14 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
5755, 56vtoclg 3454 . . . . . . . . . . . . 13 ((𝐺‘1) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘1) ∈ 𝐴) → (𝐺‘1):𝑇⟶ℝ))
5850, 51, 57sylc 65 . . . . . . . . . . . 12 (𝜑 → (𝐺‘1):𝑇⟶ℝ)
5958ffvelrnda 6586 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → ((𝐺‘1)‘𝑡) ∈ ℝ)
6059recnd 10358 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝐺‘1)‘𝑡) ∈ ℂ)
61 fveq2 6412 . . . . . . . . . . . 12 (𝑖 = 1 → (𝐺𝑖) = (𝐺‘1))
6261fveq1d 6414 . . . . . . . . . . 11 (𝑖 = 1 → ((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6362fsum1 14816 . . . . . . . . . 10 ((1 ∈ ℤ ∧ ((𝐺‘1)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6444, 60, 63sylancr 582 . . . . . . . . 9 ((𝜑𝑡𝑇) → Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6543, 64mpteq2da 4937 . . . . . . . 8 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ ((𝐺‘1)‘𝑡)))
6658feqmptd 6475 . . . . . . . 8 (𝜑 → (𝐺‘1) = (𝑡𝑇 ↦ ((𝐺‘1)‘𝑡)))
6765, 66eqtr4d 2837 . . . . . . 7 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) = (𝐺‘1))
6867, 50eqeltrd 2879 . . . . . 6 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
6968adantr 473 . . . . 5 ((𝜑 ∧ 1 ≤ 𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
70 simprl 788 . . . . . . 7 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → 𝜑)
71 simpll 784 . . . . . . 7 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → 𝑦 ∈ ℕ)
72 simprr 790 . . . . . . 7 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑦 + 1) ≤ 𝑀)
73 simp1 1167 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝜑)
74 nnre 11321 . . . . . . . . . . . 12 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
75743ad2ant2 1165 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ∈ ℝ)
76 1red 10330 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℝ)
7775, 76readdcld 10359 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℝ)
7823ad2ant1 1164 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℕ)
7978nnred 11330 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℝ)
8075lep1d 11248 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ≤ (𝑦 + 1))
81 simp3 1169 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ≤ 𝑀)
8275, 77, 79, 80, 81letrd 10485 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦𝑀)
8373, 82jca 508 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝜑𝑦𝑀))
8470, 71, 72, 83syl3anc 1491 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝜑𝑦𝑀))
85 simplr 786 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
8684, 85mpd 15 . . . . . . 7 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
87 nfv 2010 . . . . . . . . . . 11 𝑡 𝑦 ∈ ℕ
88 nfv 2010 . . . . . . . . . . 11 𝑡(𝑦 + 1) ≤ 𝑀
8943, 87, 88nf3an 2001 . . . . . . . . . 10 𝑡(𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀)
90 simpl2 1245 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑦 ∈ ℕ)
9190, 46syl6eleq 2889 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑦 ∈ (ℤ‘1))
92 simpll1 1270 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝜑)
93 1zzd 11697 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ∈ ℤ)
942nnzd 11770 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℤ)
95943ad2ant1 1164 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℤ)
9695ad2antrr 718 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑀 ∈ ℤ)
97 elfzelz 12595 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℤ)
9897adantl 474 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ ℤ)
99 elfzle1 12597 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...(𝑦 + 1)) → 1 ≤ 𝑖)
10099adantl 474 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ≤ 𝑖)
10197zred 11771 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℝ)
102101adantl 474 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ ℝ)
10377ad2antrr 718 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → (𝑦 + 1) ∈ ℝ)
10479ad2antrr 718 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑀 ∈ ℝ)
105 elfzle2 12598 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ≤ (𝑦 + 1))
106105adantl 474 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ≤ (𝑦 + 1))
107 simpll3 1274 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → (𝑦 + 1) ≤ 𝑀)
108102, 103, 104, 106, 107letrd 10485 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖𝑀)
109 elfz4 12588 . . . . . . . . . . . . . 14 (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑀)) → 𝑖 ∈ (1...𝑀))
11093, 96, 98, 100, 108, 109syl32anc 1498 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ (1...𝑀))
111 simplr 786 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑡𝑇)
11245ffvelrnda 6586 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺𝑖) ∈ 𝐴)
1131123adant3 1163 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝐺𝑖) ∈ 𝐴)
114 simp1 1167 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → 𝜑)
115114, 113jca 508 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝜑 ∧ (𝐺𝑖) ∈ 𝐴))
116 eleq1 2867 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝐺𝑖) → (𝑓𝐴 ↔ (𝐺𝑖) ∈ 𝐴))
