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Theorem stoweidlem20 39541
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 10979 . . . . 5 (𝜑𝑀 ∈ ℝ)
43leidd 10538 . . . 4 (𝜑𝑀𝑀)
54ancli 573 . . 3 (𝜑 → (𝜑𝑀𝑀))
6 eleq1 2686 . . . . 5 (𝑛 = 𝑀 → (𝑛 ∈ ℕ ↔ 𝑀 ∈ ℕ))
7 breq1 4616 . . . . . . 7 (𝑛 = 𝑀 → (𝑛𝑀𝑀𝑀))
87anbi2d 739 . . . . . 6 (𝑛 = 𝑀 → ((𝜑𝑛𝑀) ↔ (𝜑𝑀𝑀)))
9 oveq2 6612 . . . . . . . . 9 (𝑛 = 𝑀 → (1...𝑛) = (1...𝑀))
109sumeq1d 14365 . . . . . . . 8 (𝑛 = 𝑀 → Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
1110mpteq2dv 4705 . . . . . . 7 (𝑛 = 𝑀 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1211eleq1d 2683 . . . . . 6 (𝑛 = 𝑀 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
138, 12imbi12d 334 . . . . 5 (𝑛 = 𝑀 → (((𝜑𝑛𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑𝑀𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
146, 13imbi12d 334 . . . 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 4616 . . . . . . 7 (𝑥 = 1 → (𝑥𝑀 ↔ 1 ≤ 𝑀))
1615anbi2d 739 . . . . . 6 (𝑥 = 1 → ((𝜑𝑥𝑀) ↔ (𝜑 ∧ 1 ≤ 𝑀)))
17 oveq2 6612 . . . . . . . . 9 (𝑥 = 1 → (1...𝑥) = (1...1))
1817sumeq1d 14365 . . . . . . . 8 (𝑥 = 1 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡))
1918mpteq2dv 4705 . . . . . . 7 (𝑥 = 1 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)))
2019eleq1d 2683 . . . . . 6 (𝑥 = 1 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
2116, 20imbi12d 334 . . . . 5 (𝑥 = 1 → (((𝜑𝑥𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑 ∧ 1 ≤ 𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
22 breq1 4616 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑀𝑦𝑀))
2322anbi2d 739 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥𝑀) ↔ (𝜑𝑦𝑀)))
24 oveq2 6612 . . . . . . . . 9 (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
2524sumeq1d 14365 . . . . . . . 8 (𝑥 = 𝑦 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
2625mpteq2dv 4705 . . . . . . 7 (𝑥 = 𝑦 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)))
2726eleq1d 2683 . . . . . 6 (𝑥 = 𝑦 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
2823, 27imbi12d 334 . . . . 5 (𝑥 = 𝑦 → (((𝜑𝑥𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
29 breq1 4616 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑥𝑀 ↔ (𝑦 + 1) ≤ 𝑀))
3029anbi2d 739 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝜑𝑥𝑀) ↔ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)))
31 oveq2 6612 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
3231sumeq1d 14365 . . . . . . . 8 (𝑥 = (𝑦 + 1) → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡))
3332mpteq2dv 4705 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)))
3433eleq1d 2683 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴))
3530, 34imbi12d 334 . . . . 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 4616 . . . . . . 7 (𝑥 = 𝑛 → (𝑥𝑀𝑛𝑀))
3736anbi2d 739 . . . . . 6 (𝑥 = 𝑛 → ((𝜑𝑥𝑀) ↔ (𝜑𝑛𝑀)))
38 oveq2 6612 . . . . . . . . 9 (𝑥 = 𝑛 → (1...𝑥) = (1...𝑛))
3938sumeq1d 14365 . . . . . . . 8 (𝑥 = 𝑛 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡))
4039mpteq2dv 4705 . . . . . . 7 (𝑥 = 𝑛 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)))
4140eleq1d 2683 . . . . . 6 (𝑥 = 𝑛 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
4237, 41imbi12d 334 . . . . 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 11351 . . . . . . . . . 10 1 ∈ ℤ
45 stoweidlem20.4 . . . . . . . . . . . . . 14 (𝜑𝐺:(1...𝑀)⟶𝐴)
46 nnuz 11667 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
472, 46syl6eleq 2708 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (ℤ‘1))
48 eluzfz1 12290 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ‘1) → 1 ∈ (1...𝑀))
4947, 48syl 17 . . . . . . . . . . . . . 14 (𝜑 → 1 ∈ (1...𝑀))
5045, 49ffvelrnd 6316 . . . . . . . . . . . . 13 (𝜑 → (𝐺‘1) ∈ 𝐴)
5150ancli 573 . . . . . . . . . . . . 13 (𝜑 → (𝜑 ∧ (𝐺‘1) ∈ 𝐴))
52 eleq1 2686 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐺‘1) → (𝑓𝐴 ↔ (𝐺‘1) ∈ 𝐴))
5352anbi2d 739 . . . . . . . . . . . . . . 15 (𝑓 = (𝐺‘1) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺‘1) ∈ 𝐴)))
54 feq1 5983 . . . . . . . . . . . . . . 15 (𝑓 = (𝐺‘1) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘1):𝑇⟶ℝ))
5553, 54imbi12d 334 . . . . . . . . . . . . . 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 3252 . . . . . . . . . . . . 13 ((𝐺‘1) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘1) ∈ 𝐴) → (𝐺‘1):𝑇⟶ℝ))
5850, 51, 57sylc 65 . . . . . . . . . . . 12 (𝜑 → (𝐺‘1):𝑇⟶ℝ)
5958ffvelrnda 6315 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → ((𝐺‘1)‘𝑡) ∈ ℝ)
6059recnd 10012 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝐺‘1)‘𝑡) ∈ ℂ)
61 fveq2 6148 . . . . . . . . . . . 12 (𝑖 = 1 → (𝐺𝑖) = (𝐺‘1))
6261fveq1d 6150 . . . . . . . . . . 11 (𝑖 = 1 → ((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6362fsum1 14406 . . . . . . . . . 10 ((1 ∈ ℤ ∧ ((𝐺‘1)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6444, 60, 63sylancr 694 . . . . . . . . 9 ((𝜑𝑡𝑇) → Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6543, 64mpteq2da 4703 . . . . . . . 8 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ ((𝐺‘1)‘𝑡)))
6658feqmptd 6206 . . . . . . . 8 (𝜑 → (𝐺‘1) = (𝑡𝑇 ↦ ((𝐺‘1)‘𝑡)))
6765, 66eqtr4d 2658 . . . . . . 7 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) = (𝐺‘1))
6867, 50eqeltrd 2698 . . . . . 6 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
6968adantr 481 . . . . 5 ((𝜑 ∧ 1 ≤ 𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
70 simprl 793 . . . . . . 7 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → 𝜑)
71 simpll 789 . . . . . . 7 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → 𝑦 ∈ ℕ)
72 simprr 795 . . . . . . 7 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑦 + 1) ≤ 𝑀)
