Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem20 Structured version   Visualization version   GIF version

Theorem stoweidlem20 42290
 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 11645 . . . . 5 (𝜑𝑀 ∈ ℝ)
43leidd 11198 . . . 4 (𝜑𝑀𝑀)
54ancli 551 . . 3 (𝜑 → (𝜑𝑀𝑀))
6 eleq1 2898 . . . . 5 (𝑛 = 𝑀 → (𝑛 ∈ ℕ ↔ 𝑀 ∈ ℕ))
7 breq1 5060 . . . . . . 7 (𝑛 = 𝑀 → (𝑛𝑀𝑀𝑀))
87anbi2d 630 . . . . . 6 (𝑛 = 𝑀 → ((𝜑𝑛𝑀) ↔ (𝜑𝑀𝑀)))
9 oveq2 7156 . . . . . . . . 9 (𝑛 = 𝑀 → (1...𝑛) = (1...𝑀))
109sumeq1d 15050 . . . . . . . 8 (𝑛 = 𝑀 → Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))
1110mpteq2dv 5153 . . . . . . 7 (𝑛 = 𝑀 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))
1211eleq1d 2895 . . . . . 6 (𝑛 = 𝑀 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
138, 12imbi12d 347 . . . . 5 (𝑛 = 𝑀 → (((𝜑𝑛𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑𝑀𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
146, 13imbi12d 347 . . . 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 5060 . . . . . . 7 (𝑥 = 1 → (𝑥𝑀 ↔ 1 ≤ 𝑀))
1615anbi2d 630 . . . . . 6 (𝑥 = 1 → ((𝜑𝑥𝑀) ↔ (𝜑 ∧ 1 ≤ 𝑀)))
17 oveq2 7156 . . . . . . . . 9 (𝑥 = 1 → (1...𝑥) = (1...1))
1817sumeq1d 15050 . . . . . . . 8 (𝑥 = 1 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡))
1918mpteq2dv 5153 . . . . . . 7 (𝑥 = 1 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)))
2019eleq1d 2895 . . . . . 6 (𝑥 = 1 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
2116, 20imbi12d 347 . . . . 5 (𝑥 = 1 → (((𝜑𝑥𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑 ∧ 1 ≤ 𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
22 breq1 5060 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑀𝑦𝑀))
2322anbi2d 630 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥𝑀) ↔ (𝜑𝑦𝑀)))
24 oveq2 7156 . . . . . . . . 9 (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
2524sumeq1d 15050 . . . . . . . 8 (𝑥 = 𝑦 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
2625mpteq2dv 5153 . . . . . . 7 (𝑥 = 𝑦 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)))
2726eleq1d 2895 . . . . . 6 (𝑥 = 𝑦 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
2823, 27imbi12d 347 . . . . 5 (𝑥 = 𝑦 → (((𝜑𝑥𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴) ↔ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)))
29 breq1 5060 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑥𝑀 ↔ (𝑦 + 1) ≤ 𝑀))
3029anbi2d 630 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝜑𝑥𝑀) ↔ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)))
31 oveq2 7156 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
3231sumeq1d 15050 . . . . . . . 8 (𝑥 = (𝑦 + 1) → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡))
3332mpteq2dv 5153 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)))
3433eleq1d 2895 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴))
3530, 34imbi12d 347 . . . . 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 5060 . . . . . . 7 (𝑥 = 𝑛 → (𝑥𝑀𝑛𝑀))
3736anbi2d 630 . . . . . 6 (𝑥 = 𝑛 → ((𝜑𝑥𝑀) ↔ (𝜑𝑛𝑀)))
38 oveq2 7156 . . . . . . . . 9 (𝑥 = 𝑛 → (1...𝑥) = (1...𝑛))
3938sumeq1d 15050 . . . . . . . 8 (𝑥 = 𝑛 → Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡))
4039mpteq2dv 5153 . . . . . . 7 (𝑥 = 𝑛 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)))
4140eleq1d 2895 . . . . . 6 (𝑥 = 𝑛 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑥)((𝐺𝑖)‘𝑡)) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
4237, 41imbi12d 347 . . . . 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 12004 . . . . . . . . . 10 1 ∈ ℤ
45 stoweidlem20.4 . . . . . . . . . . . . . 14 (𝜑𝐺:(1...𝑀)⟶𝐴)
46 nnuz 12273 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
472, 46eleqtrdi 2921 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (ℤ‘1))
48 eluzfz1 12906 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ‘1) → 1 ∈ (1...𝑀))
4947, 48syl 17 . . . . . . . . . . . . . 14 (𝜑 → 1 ∈ (1...𝑀))
5045, 49ffvelrnd 6845 . . . . . . . . . . . . 13 (𝜑 → (𝐺‘1) ∈ 𝐴)
5150ancli 551 . . . . . . . . . . . . 13 (𝜑 → (𝜑 ∧ (𝐺‘1) ∈ 𝐴))
52 eleq1 2898 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐺‘1) → (𝑓𝐴 ↔ (𝐺‘1) ∈ 𝐴))
5352anbi2d 630 . . . . . . . . . . . . . . 15 (𝑓 = (𝐺‘1) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺‘1) ∈ 𝐴)))
54 feq1 6488 . . . . . . . . . . . . . . 15 (𝑓 = (𝐺‘1) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘1):𝑇⟶ℝ))
