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Theorem stoweidlem17 38710
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 11194 . 2 (𝜑𝑁 ∈ ℕ0)
3 nn0uz 11550 . . . . 5 0 = (ℤ‘0)
42, 3syl6eleq 2693 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
5 eluzfz2 12171 . . . 4 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
64, 5syl 17 . . 3 (𝜑𝑁 ∈ (0...𝑁))
76ancli 571 . 2 (𝜑 → (𝜑𝑁 ∈ (0...𝑁)))
8 eleq1 2671 . . . . 5 (𝑛 = 0 → (𝑛 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
98anbi2d 735 . . . 4 (𝑛 = 0 → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
10 oveq2 6531 . . . . . . 7 (𝑛 = 0 → (0...𝑛) = (0...0))
1110sumeq1d 14221 . . . . . 6 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡)))
1211mpteq2dv 4663 . . . . 5 (𝑛 = 0 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))))
1312eleq1d 2667 . . . 4 (𝑛 = 0 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
149, 13imbi12d 332 . . 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 2671 . . . . 5 (𝑛 = 𝑚 → (𝑛 ∈ (0...𝑁) ↔ 𝑚 ∈ (0...𝑁)))
1615anbi2d 735 . . . 4 (𝑛 = 𝑚 → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑𝑚 ∈ (0...𝑁))))
17 oveq2 6531 . . . . . . 7 (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
1817sumeq1d 14221 . . . . . 6 (𝑛 = 𝑚 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
1918mpteq2dv 4663 . . . . 5 (𝑛 = 𝑚 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))))
2019eleq1d 2667 . . . 4 (𝑛 = 𝑚 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
2116, 20imbi12d 332 . . 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 2671 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑛 ∈ (0...𝑁) ↔ (𝑚 + 1) ∈ (0...𝑁)))
2322anbi2d 735 . . . 4 (𝑛 = (𝑚 + 1) → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))))
24 oveq2 6531 . . . . . . 7 (𝑛 = (𝑚 + 1) → (0...𝑛) = (0...(𝑚 + 1)))
2524sumeq1d 14221 . . . . . 6 (𝑛 = (𝑚 + 1) → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡)))
2625mpteq2dv 4663 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))))
2726eleq1d 2667 . . . 4 (𝑛 = (𝑚 + 1) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
2823, 27imbi12d 332 . . 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 2671 . . . . 5 (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁)))
3029anbi2d 735 . . . 4 (𝑛 = 𝑁 → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑𝑁 ∈ (0...𝑁))))
31 oveq2 6531 . . . . . . 7 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
3231sumeq1d 14221 . . . . . 6 (𝑛 = 𝑁 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡)))
3332mpteq2dv 4663 . . . . 5 (𝑛 = 𝑁 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))))
3433eleq1d 2667 . . . 4 (𝑛 = 𝑁 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
3530, 34imbi12d 332 . . 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 11217 . . . . . . . . 9 0 ∈ ℤ
37 fzsn 12205 . . . . . . . . 9 (0 ∈ ℤ → (0...0) = {0})
3836, 37ax-mp 5 . . . . . . . 8 (0...0) = {0}
3938sumeq1i 14218 . . . . . . 7 Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡))
4039mpteq2i 4659 . . . . . 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 479 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 𝐸 ∈ ℝ)
4443recnd 9920 . . . . . . . . 9 ((𝜑𝑡𝑇) → 𝐸 ∈ ℂ)
45 stoweidlem17.3 . . . . . . . . . . . . 13 (𝜑𝑋:(0...𝑁)⟶𝐴)
46 nnz 11228 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
47 nngt0 10892 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → 0 < 𝑁)
48 0re 9892 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℝ
49 nnre 10870 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
50 ltle 9973 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁))
5148, 49, 50sylancr 693 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → (0 < 𝑁 → 0 ≤ 𝑁))
5247, 51mpd 15 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → 0 ≤ 𝑁)
5346, 52jca 552 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
541, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
5536eluz1i 11523 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘0) ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
5654, 55sylibr 222 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ (ℤ‘0))
57 eluzfz1 12170 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
5856, 57syl 17 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ (0...𝑁))
