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Theorem stoweidlem17 39997
Description: This lemma proves that the function 𝑔 (as defined in [BrosowskiDeutsh] p. 91, at the end of page 91) belongs to the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem17.1 𝑡𝜑
stoweidlem17.2 (𝜑𝑁 ∈ ℕ)
stoweidlem17.3 (𝜑𝑋:(0...𝑁)⟶𝐴)
stoweidlem17.4 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stoweidlem17.5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem17.6 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
stoweidlem17.7 (𝜑𝐸 ∈ ℝ)
stoweidlem17.8 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
Assertion
Ref Expression
stoweidlem17 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
Distinct variable groups:   𝑓,𝑔,𝑖,𝑡,𝐸   𝐴,𝑓,𝑔   𝑇,𝑓,𝑔,𝑖,𝑡   𝑓,𝑋,𝑔,𝑖,𝑡   𝜑,𝑓,𝑔,𝑖   𝑖,𝑁,𝑡   𝑥,𝑡,𝐸   𝑥,𝐴   𝑥,𝑇   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑡)   𝐴(𝑡,𝑖)   𝑁(𝑥,𝑓,𝑔)   𝑋(𝑥)

Proof of Theorem stoweidlem17
Dummy variables 𝑚 𝑟 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 stoweidlem17.2 . . 3 (𝜑𝑁 ∈ ℕ)
21nnnn0d 11336 . 2 (𝜑𝑁 ∈ ℕ0)
3 nn0uz 11707 . . . . 5 0 = (ℤ‘0)
42, 3syl6eleq 2709 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
5 eluzfz2 12334 . . . 4 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
64, 5syl 17 . . 3 (𝜑𝑁 ∈ (0...𝑁))
76ancli 573 . 2 (𝜑 → (𝜑𝑁 ∈ (0...𝑁)))
8 eleq1 2687 . . . . 5 (𝑛 = 0 → (𝑛 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
98anbi2d 739 . . . 4 (𝑛 = 0 → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
10 oveq2 6643 . . . . . . 7 (𝑛 = 0 → (0...𝑛) = (0...0))
1110sumeq1d 14412 . . . . . 6 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡)))
1211mpteq2dv 4736 . . . . 5 (𝑛 = 0 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))))
1312eleq1d 2684 . . . 4 (𝑛 = 0 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
149, 13imbi12d 334 . . 3 (𝑛 = 0 → (((𝜑𝑛 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)))
15 eleq1 2687 . . . . 5 (𝑛 = 𝑚 → (𝑛 ∈ (0...𝑁) ↔ 𝑚 ∈ (0...𝑁)))
1615anbi2d 739 . . . 4 (𝑛 = 𝑚 → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑𝑚 ∈ (0...𝑁))))
17 oveq2 6643 . . . . . . 7 (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
1817sumeq1d 14412 . . . . . 6 (𝑛 = 𝑚 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
1918mpteq2dv 4736 . . . . 5 (𝑛 = 𝑚 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))))
2019eleq1d 2684 . . . 4 (𝑛 = 𝑚 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
2116, 20imbi12d 334 . . 3 (𝑛 = 𝑚 → (((𝜑𝑛 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)))
22 eleq1 2687 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑛 ∈ (0...𝑁) ↔ (𝑚 + 1) ∈ (0...𝑁)))
2322anbi2d 739 . . . 4 (𝑛 = (𝑚 + 1) → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))))
24 oveq2 6643 . . . . . . 7 (𝑛 = (𝑚 + 1) → (0...𝑛) = (0...(𝑚 + 1)))
2524sumeq1d 14412 . . . . . 6 (𝑛 = (𝑚 + 1) → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡)))
2625mpteq2dv 4736 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))))
2726eleq1d 2684 . . . 4 (𝑛 = (𝑚 + 1) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
2823, 27imbi12d 334 . . 3 (𝑛 = (𝑚 + 1) → (((𝜑𝑛 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)))
29 eleq1 2687 . . . . 5 (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁)))
3029anbi2d 739 . . . 4 (𝑛 = 𝑁 → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑𝑁 ∈ (0...𝑁))))
31 oveq2 6643 . . . . . . 7 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
3231sumeq1d 14412 . . . . . 6 (𝑛 = 𝑁 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡)))
3332mpteq2dv 4736 . . . . 5 (𝑛 = 𝑁 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))))
3433eleq1d 2684 . . . 4 (𝑛 = 𝑁 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
3530, 34imbi12d 334 . . 3 (𝑛 = 𝑁 → (((𝜑𝑛 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑𝑁 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)))
36 0z 11373 . . . . . . . . 9 0 ∈ ℤ
37 fzsn 12368 . . . . . . . . 9 (0 ∈ ℤ → (0...0) = {0})
3836, 37ax-mp 5 . . . . . . . 8 (0...0) = {0}
3938sumeq1i 14409 . . . . . . 7 Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡))
4039mpteq2i 4732 . . . . . 6 (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡)))
41 stoweidlem17.1 . . . . . . 7 𝑡𝜑
42 stoweidlem17.7 . . . . . . . . . . 11 (𝜑𝐸 ∈ ℝ)
4342adantr 481 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 𝐸 ∈ ℝ)
4443recnd 10053 . . . . . . . . 9 ((𝜑𝑡𝑇) → 𝐸 ∈ ℂ)
45 stoweidlem17.3 . . . . . . . . . . . . 13 (𝜑𝑋:(0...𝑁)⟶𝐴)
46 nnz 11384 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
47 nngt0 11034 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → 0 < 𝑁)