5553, 54imbi12d 347 . . . . . . . . . . . . . 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 3566 . . . . . . . . . . . . 13 ((𝐺‘1) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘1) ∈ 𝐴) → (𝐺‘1):𝑇⟶ℝ))
5850, 51, 57sylc 65 . . . . . . . . . . . 12 (𝜑 → (𝐺‘1):𝑇⟶ℝ)
5958ffvelrnda 6844 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → ((𝐺‘1)‘𝑡) ∈ ℝ)
6059recnd 10661 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝐺‘1)‘𝑡) ∈ ℂ)
61 fveq2 6663 . . . . . . . . . . . 12 (𝑖 = 1 → (𝐺𝑖) = (𝐺‘1))
6261fveq1d 6665 . . . . . . . . . . 11 (𝑖 = 1 → ((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6362fsum1 15094 . . . . . . . . . 10 ((1 ∈ ℤ ∧ ((𝐺‘1)‘𝑡) ∈ ℂ) → Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6444, 60, 63sylancr 589 . . . . . . . . 9 ((𝜑𝑡𝑇) → Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡) = ((𝐺‘1)‘𝑡))
6543, 64mpteq2da 5151 . . . . . . . 8 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ ((𝐺‘1)‘𝑡)))
6658feqmptd 6726 . . . . . . . 8 (𝜑 → (𝐺‘1) = (𝑡𝑇 ↦ ((𝐺‘1)‘𝑡)))
6765, 66eqtr4d 2857 . . . . . . 7 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) = (𝐺‘1))
6867, 50eqeltrd 2911 . . . . . 6 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...1)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
6968adantr 483 . . . . 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 1130 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝜑)
74 nnre 11637 . . . . . . . . . . . 12 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
75743ad2ant2 1128 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ∈ ℝ)
76 1red 10634 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℝ)
7775, 76readdcld 10662 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℝ)
7823ad2ant1 1127 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℕ)
7978nnred 11645 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑀 ∈ ℝ)
8075lep1d 11563 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦 ≤ (𝑦 + 1))
81 simp3 1132 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ≤ 𝑀)
8275, 77, 79, 80, 81letrd 10789 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 𝑦𝑀)
8373, 82jca 514 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝜑𝑦𝑀))
8470, 71, 72, 83syl3anc 1365 . . . . . . . 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 1908 . . . . . . . . . . 11 𝑡 𝑦 ∈ ℕ
88 nfv 1908 . . . . . . . . . . 11 𝑡(𝑦 + 1) ≤ 𝑀
8943, 87, 88nf3an 1895 . . . . . . . . . 10 𝑡(𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀)
90 simpl2 1186 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑦 ∈ ℕ)
9190, 46eleqtrdi 2921 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑦 ∈ (ℤ‘1))
92 simpll1 1206 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝜑)
93 1zzd 12005 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ∈ ℤ)
942nnzd 12078 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℤ)
95943ad2ant1 1127 . . . . . . . . . . . . . . 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 12900 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℤ)
9897adantl 484 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ ℤ)
99 elfzle1 12902 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...(𝑦 + 1)) → 1 ≤ 𝑖)
10099adantl 484 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 1 ≤ 𝑖)
10197zred 12079 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ∈ ℝ)
102101adantl 484 . . . . . . . . . . . . . . 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 12903 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (1...(𝑦 + 1)) → 𝑖 ≤ (𝑦 + 1))
106105adantl 484 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ≤ (𝑦 + 1))
107 simpll3 1208 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → (𝑦 + 1) ≤ 𝑀)
108102, 103, 104, 106, 107letrd 10789 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖𝑀)
109 elfz4 12893 . . . . . . . . . . . . . 14 (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑀)) → 𝑖 ∈ (1...𝑀))
11093, 96, 98, 100, 108, 109syl32anc 1372 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑖 ∈ (1...𝑀))
111 simplr 767 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → 𝑡𝑇)
11245ffvelrnda 6844 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺𝑖) ∈ 𝐴)
1131123adant3 1126 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝐺𝑖) ∈ 𝐴)
114 simp1 1130 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → 𝜑)
115114, 113jca 514 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝜑 ∧ (𝐺𝑖) ∈ 𝐴))
116 eleq1 2898 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝐺𝑖) → (𝑓𝐴 ↔ (𝐺𝑖) ∈ 𝐴))
