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Theorem stoweidlem20 44251
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 12168 . . . . 5 (𝜑𝑀 ∈ ℝ)
43leidd 11721 . . . 4 (𝜑𝑀𝑀)
54ancli 549 . . 3 (𝜑 → (𝜑𝑀𝑀))
6 eleq1 2825 . . . . 5 (𝑛 = 𝑀 → (𝑛 ∈ ℕ ↔ 𝑀 ∈ ℕ))
7 breq1 5108 . . . . . . 7 (𝑛 = 𝑀 → (𝑛𝑀𝑀𝑀))
87anbi2d 629 . . . . . 6 (𝑛 = 𝑀 → ((𝜑𝑛𝑀) ↔ (𝜑𝑀𝑀)))
9 oveq2 7365 . . . . . . . . 9 (𝑛 = 𝑀 → (1...𝑛) = (1...𝑀))
109sumeq1d 15586 . . . . . . . 8 (𝑛 = 𝑀 → Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
1110mpteq2dv 5207 . . . . . . 7 (𝑛 = 𝑀 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1211eleq1d 2822 . . . . . 6 (𝑛 = 𝑀 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
138, 12imbi12d 344 . . . . 5 (𝑛 = 𝑀 → (((𝜑𝑛𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑𝑀𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
146, 13imbi12d 344 . . . 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 5108 . . . . . . 7 (𝑥 = 1 → (𝑥𝑀 ↔ 1 ≤ 𝑀))
1615anbi2d 629 . . . . . 6 (𝑥 = 1 → ((𝜑𝑥𝑀) ↔ (𝜑 ∧ 1 ≤ 𝑀)))
17 oveq2 7365 . . . . . . . . 9 (𝑥 = 1 → (1...𝑥) = (1...1))
1817sumeq1d 15586 . . . . . . . 8 (𝑥 = 1 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡))
1918mpteq2dv 5207 . . . . . . 7 (𝑥 = 1 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)))
2019eleq1d 2822 . . . . . 6 (𝑥 = 1 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
2116, 20imbi12d 344 . . . . 5 (𝑥 = 1 → (((𝜑𝑥𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑 ∧ 1 ≤ 𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
22 breq1 5108 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑀𝑦𝑀))
2322anbi2d 629 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥𝑀) ↔ (𝜑𝑦𝑀)))
24 oveq2 7365 . . . . . . . . 9 (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
2524sumeq1d 15586 . . . . . . . 8 (𝑥 = 𝑦 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
2625mpteq2dv 5207 . . . . . . 7 (𝑥 = 𝑦 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)))
2726eleq1d 2822 . . . . . 6 (𝑥 = 𝑦 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
2823, 27imbi12d 344 . . . . 5 (𝑥 = 𝑦 → (((𝜑𝑥𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
29 breq1 5108 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑥𝑀 ↔ (𝑦 + 1) ≤ 𝑀))
3029anbi2d 629 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝜑𝑥𝑀) ↔ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)))
31 oveq2 7365 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
3231sumeq1d 15586 . . . . . . . 8 (𝑥 = (𝑦 + 1) → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡))
3332mpteq2dv 5207 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)))
3433eleq1d 2822 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴))
3530, 34imbi12d 344 . . . . 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 5108 . . . . . . 7 (𝑥 = 𝑛 → (𝑥𝑀𝑛𝑀))
3736anbi2d 629 . . . . . 6 (𝑥 = 𝑛 → ((𝜑𝑥𝑀) ↔ (𝜑𝑛𝑀)))
38 oveq2 7365 . . . . . . . . 9 (𝑥 = 𝑛 → (1...𝑥) = (1...𝑛))
3938sumeq1d 15586 . . . . . . . 8 (𝑥 = 𝑛 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡))
4039mpteq2dv 5207 . . . . . . 7 (𝑥 = 𝑛 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)))
4140eleq1d 2822 . . . . . 6 (𝑥 = 𝑛 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
4237, 41imbi12d 344 . . . . 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 12533 . . . . . . . . . 10 1 ∈ ℤ
45 stoweidlem20.4 . . . . . . . . . . . . . 14 (𝜑𝐺:(1...𝑀)⟶𝐴)
46 nnuz 12806 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
472, 46eleqtrdi 2848 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (ℤ‘1))
48 eluzfz1 13448 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ‘1) → 1 ∈ (1...𝑀))
4947, 48syl 17 . . . . . . . . . . . . . 14 (𝜑 → 1 ∈ (1...𝑀))
5045, 49ffvelcdmd 7036 . . . . . . . . . . . . 13 (𝜑 → (𝐺‘1) ∈ 𝐴)
5150ancli 549 . . . . . . . . . . . . 13 (𝜑 → (𝜑 ∧ (𝐺‘1) ∈ 𝐴))
52 eleq1 2825 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐺‘1) → (𝑓𝐴 ↔ (𝐺‘1) ∈ 𝐴))
5352anbi2d 629 . . . . . . . . . . . . . . 15 (𝑓 = (𝐺‘1) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺‘1) ∈ 𝐴)))
54 feq1 6649 . . . . . . . . . . . . . . 15 (𝑓 = (𝐺‘1) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘1):𝑇⟶ℝ))
5553, 54imbi12d 344 . . . . . . . . . . . . . 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 3525 . . . . . . . . . . . . 13 ((𝐺‘1) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘1) ∈ 𝐴) → (𝐺‘1):𝑇⟶ℝ))