48 0re 10025 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℝ
49 nnre 11012 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
50 ltle 10111 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁))
5148, 49, 50sylancr 694 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → (0 < 𝑁 → 0 ≤ 𝑁))
5247, 51mpd 15 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → 0 ≤ 𝑁)
5346, 52jca 554 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
541, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
5536eluz1i 11680 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘0) ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
5654, 55sylibr 224 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ (ℤ‘0))
57 eluzfz1 12333 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
5856, 57syl 17 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ (0...𝑁))
5945, 58ffvelrnd 6346 . . . . . . . . . . . 12 (𝜑 → (𝑋‘0) ∈ 𝐴)
60 feq1 6013 . . . . . . . . . . . . . 14 (𝑓 = (𝑋‘0) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘0):𝑇⟶ℝ))
6160imbi2d 330 . . . . . . . . . . . . 13 (𝑓 = (𝑋‘0) → ((𝜑𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘0):𝑇⟶ℝ)))
62 stoweidlem17.8 . . . . . . . . . . . . . 14 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
6362expcom 451 . . . . . . . . . . . . 13 (𝑓𝐴 → (𝜑𝑓:𝑇⟶ℝ))
6461, 63vtoclga 3267 . . . . . . . . . . . 12 ((𝑋‘0) ∈ 𝐴 → (𝜑 → (𝑋‘0):𝑇⟶ℝ))
6559, 64mpcom 38 . . . . . . . . . . 11 (𝜑 → (𝑋‘0):𝑇⟶ℝ)
6665ffvelrnda 6345 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝑋‘0)‘𝑡) ∈ ℝ)
6766recnd 10053 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑋‘0)‘𝑡) ∈ ℂ)
6844, 67mulcld 10045 . . . . . . . 8 ((𝜑𝑡𝑇) → (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ)
69 fveq2 6178 . . . . . . . . . . 11 (𝑖 = 0 → (𝑋𝑖) = (𝑋‘0))
7069fveq1d 6180 . . . . . . . . . 10 (𝑖 = 0 → ((𝑋𝑖)‘𝑡) = ((𝑋‘0)‘𝑡))
7170oveq2d 6651 . . . . . . . . 9 (𝑖 = 0 → (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
7271sumsn 14456 . . . . . . . 8 ((0 ∈ ℤ ∧ (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
7336, 68, 72sylancr 694 . . . . . . 7 ((𝜑𝑡𝑇) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
7441, 73mpteq2da 4734 . . . . . 6 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))))
7540, 74syl5eq 2666 . . . . 5 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))))
76 stoweidlem17.5 . . . . . 6 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
77 stoweidlem17.6 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
7841, 76, 77, 62, 42, 59stoweidlem2 39982 . . . . 5 (𝜑 → (𝑡𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))) ∈ 𝐴)
7975, 78eqeltrd 2699 . . . 4 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
8079adantr 481 . . 3 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
81 eqidd 2621 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑡𝐸 = 𝐸)
8281cbvmptv 4741 . . . . . . . . . . . . . . 15 (𝑟𝑇𝐸) = (𝑡𝑇𝐸)
8382eqcomi 2629 . . . . . . . . . . . . . 14 (𝑡𝑇𝐸) = (𝑟𝑇𝐸)
8483a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝑡𝑇𝐸) = (𝑟𝑇𝐸))
85 eqidd 2621 . . . . . . . . . . . . 13 (((𝜑𝑡𝑇) ∧ 𝑟 = 𝑡) → 𝐸 = 𝐸)
86 simpr 477 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → 𝑡𝑇)
8784, 85, 86, 43fvmptd 6275 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → ((𝑡𝑇𝐸)‘𝑡) = 𝐸)
8887oveq1d 6650 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
8941, 88mpteq2da 4734 . . . . . . . . . 10 (𝜑 → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
9089adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
9145ffvelrnda 6345 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑋‘(𝑚 + 1)) ∈ 𝐴)
92 simpl 473 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝜑)
93 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐸𝑥 = 𝐸)
9493mpteq2dv 4736 . . . . . . . . . . . . . . 15 (𝑥 = 𝐸 → (𝑡𝑇𝑥) = (𝑡𝑇𝐸))
9594eleq1d 2684 . . . . . . . . . . . . . 14 (𝑥 = 𝐸 → ((𝑡𝑇𝑥) ∈ 𝐴 ↔ (𝑡𝑇𝐸) ∈ 𝐴))
9695imbi2d 330 . . . . . . . . . . . . 13 (𝑥 = 𝐸 → ((𝜑 → (𝑡𝑇𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)))
9777expcom 451 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → (𝜑 → (𝑡𝑇𝑥) ∈ 𝐴))
9896, 97vtoclga 3267 . . . . . . . . . . . 12 (𝐸 ∈ ℝ → (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴))
9942, 98mpcom 38 . . . . . . . . . . 11 (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)
10099adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇𝐸) ∈ 𝐴)
101 fveq1 6177 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑔𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