117116anbi2d 623 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺𝑖) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺𝑖) ∈ 𝐴)))
118 feq1 6238 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝐺𝑖):𝑇⟶ℝ))
119117, 118imbi12d 336 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝐺𝑖) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺𝑖) ∈ 𝐴) → (𝐺𝑖):𝑇⟶ℝ)))
120119, 56vtoclg 3454 . . . . . . . . . . . . . . . 16 ((𝐺𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝐺𝑖) ∈ 𝐴) → (𝐺𝑖):𝑇⟶ℝ))
121113, 115, 120sylc 65 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝐺𝑖):𝑇⟶ℝ)
122 simp3 1169 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → 𝑡𝑇)
123121, 122ffvelrnd 6587 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
124123recnd 10358 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
12592, 110, 111, 124syl3anc 1491 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
126 fveq2 6412 . . . . . . . . . . . . 13 (𝑖 = (𝑦 + 1) → (𝐺𝑖) = (𝐺‘(𝑦 + 1)))
127126fveq1d 6414 . . . . . . . . . . . 12 (𝑖 = (𝑦 + 1) → ((𝐺𝑖)‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
12891, 125, 127fsump1 14825 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡) = (Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
129 simpr 478 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑡𝑇)
130 fzfid 13026 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → (1...𝑦) ∈ Fin)
131 simpll1 1270 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝜑)
132 1zzd 11697 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ∈ ℤ)
13395ad2antrr 718 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℤ)
134 elfzelz 12595 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℤ)
135134adantl 474 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℤ)
136 elfzle1 12597 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑦) → 1 ≤ 𝑖)
137136adantl 474 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ≤ 𝑖)
138134zred 11771 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℝ)
139138adantl 474 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℝ)
14077adantr 473 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ∈ ℝ)
14179adantr 473 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℝ)
14275adantr 473 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑦 ∈ ℝ)
143 elfzle2 12598 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑦) → 𝑖𝑦)
144143adantl 474 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑦)
145 letrp1 11158 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑖𝑦) → 𝑖 ≤ (𝑦 + 1))
146139, 142, 144, 145syl3anc 1491 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ (𝑦 + 1))
147 simpl3 1247 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ≤ 𝑀)
148139, 140, 141, 146, 147letrd 10485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑀)
149148adantlr 707 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑀)
150132, 133, 135, 137, 149, 109syl32anc 1498 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ (1...𝑀))
151 simplr 786 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑡𝑇)
152131, 150, 151, 123syl3anc 1491 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
153130, 152fsumrecl 14805 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) ∈ ℝ)
154 eqid 2800 . . . . . . . . . . . . . 14 (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
155154fvmpt2 6517 . . . . . . . . . . . . 13 ((𝑡𝑇 ∧ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) ∈ ℝ) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
156129, 153, 155syl2anc 580 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
157156oveq1d 6894 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)) = (Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
158128, 157eqtr4d 2837 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡) = (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
15989, 158mpteq2da 4937 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡))))
160159adantr 473 . . . . . . . 8 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡))))
161 1zzd 11697 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℤ)
162 peano2nn 11327 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
163162nnzd 11770 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℤ)
1641633ad2ant2 1165 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℤ)
165162nnge1d 11360 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → 1 ≤ (𝑦 + 1))
1661653ad2ant2 1165 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ≤ (𝑦 + 1))
167 elfz4 12588 . . . . . . . . . . . . . . . . 17 (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ) ∧ (1 ≤ (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑦 + 1) ∈ (1...𝑀))
168161, 95, 164, 166, 81, 167syl32anc 1498 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ (1...𝑀))
16945ffvelrnda 6586 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
17073, 168, 169syl2anc 580 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