5850, 51, 57sylc 65 . . . . . . . . . . . 12 (𝜑 → (𝐺‘1):𝑇⟶ℝ)
5958ffvelcdmda 7035 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → ((𝐺‘1)‘𝑡) ∈ ℝ)
6059recnd 11183 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝐺‘1)‘𝑡) ∈ ℂ)
61 fveq2 6842 . . . . . . . . . . . 12 (𝑖 = 1 → (𝐺𝑖) = (𝐺‘1))
6261fveq1d 6844 . . . . . . . . . . 11 (𝑖 = 1 → ((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6362fsum1 15632 . . . . . . . . . 10 ((1 ∈ ℤ ∧ ((𝐺‘1)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6444, 60, 63sylancr 587 . . . . . . . . 9 ((𝜑𝑡𝑇) → Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6543, 64mpteq2da 5203 . . . . . . . 8 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ ((𝐺‘1)‘𝑡)))
6658feqmptd 6910 . . . . . . . 8 (𝜑 → (𝐺‘1) = (𝑡𝑇 ↦ ((𝐺‘1)‘𝑡)))
6765, 66eqtr4d 2779 . . . . . . 7 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) = (𝐺‘1))
6867, 50eqeltrd 2838 . . . . . 6 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
6968adantr 481 . . . . 5 ((𝜑 ∧ 1 ≤ 𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
70 simprl 769 . . . . . . 7 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → 𝜑)
71 simpll 765 . . . . . . 7 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → 𝑦 ∈ ℕ)
72 simprr 771 . . . . . . 7 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑦 + 1) ≤ 𝑀)
73 simp1 1136 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝜑)
74 nnre 12160 . . . . . . . . . . . 12 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
75743ad2ant2 1134 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ∈ ℝ)
76 1red 11156 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℝ)
7775, 76readdcld 11184 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℝ)
7823ad2ant1 1133 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℕ)
7978nnred 12168 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℝ)
8075lep1d 12086 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ≤ (𝑦 + 1))
81 simp3 1138 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ≤ 𝑀)
8275, 77, 79, 80, 81letrd 11312 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦𝑀)
8373, 82jca 512 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝜑𝑦𝑀))
8470, 71, 72, 83syl3anc 1371 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝜑𝑦𝑀))
85 simplr 767 . . . . . . . 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 1917 . . . . . . . . . . 11 𝑡 𝑦 ∈ ℕ
88 nfv 1917 . . . . . . . . . . 11 𝑡(𝑦 + 1) ≤ 𝑀
8943, 87, 88nf3an 1904 . . . . . . . . . 10 𝑡(𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀)
90 simpl2 1192 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑦 ∈ ℕ)
9190, 46eleqtrdi 2848 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑦 ∈ (ℤ‘1))
92 simpll1 1212 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝜑)
93 1zzd 12534 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ∈ ℤ)
942nnzd 12526 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℤ)
95943ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℤ)
9695ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑀 ∈ ℤ)
97 elfzelz 13441 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℤ)
9897adantl 482 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ ℤ)
99 elfzle1 13444 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...(𝑦 + 1)) → 1 ≤ 𝑖)
10099adantl 482 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ≤ 𝑖)
10197zred 12607 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℝ)
102101adantl 482 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ ℝ)
10377ad2antrr 724 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → (𝑦 + 1) ∈ ℝ)
10479ad2antrr 724 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑀 ∈ ℝ)
105 elfzle2 13445 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ≤ (𝑦 + 1))
106105adantl 482 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ≤ (𝑦 + 1))
107 simpll3 1214 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → (𝑦 + 1) ≤ 𝑀)
108102, 103, 104, 106, 107letrd 11312 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖𝑀)
10993, 96, 98, 100, 108elfzd 13432 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ (1...𝑀))
110 simplr 767 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑡𝑇)
11145ffvelcdmda 7035 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺𝑖) ∈ 𝐴)