102101oveq2d 6651 . . . . . . . . . . . . . . 15 (𝑔 = (𝑋‘(𝑚 + 1)) → (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡)) = (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)))
103102mpteq2dv 4736 . . . . . . . . . . . . . 14 (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))))
104103eleq1d 2684 . . . . . . . . . . . . 13 (𝑔 = (𝑋‘(𝑚 + 1)) → ((𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴))
105104imbi2d 330 . . . . . . . . . . . 12 (𝑔 = (𝑋‘(𝑚 + 1)) → (((𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)))
10682eleq1i 2690 . . . . . . . . . . . . . . . 16 ((𝑟𝑇𝐸) ∈ 𝐴 ↔ (𝑡𝑇𝐸) ∈ 𝐴)
107 fveq1 6177 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = (𝑟𝑇𝐸) → (𝑓𝑡) = ((𝑟𝑇𝐸)‘𝑡))
10882fveq1i 6179 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑟𝑇𝐸)‘𝑡) = ((𝑡𝑇𝐸)‘𝑡)
109107, 108syl6eq 2670 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑟𝑇𝐸) → (𝑓𝑡) = ((𝑡𝑇𝐸)‘𝑡))
110109oveq1d 6650 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑟𝑇𝐸) → ((𝑓𝑡) · (𝑔𝑡)) = (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡)))
111110mpteq2dv 4736 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑟𝑇𝐸) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))))
112111eleq1d 2684 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑟𝑇𝐸) → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
113112imbi2d 330 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑟𝑇𝐸) → (((𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴)))
114763com12 1267 . . . . . . . . . . . . . . . . . 18 ((𝑓𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
1151143expib 1266 . . . . . . . . . . . . . . . . 17 (𝑓𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴))
116113, 115vtoclga 3267 . . . . . . . . . . . . . . . 16 ((𝑟𝑇𝐸) ∈ 𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
117106, 116sylbir 225 . . . . . . . . . . . . . . 15 ((𝑡𝑇𝐸) ∈ 𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
1181173impib 1260 . . . . . . . . . . . . . 14 (((𝑡𝑇𝐸) ∈ 𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴)
1191183com13 1268 . . . . . . . . . . . . 13 ((𝑔𝐴𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴)
1201193expib 1266 . . . . . . . . . . . 12 (𝑔𝐴 → ((𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
121105, 120vtoclga 3267 . . . . . . . . . . 11 ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴))
1221213impib 1260 . . . . . . . . . 10 (((𝑋‘(𝑚 + 1)) ∈ 𝐴𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
12391, 92, 100, 122syl3anc 1324 . . . . . . . . 9 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
12490, 123eqeltrrd 2700 . . . . . . . 8 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
125124ad2antll 764 . . . . . . 7 (((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
126 simprrl 803 . . . . . . 7 (((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → 𝜑)
127 simpl 473 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℕ0)
128 simprl 793 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝜑)
1291ad2antrl 763 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ)
130129nnnn0d 11336 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ0)
131 nn0re 11286 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
132131adantr 481 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℝ)
133 peano2nn0 11318 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
134133nn0red 11337 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ)
135134adantr 481 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ∈ ℝ)
1361nnred 11020 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℝ)
137136ad2antrl 763 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℝ)
138 lep1 10847 . . . . . . . . . . . . 13 (𝑚 ∈ ℝ → 𝑚 ≤ (𝑚 + 1))
139127, 131, 1383syl 18 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ≤ (𝑚 + 1))
140 elfzle2 12330 . . . . . . . . . . . . 13 ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ≤ 𝑁)
141140ad2antll 764 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ≤ 𝑁)
142132, 135, 137, 139, 141letrd 10179 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚𝑁)
143 elfz2nn0 12415 . . . . . . . . . . 11 (𝑚 ∈ (0...𝑁) ↔ (𝑚 ∈ ℕ0𝑁 ∈ ℕ0𝑚𝑁))
144127, 130, 142, 143syl3anbrc 1244 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ (0...𝑁))
145127, 128, 144jca32 557 . . . . . . . . 9 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ0 ∧ (𝜑𝑚 ∈ (0...𝑁))))
146145adantl 482 . . . . . . . 8 (((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑚 ∈ ℕ0 ∧ (𝜑𝑚 ∈ (0...𝑁))))
147 pm3.31 461 . . . . . . . . 9 ((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) → ((𝑚 ∈ ℕ0 ∧ (𝜑𝑚 ∈ (0...𝑁))) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