171 eleq1 2867 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓𝐴 ↔ (𝐺‘(𝑦 + 1)) ∈ 𝐴))
172171anbi2d 623 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺‘(𝑦 + 1)) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴)))
173 feq1 6238 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
174172, 173imbi12d 336 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝐺‘(𝑦 + 1)) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)))
175174, 56vtoclg 3454 . . . . . . . . . . . . . . . 16 ((𝐺‘(𝑦 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
176175anabsi7 662 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
17773, 170, 176syl2anc 580 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
178177ffvelrnda 6586 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ)
179 eqid 2800 . . . . . . . . . . . . . 14 (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
180179fvmpt2 6517 . . . . . . . . . . . . 13 ((𝑡𝑇 ∧ ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ) → ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
181129, 178, 180syl2anc 580 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
182181oveq2d 6895 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡)) = (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
18389, 182mpteq2da 4937 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡))))
184183adantr 473 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡))))
185 simpl1 1243 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → 𝜑)
186 simpr 478 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
187168adantr 473 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑦 + 1) ∈ (1...𝑀))
188176feqmptd 6475 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
189169, 188syldan 586 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
190189, 169eqeltrrd 2880 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴)
191185, 187, 190syl2anc 580 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴)
192 stoweidlem20.5 . . . . . . . . . . 11 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
193 nfmpt1 4941 . . . . . . . . . . 11 𝑡(𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
194 nfmpt1 4941 . . . . . . . . . . 11 𝑡(𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
195192, 193, 194stoweidlem8 40963 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ∧ (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡))) ∈ 𝐴)
196185, 186, 191, 195syl3anc 1491 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡))) ∈ 𝐴)
197184, 196eqeltrrd 2880 . . . . . . . 8 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡))) ∈ 𝐴)
198160, 197eqeltrd 2879 . . . . . . 7 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴)
19970, 71, 72, 86, 198syl31anc 1493 . . . . . 6 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴)
200199exp31 411 . . . . 5 (𝑦 ∈ ℕ → (((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → ((𝜑 ∧ (𝑦 + 1) ≤ 𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
20121, 28, 35, 42, 69, 200nnind 11333 . . . 4 (𝑛 ∈ ℕ → ((𝜑𝑛𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
20214, 201vtoclg 3454 . . 3 (𝑀 ∈ ℕ → (𝑀 ∈ ℕ → ((𝜑𝑀𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
2032, 2, 5, 202syl3c 66 . 2 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
2041, 203syl5eqel 2883 1 (𝜑𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wnf 1879  wcel 2157   class class class wbr 4844  cmpt 4923  wf 6098  cfv 6102  (class class class)co 6879  cc 10223  cr 10224  1c1 10226   + caddc 10228  cle 10365  cn 11313  cz 11665  cuz 11929  ...cfz 12579  Σcsu 14756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184  ax-inf2 8789  ax-cnex 10281  ax-resscn 10282  ax-1cn 10283  ax-icn 10284  ax-addcl 10285  ax-addrcl 10286  ax-mulcl 10287  ax-mulrcl 10288  ax-mulcom 10289  ax-addass 10290  ax-mulass 10291  ax-distr 10292  ax-i2m1 10293  ax-1ne0 10294  ax-1rid 10295  ax-rnegex 10296  ax-rrecex 10297  ax-cnre 10298  ax-pre-lttri 10299  ax-pre-lttrn 10300  ax-pre-ltadd 10301  ax-pre-mulgt0 10302  ax-pre-sup 10303
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-se 5273  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899  df-ord 5945  df-on 5946  df-lim 5947  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-isom 6111  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-om 7301  df-1st 7402  df-2nd 7403  df-wrecs 7646  df-recs 7708  df-rdg 7746  df-1o 7800  df-oadd 7804  df-er 7983  df-en 8197  df-dom 8198  df-sdom 8199  df-fin 8200  df-sup 8591  df-oi 8658  df-card 9052  df-pnf 10366  df-mnf 10367  df-xr 10368  df-ltxr 10369  df-le 10370  df-sub 10559  df-neg 10560  df-div 10978  df-nn 11314  df-2 11375  df-3 11376  df-n0 11580  df-z 11666  df-uz 11930  df-rp 12074  df-fz 12580  df-fzo 12720  df-seq 13055  df-exp 13114  df-hash 13370  df-cj 14179  df-re 14180  df-im 14181  df-sqrt 14315  df-abs 14316  df-clim 14559  df-sum 14757
This theorem is referenced by:  stoweidlem32  40987
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