1121113adant3 1132 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝐺𝑖) ∈ 𝐴)
113 simp1 1136 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → 𝜑)
114113, 112jca 512 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝜑 ∧ (𝐺𝑖) ∈ 𝐴))
115 eleq1 2825 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝐺𝑖) → (𝑓𝐴 ↔ (𝐺𝑖) ∈ 𝐴))
116115anbi2d 629 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺𝑖) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺𝑖) ∈ 𝐴)))
117 feq1 6649 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝐺𝑖):𝑇⟶ℝ))
118116, 117imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝐺𝑖) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺𝑖) ∈ 𝐴) → (𝐺𝑖):𝑇⟶ℝ)))
119118, 56vtoclg 3525 . . . . . . . . . . . . . . . 16 ((𝐺𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝐺𝑖) ∈ 𝐴) → (𝐺𝑖):𝑇⟶ℝ))
120112, 114, 119sylc 65 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝐺𝑖):𝑇⟶ℝ)
121 simp3 1138 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → 𝑡𝑇)
122120, 121ffvelcdmd 7036 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
123122recnd 11183 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
12492, 109, 110, 123syl3anc 1371 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
125 fveq2 6842 . . . . . . . . . . . . 13 (𝑖 = (𝑦 + 1) → (𝐺𝑖) = (𝐺‘(𝑦 + 1)))
126125fveq1d 6844 . . . . . . . . . . . 12 (𝑖 = (𝑦 + 1) → ((𝐺𝑖)‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
12791, 124, 126fsump1 15641 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡) = (Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
128 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑡𝑇)
129 fzfid 13878 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → (1...𝑦) ∈ Fin)
130 simpll1 1212 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝜑)
131 1zzd 12534 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ∈ ℤ)
13295ad2antrr 724 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℤ)
133 elfzelz 13441 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℤ)
134133adantl 482 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℤ)
135 elfzle1 13444 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑦) → 1 ≤ 𝑖)
136135adantl 482 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ≤ 𝑖)
137133zred 12607 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℝ)
138137adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℝ)
13977adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ∈ ℝ)
14079adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℝ)
14175adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑦 ∈ ℝ)
142 elfzle2 13445 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑦) → 𝑖𝑦)
143142adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑦)
144 letrp1 11999 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑖𝑦) → 𝑖 ≤ (𝑦 + 1))
145138, 141, 143, 144syl3anc 1371 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ (𝑦 + 1))
146 simpl3 1193 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ≤ 𝑀)
147138, 139, 140, 145, 146letrd 11312 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑀)
148147adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑀)
149131, 132, 134, 136, 148elfzd 13432 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ (1...𝑀))
150 simplr 767 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑡𝑇)
151130, 149, 150, 122syl3anc 1371 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
152129, 151fsumrecl 15619 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) ∈ ℝ)
153 eqid 2736 . . . . . . . . . . . . . 14 (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
154153fvmpt2 6959 . . . . . . . . . . . . 13 ((𝑡𝑇 ∧ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) ∈ ℝ) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
155128, 152, 154syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
156155oveq1d 7372 . . . . . . . . . . 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)))
157127, 156eqtr4d 2779 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡) = (((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
15889, 157mpteq2da 5203 . . . . . . . . 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))))
159158adantr 481 . . . . . . . 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))))
160 1zzd 12534 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℤ)