148147adantr 481 . . . . . . . 8 (((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → ((𝑚 ∈ ℕ0 ∧ (𝜑𝑚 ∈ (0...𝑁))) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
149146, 148mpd 15 . . . . . . 7 (((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
150 fveq2 6178 . . . . . . . . . . . 12 (𝑟 = 𝑡 → ((𝑋‘(𝑚 + 1))‘𝑟) = ((𝑋‘(𝑚 + 1))‘𝑡))
151150oveq2d 6651 . . . . . . . . . . 11 (𝑟 = 𝑡 → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
152151cbvmptv 4741 . . . . . . . . . 10 (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
153152eleq1i 2690 . . . . . . . . 9 ((𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
154 fveq1 6177 . . . . . . . . . . . . . . 15 (𝑔 = (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔𝑡) = ((𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡))
155152fveq1i 6179 . . . . . . . . . . . . . . 15 ((𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡) = ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)
156154, 155syl6eq 2670 . . . . . . . . . . . . . 14 (𝑔 = (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔𝑡) = ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))
157156oveq2d 6651 . . . . . . . . . . . . 13 (𝑔 = (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡)) = (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)))
158157mpteq2dv 4736 . . . . . . . . . . . 12 (𝑔 = (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))))
159158eleq1d 2684 . . . . . . . . . . 11 (𝑔 = (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → ((𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴))
160159imbi2d 330 . . . . . . . . . 10 (𝑔 = (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (((𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴)))
161 fveq2 6178 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑡 → ((𝑋𝑖)‘𝑟) = ((𝑋𝑖)‘𝑡))
162161oveq2d 6651 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑡 → (𝐸 · ((𝑋𝑖)‘𝑟)) = (𝐸 · ((𝑋𝑖)‘𝑡)))
163162sumeq2sdv 14416 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑡 → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
164163cbvmptv 4741 . . . . . . . . . . . . . . 15 (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
165164eleq1i 2690 . . . . . . . . . . . . . 14 ((𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
166 fveq1 6177 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → (𝑓𝑡) = ((𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟)))‘𝑡))
167164fveq1i 6179 . . . . . . . . . . . . . . . . . . . 20 ((𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟)))‘𝑡) = ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡)
168166, 167syl6eq 2670 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → (𝑓𝑡) = ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡))
169168oveq1d 6650 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → ((𝑓𝑡) + (𝑔𝑡)) = (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡)))
170169mpteq2dv 4736 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))))
171170eleq1d 2684 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → ((𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴))
172171imbi2d 330 . . . . . . . . . . . . . . 15 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → (((𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴)))
173 stoweidlem17.4 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
1741733com12 1267 . . . . . . . . . . . . . . . 16 ((𝑓𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
1751743expib 1266 . . . . . . . . . . . . . . 15 (𝑓𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴))
176172, 175vtoclga 3267 . . . . . . . . . . . . . 14 ((𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) ∈ 𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴))
177165, 176sylbir 225 . . . . . . . . . . . . 13 ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴))
1781773impib 1260 . . . . . . . . . . . 12 (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴)
1791783com13 1268 . . . . . . . . . . 11 ((𝑔𝐴𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴)
1801793expib 1266 . . . . . . . . . 10 (𝑔𝐴 → ((𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴))
181160, 180vtoclga 3267 . . . . . . . . 9 ((𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) ∈ 𝐴 → ((𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴))
182153, 181sylbir 225 . . . . . . . 8 ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴 → ((𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴))