73 simp1 1059 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝜑)
74 nnre 10971 . . . . . . . . . . . 12 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
75743ad2ant2 1081 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ∈ ℝ)
76 1red 9999 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℝ)
7775, 76readdcld 10013 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℝ)
7823ad2ant1 1080 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℕ)
7978nnred 10979 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℝ)
8075lep1d 10899 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ≤ (𝑦 + 1))
81 simp3 1061 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ≤ 𝑀)
8275, 77, 79, 80, 81letrd 10138 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦𝑀)
8373, 82jca 554 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝜑𝑦𝑀))
8470, 71, 72, 83syl3anc 1323 . . . . . . . 8 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝜑𝑦𝑀))
85 simplr 791 . . . . . . . 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 1840 . . . . . . . . . . 11 𝑡 𝑦 ∈ ℕ
88 nfv 1840 . . . . . . . . . . 11 𝑡(𝑦 + 1) ≤ 𝑀
8943, 87, 88nf3an 1828 . . . . . . . . . 10 𝑡(𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀)
90 simpl2 1063 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑦 ∈ ℕ)
9190, 46syl6eleq 2708 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑦 ∈ (ℤ‘1))
92 simpll1 1098 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝜑)
93 1zzd 11352 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ∈ ℤ)
942nnzd 11425 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℤ)
95943ad2ant1 1080 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℤ)
9695ad2antrr 761 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑀 ∈ ℤ)
97 elfzelz 12284 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℤ)
9897adantl 482 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ ℤ)
99 elfzle1 12286 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...(𝑦 + 1)) → 1 ≤ 𝑖)
10099adantl 482 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ≤ 𝑖)
10197zred 11426 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℝ)
102101adantl 482 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ ℝ)
10377ad2antrr 761 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → (𝑦 + 1) ∈ ℝ)
10479ad2antrr 761 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑀 ∈ ℝ)
105 elfzle2 12287 . . . . . . . . . . . . . . . 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 1100 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → (𝑦 + 1) ≤ 𝑀)
108102, 103, 104, 106, 107letrd 10138 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖𝑀)
109 elfz4 12277 . . . . . . . . . . . . . 14 (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑀)) → 𝑖 ∈ (1...𝑀))
11093, 96, 98, 100, 108, 109syl32anc 1331 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ (1...𝑀))
111 simplr 791 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑡𝑇)
11245ffvelrnda 6315 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺𝑖) ∈ 𝐴)
1131123adant3 1079 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝐺𝑖) ∈ 𝐴)
114 simp1 1059 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → 𝜑)
115114, 113jca 554 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝜑 ∧ (𝐺𝑖) ∈ 𝐴))
116 eleq1 2686 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝐺𝑖) → (𝑓𝐴 ↔ (𝐺𝑖) ∈ 𝐴))
117116anbi2d 739 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺𝑖) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺𝑖) ∈ 𝐴)))
118 feq1 5983 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝐺𝑖):𝑇⟶ℝ))
119117, 118imbi12d 334 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝐺𝑖) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺𝑖) ∈ 𝐴) → (𝐺𝑖):𝑇⟶ℝ)))
120119, 56vtoclg 3252 . . . . . . . . . . . . . . . 16 ((𝐺𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝐺𝑖) ∈ 𝐴) → (𝐺𝑖):𝑇⟶ℝ))
121113, 115, 120sylc 65 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝐺𝑖):𝑇⟶ℝ)
122 simp3 1061 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → 𝑡𝑇)
123121, 122ffvelrnd 6316 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
124123recnd 10012 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
12592, 110, 111, 124syl3anc 1323 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
126 fveq2 6148 . . . . . . . . . . . . 13 (𝑖 = (𝑦 + 1) → (𝐺𝑖) = (𝐺‘(𝑦 + 1)))
127126fveq1d 6150 . . . . . . . . . . . 12 (𝑖 = (𝑦 + 1) → ((𝐺𝑖)‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
12891, 125, 127fsump1 14415 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡) = (Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
129 simpr 477 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑡𝑇)
130 fzfid 12712 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → (1...𝑦) ∈ Fin)
131 simpll1 1098 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝜑)
132 1zzd 11352 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ∈ ℤ)
13395ad2antrr 761 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℤ)
134 elfzelz 12284 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℤ)
135134adantl 482 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℤ)
136 elfzle1 12286 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑦) → 1 ≤ 𝑖)
137136adantl 482 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ≤ 𝑖)
138134zred 11426 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℝ)
139138adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℝ)
14077adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ∈ ℝ)
14179adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℝ)
14275adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑦 ∈ ℝ)