117116anbi2d 630 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺𝑖) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺𝑖) ∈ 𝐴)))
118 feq1 6488 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝐺𝑖):𝑇⟶ℝ))
119117, 118imbi12d 347 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝐺𝑖) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺𝑖) ∈ 𝐴) → (𝐺𝑖):𝑇⟶ℝ)))
120119, 56vtoclg 3566 . . . . . . . . . . . . . . . 16 ((𝐺𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝐺𝑖) ∈ 𝐴) → (𝐺𝑖):𝑇⟶ℝ))
121113, 115, 120sylc 65 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → (𝐺𝑖):𝑇⟶ℝ)
122 simp3 1132 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → 𝑡𝑇)
123121, 122ffvelrnd 6845 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
124123recnd 10661 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡𝑇) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
12592, 110, 111, 124syl3anc 1365 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...(𝑦 + 1))) → ((𝐺𝑖)‘𝑡) ∈ ℂ)
126 fveq2 6663 . . . . . . . . . . . . 13 (𝑖 = (𝑦 + 1) → (𝐺𝑖) = (𝐺‘(𝑦 + 1)))
127126fveq1d 6665 . . . . . . . . . . . 12 (𝑖 = (𝑦 + 1) → ((𝐺𝑖)‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
12891, 125, 127fsump1 15103 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡) = (Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) + ((𝐺‘(𝑦 + 1))‘𝑡)))
129 simpr 487 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → 𝑡𝑇)
130 fzfid 13333 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → (1...𝑦) ∈ Fin)
131 simpll1 1206 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝜑)
132 1zzd 12005 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ∈ ℤ)
13395ad2antrr 724 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℤ)
134 elfzelz 12900 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℤ)
135134adantl 484 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℤ)
136 elfzle1 12902 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑦) → 1 ≤ 𝑖)
137136adantl 484 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 1 ≤ 𝑖)
138134zred 12079 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1...𝑦) → 𝑖 ∈ ℝ)
139138adantl 484 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ ℝ)
14077adantr 483 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ∈ ℝ)
14179adantr 483 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑀 ∈ ℝ)
14275adantr 483 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑦 ∈ ℝ)
143 elfzle2 12903 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑦) → 𝑖𝑦)
144143adantl 484 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑦)
145 letrp1 11476 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑖𝑦) → 𝑖 ≤ (𝑦 + 1))
146139, 142, 144, 145syl3anc 1365 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ≤ (𝑦 + 1))
147 simpl3 1187 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → (𝑦 + 1) ≤ 𝑀)
148139, 140, 141, 146, 147letrd 10789 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑀)
149148adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖𝑀)
150132, 133, 135, 137, 149, 109syl32anc 1372 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑖 ∈ (1...𝑀))
151 simplr 767 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → 𝑡𝑇)
152131, 150, 151, 123syl3anc 1365 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (1...𝑦)) → ((𝐺𝑖)‘𝑡) ∈ ℝ)
153130, 152fsumrecl 15083 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) ∈ ℝ)
154 eqid 2819 . . . . . . . . . . . . . 14 (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
155154fvmpt2 6772 . . . . . . . . . . . . 13 ((𝑡𝑇 ∧ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡) ∈ ℝ) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
156129, 153, 155syl2anc 586 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))‘𝑡) = Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
157156oveq1d 7163 . . . . . . . . . . 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 2857 . . . . . . . . . 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 5151 . . . . . . . . 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 483 . . . . . . . 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 12005 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ∈ ℤ)
162 peano2nn 11642 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
163162nnzd 12078 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℤ)
1641633ad2ant2 1128 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ ℤ)
165162nnge1d 11677 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → 1 ≤ (𝑦 + 1))