1831823impib 1260 . . . . . . 7 (((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴)
184125, 126, 149, 183syl3anc 1324 . . . . . 6 (((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴)
185 3anass 1040 . . . . . . . . 9 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ↔ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))))
186185biimpri 218 . . . . . . . 8 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
187186adantl 482 . . . . . . 7 (((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
188 nfv 1841 . . . . . . . . . 10 𝑡 𝑚 ∈ ℕ0
189 nfv 1841 . . . . . . . . . 10 𝑡(𝑚 + 1) ∈ (0...𝑁)
190188, 41, 189nf3an 1829 . . . . . . . . 9 𝑡(𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))
191 simpr 477 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 𝑡𝑇)
192 fzfid 12755 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (0...𝑚) ∈ Fin)
193423ad2ant2 1081 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝐸 ∈ ℝ)
194193adantr 481 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 𝐸 ∈ ℝ)
195194adantr 481 . . . . . . . . . . . . . 14 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → 𝐸 ∈ ℝ)
196 fzelp1 12378 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0...𝑚) → 𝑖 ∈ (0...(𝑚 + 1)))
197196anim2i 592 . . . . . . . . . . . . . . . 16 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))))
198 an32 838 . . . . . . . . . . . . . . . 16 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ↔ (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ∧ 𝑡𝑇))
199197, 198sylib 208 . . . . . . . . . . . . . . 15 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ∧ 𝑡𝑇))
200453ad2ant2 1081 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑋:(0...𝑁)⟶𝐴)
201200adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑋:(0...𝑁)⟶𝐴)
202 elfzuz3 12324 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 + 1) ∈ (0...𝑁) → 𝑁 ∈ (ℤ‘(𝑚 + 1)))
203 fzss2 12366 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘(𝑚 + 1)) → (0...(𝑚 + 1)) ⊆ (0...𝑁))
204202, 203syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 + 1) ∈ (0...𝑁) → (0...(𝑚 + 1)) ⊆ (0...𝑁))
205204sselda 3595 . . . . . . . . . . . . . . . . . . 19 (((𝑚 + 1) ∈ (0...𝑁) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
2062053ad2antl3 1223 . . . . . . . . . . . . . . . . . 18 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
207201, 206ffvelrnd 6346 . . . . . . . . . . . . . . . . 17 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋𝑖) ∈ 𝐴)
208 simpl2 1063 . . . . . . . . . . . . . . . . 17 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝜑)
209 feq1 6013 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑋𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑋𝑖):𝑇⟶ℝ))
210209imbi2d 330 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑋𝑖) → ((𝜑𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋𝑖):𝑇⟶ℝ)))
211210, 63vtoclga 3267 . . . . . . . . . . . . . . . . 17 ((𝑋𝑖) ∈ 𝐴 → (𝜑 → (𝑋𝑖):𝑇⟶ℝ))
212207, 208, 211sylc 65 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋𝑖):𝑇⟶ℝ)
213212ffvelrnda 6345 . . . . . . . . . . . . . . 15 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ∧ 𝑡𝑇) → ((𝑋𝑖)‘𝑡) ∈ ℝ)
214199, 213syl 17 . . . . . . . . . . . . . 14 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑋𝑖)‘𝑡) ∈ ℝ)
215195, 214remulcld 10055 . . . . . . . . . . . . 13 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℝ)
216192, 215fsumrecl 14446 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℝ)
217 eqid 2620 . . . . . . . . . . . . 13 (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
218217fvmpt2 6278 . . . . . . . . . . . 12 ((𝑡𝑇 ∧ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℝ) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
219191, 216, 218syl2anc 692 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
220219oveq1d 6650 . . . . . . . . . 10 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
221 3simpc 1058 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
222221adantr 481 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
223 feq1 6013 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑋‘(𝑚 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘(𝑚 + 1)):𝑇⟶ℝ))
224223imbi2d 330 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑋‘(𝑚 + 1)) → ((𝜑𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)))
225224, 63vtoclga 3267 . . . . . . . . . . . . . . . 16 ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ))
22691, 92, 225sylc 65 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)
227222, 226syl 17 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)