143 elfzle2 12287 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑦) → 𝑖𝑦)
144143adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑦)
145 letrp1 10809 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑖𝑦) → 𝑖 ≤ (𝑦 + 1))
146139, 142, 144, 145syl3anc 1323 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ (𝑦 + 1))
147 simpl3 1064 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ≤ 𝑀)
148139, 140, 141, 146, 147letrd 10138 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑀)
149148adantlr 750 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑀)
150132, 133, 135, 137, 149, 109syl32anc 1331 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ (1...𝑀))
151 simplr 791 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑡𝑇)
152131, 150, 151, 123syl3anc 1323 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
153130, 152fsumrecl 14398 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) ∈ ℝ)
154 eqid 2621 . . . . . . . . . . . . . 14 (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
155154fvmpt2 6248 . . . . . . . . . . . . 13 ((𝑡𝑇 ∧ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) ∈ ℝ) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
156129, 153, 155syl2anc 692 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
157156oveq1d 6619 . . . . . . . . . . 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 2658 . . . . . . . . . 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 4703 . . . . . . . . 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 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))))
161 1zzd 11352 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℤ)
162 peano2nn 10976 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
163162nnzd 11425 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℤ)
1641633ad2ant2 1081 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℤ)
165162nnge1d 11007 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → 1 ≤ (𝑦 + 1))
1661653ad2ant2 1081 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ≤ (𝑦 + 1))
167 elfz4 12277 . . . . . . . . . . . . . . . . 17 (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ) ∧ (1 ≤ (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑦 + 1) ∈ (1...𝑀))
168161, 95, 164, 166, 81, 167syl32anc 1331 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ (1...𝑀))
16945ffvelrnda 6315 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
17073, 168, 169syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
171 eleq1 2686 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓𝐴 ↔ (𝐺‘(𝑦 + 1)) ∈ 𝐴))
172171anbi2d 739 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺‘(𝑦 + 1)) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴)))
173 feq1 5983 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
174172, 173imbi12d 334 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝐺‘(𝑦 + 1)) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)))
175174, 56vtoclg 3252 . . . . . . . . . . . . . . . 16 ((𝐺‘(𝑦 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
176175anabsi7 859 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
17773, 170, 176syl2anc 692 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
178177ffvelrnda 6315 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ)
179 eqid 2621 . . . . . . . . . . . . . 14 (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
180179fvmpt2 6248 . . . . . . . . . . . . 13 ((𝑡𝑇 ∧ ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ) → ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
181129, 178, 180syl2anc 692 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
182181oveq2d 6620 . . . . . . . . . . 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 4703 . . . . . . . . . 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 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))))
185 simpl1 1062 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → 𝜑)
186 simpr 477 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
187168adantr 481 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑦 + 1) ∈ (1...𝑀))
188176feqmptd 6206 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
189169, 188syldan 487 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
190189, 169eqeltrrd 2699 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴)
191185, 187, 190syl2anc 692 . . . . . . . . . 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 4707 . . . . . . . . . . 11 𝑡(𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
194 nfmpt1 4707 . . . . . . . . . . 11 𝑡(𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
195192, 193, 194stoweidlem8 39529 . . . . . . . . . 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 1323 . . . . . . . . 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 2699 . . . . . . . 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 2698 . . . . . . 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 1326 . . . . . 6 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴)
200199exp31 629 . . . . 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 10982 . . . 4 (𝑛 ∈ ℕ → ((𝜑𝑛𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
20214, 201vtoclg 3252 . . 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 2702 1 (𝜑𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wnf 1705  wcel 1987   class class class wbr 4613  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  cc 9878  cr 9879  1c1 9881   + caddc 9883  cle 10019  cn 10964  cz 11321  cuz 11631  ...cfz 12268  Σcsu 14350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351
This theorem is referenced by:  stoweidlem32  39553
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