1661653ad2ant2 1128 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → 1 ≤ (𝑦 + 1))
167 elfz4 12893 . . . . . . . . . . . . . . . . 17 (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ) ∧ (1 ≤ (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑦 + 1) ∈ (1...𝑀))
168161, 95, 164, 166, 81, 167syl32anc 1372 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝑦 + 1) ∈ (1...𝑀))
16945ffvelrnda 6844 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
17073, 168, 169syl2anc 586 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)) ∈ 𝐴)
171 eleq1 2898 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓𝐴 ↔ (𝐺‘(𝑦 + 1)) ∈ 𝐴))
172171anbi2d 630 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺‘(𝑦 + 1)) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴)))
173 feq1 6488 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐺‘(𝑦 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
174172, 173imbi12d 347 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝐺‘(𝑦 + 1)) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)))
175174, 56vtoclg 3566 . . . . . . . . . . . . . . . 16 ((𝐺‘(𝑦 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ))
176175anabsi7 669 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
17773, 170, 176syl2anc 586 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) → (𝐺‘(𝑦 + 1)):𝑇⟶ℝ)
178177ffvelrnda 6844 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ)
179 eqid 2819 . . . . . . . . . . . . . 14 (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
180179fvmpt2 6772 . . . . . . . . . . . . 13 ((𝑡𝑇 ∧ ((𝐺‘(𝑦 + 1))‘𝑡) ∈ ℝ) → ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
181129, 178, 180syl2anc 586 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))‘𝑡) = ((𝐺‘(𝑦 + 1))‘𝑡))
182181oveq2d 7164 . . . . . . . . . . 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 5151 . . . . . . . . . 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 483 . . . . . . . . 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 1185 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → 𝜑)
186 simpr 487 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)
187168adantr 483 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ ∧ (𝑦 + 1) ≤ 𝑀) ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴) → (𝑦 + 1) ∈ (1...𝑀))
188176feqmptd 6726 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐺‘(𝑦 + 1)) ∈ 𝐴) → (𝐺‘(𝑦 + 1)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
189169, 188syldan 593 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝐺‘(𝑦 + 1)) = (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)))
190189, 169eqeltrrd 2912 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 + 1) ∈ (1...𝑀)) → (𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡)) ∈ 𝐴)
191185, 187, 190syl2anc 586 . . . . . . . . . 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 5155 . . . . . . . . . . 11 𝑡(𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡))
194 nfmpt1 5155 . . . . . . . . . . 11 𝑡(𝑡𝑇 ↦ ((𝐺‘(𝑦 + 1))‘𝑡))
195192, 193, 194stoweidlem8 42278 . . . . . . . . . 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 1365 . . . . . . . . 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 2912 . . . . . . . 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 2911 . . . . . . 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 1367 . . . . . 6 (((𝑦 ∈ ℕ ∧ ((𝜑𝑦𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑦)((𝐺𝑖)‘𝑡)) ∈ 𝐴)) ∧ (𝜑 ∧ (𝑦 + 1) ≤ 𝑀)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...(𝑦 + 1))((𝐺𝑖)‘𝑡)) ∈ 𝐴)
200199exp31 422 . . . . 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 11648 . . . 4 (𝑛 ∈ ℕ → ((𝜑𝑛𝑀) → (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑛)((𝐺𝑖)‘𝑡)) ∈ 𝐴))
20214, 201vtoclg 3566 . . 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, 203eqeltrid 2915 1 (𝜑𝐹𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1081   = wceq 1530  Ⅎwnf 1777   ∈ wcel 2107   class class class wbr 5057   ↦ cmpt 5137  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148  ℂcc 10527  ℝcr 10528  1c1 10530   + caddc 10532   ≤ cle 10668  ℕcn 11630  ℤcz 11973  ℤ≥cuz 12235  ...cfz 12884  Σcsu 15034 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12885  df-fzo 13026  df-seq 13362  df-exp 13422  df-hash 13683  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035 This theorem is referenced by:  stoweidlem32  42302
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