228227, 191ffvelrnd 6346 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑋‘(𝑚 + 1))‘𝑡) ∈ ℝ)
229194, 228remulcld 10055 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ)
230 eqid 2620 . . . . . . . . . . . . 13 (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
231230fvmpt2 6278 . . . . . . . . . . . 12 ((𝑡𝑇 ∧ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ) → ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
232191, 229, 231syl2anc 692 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
233232oveq2d 6651 . . . . . . . . . 10 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)) = (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
234 elfzuz 12323 . . . . . . . . . . . . . 14 ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ∈ (ℤ‘0))
2352343ad2ant3 1082 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑚 + 1) ∈ (ℤ‘0))
236235adantr 481 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (𝑚 + 1) ∈ (ℤ‘0))
237194adantr 481 . . . . . . . . . . . . 13 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝐸 ∈ ℝ)
238213an32s 845 . . . . . . . . . . . . 13 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → ((𝑋𝑖)‘𝑡) ∈ ℝ)
239 remulcl 10006 . . . . . . . . . . . . . 14 ((𝐸 ∈ ℝ ∧ ((𝑋𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℝ)
240239recnd 10053 . . . . . . . . . . . . 13 ((𝐸 ∈ ℝ ∧ ((𝑋𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℂ)
241237, 238, 240syl2anc 692 . . . . . . . . . . . 12 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℂ)
242 fveq2 6178 . . . . . . . . . . . . . 14 (𝑖 = (𝑚 + 1) → (𝑋𝑖) = (𝑋‘(𝑚 + 1)))
243242fveq1d 6180 . . . . . . . . . . . . 13 (𝑖 = (𝑚 + 1) → ((𝑋𝑖)‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
244243oveq2d 6651 . . . . . . . . . . . 12 (𝑖 = (𝑚 + 1) → (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
245236, 241, 244fsumm1 14461 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡)) = (Σ𝑖 ∈ (0...((𝑚 + 1) − 1))(𝐸 · ((𝑋𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
246 nn0cn 11287 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
2472463ad2ant1 1080 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑚 ∈ ℂ)
248247adantr 481 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 𝑚 ∈ ℂ)
249 1cnd 10041 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 1 ∈ ℂ)
250248, 249pncand 10378 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑚 + 1) − 1) = 𝑚)
251250oveq2d 6651 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (0...((𝑚 + 1) − 1)) = (0...𝑚))
252251sumeq1d 14412 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → Σ𝑖 ∈ (0...((𝑚 + 1) − 1))(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
253252oveq1d 6650 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (Σ𝑖 ∈ (0...((𝑚 + 1) − 1))(𝐸 · ((𝑋𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
254245, 253eqtrd 2654 . . . . . . . . . 10 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡)) = (Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
255220, 233, 2543eqtr4rd 2665 . . . . . . . . 9 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡)) = (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)))
256190, 255mpteq2da 4734 . . . . . . . 8 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))))
257256eleq1d 2684 . . . . . . 7 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴))
258187, 257syl 17 . . . . . 6 (((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴))
259184, 258mpbird 247 . . . . 5 (((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
260259exp32 630 . . . 4 ((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) → (𝑚 ∈ ℕ0 → ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)))
261260pm2.86i 109 . . 3 (𝑚 ∈ ℕ0 → (((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)))
26214, 21, 28, 35, 80, 261nn0ind 11457 . 2 (𝑁 ∈ ℕ0 → ((𝜑𝑁 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
2632, 7, 262sylc 65 1 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wnf 1706  wcel 1988  wss 3567  {csn 4168   class class class wbr 4644  cmpt 4720  wf 5872  cfv 5876  (class class class)co 6635  cc 9919  cr 9920  0cc0 9921  1c1 9922   + caddc 9924   · cmul 9926   < clt 10059  cle 10060  cmin 10251  cn 11005  0cn0 11277  cz 11362  cuz 11672  ...cfz 12311  Σcsu 14397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-oi 8400  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-rp 11818  df-fz 12312  df-fzo 12450  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-sum 14398
This theorem is referenced by:  stoweidlem60  40040
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