161 peano2nn 12165 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
162161nnzd 12526 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℤ)
1631623ad2ant2 1134 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℤ)
164161nnge1d 12201 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → 1 ≤ (𝑦 + 1))
1651643ad2ant2 1134 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ≤ (𝑦 + 1))
166160, 95, 163, 165, 81elfzd 13432 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ (1...𝑀))
16745ffvelcdmda 7035 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
16873, 166, 167syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
169 eleq1 2825 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓𝐴 ↔ (𝐺‘(𝑦 + 1)) ∈ 𝐴))
170169anbi2d 629 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺‘(𝑦 + 1)) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴)))
171 feq1 6649 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
172170, 171imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝐺‘(𝑦 + 1)) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)))
173172, 56vtoclg 3525 . . . . . . . . . . . . . . . 16 ((𝐺‘(𝑦 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
174173anabsi7 669 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
17573, 168, 174syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
176175ffvelcdmda 7035 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ)
177 eqid 2736 . . . . . . . . . . . . . 14 (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
178177fvmpt2 6959 . . . . . . . . . . . . 13 ((𝑡𝑇 ∧ ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ) → ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
179128, 176, 178syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
180179oveq2d 7373 . . . . . . . . . . 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)))
18189, 180mpteq2da 5203 . . . . . . . . . 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))))
182181adantr 481 . . . . . . . . 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))))
183 simpl1 1191 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → 𝜑)
184 simpr 485 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
185166adantr 481 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑦 + 1) ∈ (1...𝑀))
186174feqmptd 6910 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
187167, 186syldan 591 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
188187, 167eqeltrrd 2839 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴)
189183, 185, 188syl2anc 584 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴)
190 stoweidlem20.5 . . . . . . . . . . 11 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
191 nfmpt1 5213 . . . . . . . . . . 11 𝑡(𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
192 nfmpt1 5213 . . . . . . . . . . 11 𝑡(𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
193190, 191, 192stoweidlem8 44239 . . . . . . . . . 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)
194183, 184, 189, 193syl3anc 1371 . . . . . . . . 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)
195182, 194eqeltrrd 2839 . . . . . . . 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)
196159, 195eqeltrd 2838 . . . . . . 7 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴)
19770, 71, 72, 86, 196syl31anc 1373 . . . . . 6 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴)
198197exp31 420 . . . . 5 (𝑦 ∈ ℕ → (((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → ((𝜑 ∧ (𝑦 + 1) ≤ 𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
19921, 28, 35, 42, 69, 198nnind 12171 . . . 4 (𝑛 ∈ ℕ → ((𝜑𝑛𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
20014, 199vtoclg 3525 . . 3 (𝑀 ∈ ℕ → (𝑀 ∈ ℕ → ((𝜑𝑀𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
2012, 2, 5, 200syl3c 66 . 2 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
2021, 201eqeltrid 2842 1 (𝜑𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wnf 1785  wcel 2106   class class class wbr 5105  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  1c1 11052   + caddc 11054  cle 11190  cn 12153  cz 12499  cuz 12763  ...cfz 13424  Σcsu 15570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571
This theorem is referenced by:  stoweidlem32  44263
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