5945, 58ffvelrnd 6249 . . . . . . . . . . . 12 (𝜑 → (𝑋‘0) ∈ 𝐴)
60 feq1 5921 . . . . . . . . . . . . . 14 (𝑓 = (𝑋‘0) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘0):𝑇⟶ℝ))
6160imbi2d 328 . . . . . . . . . . . . 13 (𝑓 = (𝑋‘0) → ((𝜑𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘0):𝑇⟶ℝ)))
62 stoweidlem17.8 . . . . . . . . . . . . . 14 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
6362expcom 449 . . . . . . . . . . . . 13 (𝑓𝐴 → (𝜑𝑓:𝑇⟶ℝ))
6461, 63vtoclga 3240 . . . . . . . . . . . 12 ((𝑋‘0) ∈ 𝐴 → (𝜑 → (𝑋‘0):𝑇⟶ℝ))
6559, 64mpcom 37 . . . . . . . . . . 11 (𝜑 → (𝑋‘0):𝑇⟶ℝ)
6665fnvinran 37995 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝑋‘0)‘𝑡) ∈ ℝ)
6766recnd 9920 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑋‘0)‘𝑡) ∈ ℂ)
6844, 67mulcld 9912 . . . . . . . 8 ((𝜑𝑡𝑇) → (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ)
69 fveq2 6084 . . . . . . . . . . 11 (𝑖 = 0 → (𝑋𝑖) = (𝑋‘0))
7069fveq1d 6086 . . . . . . . . . 10 (𝑖 = 0 → ((𝑋𝑖)‘𝑡) = ((𝑋‘0)‘𝑡))
7170oveq2d 6539 . . . . . . . . 9 (𝑖 = 0 → (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
7271sumsn 14261 . . . . . . . 8 ((0 ∈ ℤ ∧ (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
7336, 68, 72sylancr 693 . . . . . . 7 ((𝜑𝑡𝑇) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
7441, 73mpteq2da 4661 . . . . . 6 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))))
7540, 74syl5eq 2651 . . . . 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 38695 . . . . 5 (𝜑 → (𝑡𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))) ∈ 𝐴)
7975, 78eqeltrd 2683 . . . 4 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
8079adantr 479 . . 3 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
81 eqidd 2606 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑡𝐸 = 𝐸)
8281cbvmptv 4668 . . . . . . . . . . . . . . 15 (𝑟𝑇𝐸) = (𝑡𝑇𝐸)
8382eqcomi 2614 . . . . . . . . . . . . . 14 (𝑡𝑇𝐸) = (𝑟𝑇𝐸)
8483a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝑡𝑇𝐸) = (𝑟𝑇𝐸))
85 eqidd 2606 . . . . . . . . . . . . 13 (((𝜑𝑡𝑇) ∧ 𝑟 = 𝑡) → 𝐸 = 𝐸)
86 simpr 475 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → 𝑡𝑇)
8784, 85, 86, 43fvmptd 6178 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → ((𝑡𝑇𝐸)‘𝑡) = 𝐸)
8887oveq1d 6538 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
8941, 88mpteq2da 4661 . . . . . . . . . 10 (𝜑 → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
9089adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
9145fnvinran 37995 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑋‘(𝑚 + 1)) ∈ 𝐴)
92 simpl 471 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝜑)
93 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐸𝑥 = 𝐸)
9493mpteq2dv 4663 . . . . . . . . . . . . . . 15 (𝑥 = 𝐸 → (𝑡𝑇𝑥) = (𝑡𝑇𝐸))
9594eleq1d 2667 . . . . . . . . . . . . . 14 (𝑥 = 𝐸 → ((𝑡𝑇𝑥) ∈ 𝐴 ↔ (𝑡𝑇𝐸) ∈ 𝐴))
9695imbi2d 328 . . . . . . . . . . . . 13 (𝑥 = 𝐸 → ((𝜑 → (𝑡𝑇𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)))
9777expcom 449 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → (𝜑 → (𝑡𝑇𝑥) ∈ 𝐴))
9896, 97vtoclga 3240 . . . . . . . . . . . 12 (𝐸 ∈ ℝ → (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴))
9942, 98mpcom 37 . . . . . . . . . . 11 (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)
10099adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇𝐸) ∈ 𝐴)
101 fveq1 6083 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑔𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
102101oveq2d 6539 . . . . . . . . . . . . . . 15 (𝑔 = (𝑋‘(𝑚 + 1)) → (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡)) = (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)))
103102mpteq2dv 4663 . . . . . . . . . . . . . 14 (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))))
104103eleq1d 2667 . . . . . . . . . . . . 13 (𝑔 = (𝑋‘(𝑚 + 1)) → ((𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴))
105104imbi2d 328 . . . . . . . . . . . 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 2674 . . . . . . . . . . . . . . . 16 ((𝑟𝑇𝐸) ∈ 𝐴 ↔ (𝑡𝑇𝐸) ∈ 𝐴)
107 fveq1 6083 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = (𝑟𝑇𝐸) → (𝑓𝑡) = ((𝑟𝑇𝐸)‘𝑡))
10882fveq1i 6085 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑟𝑇𝐸)‘𝑡) = ((𝑡𝑇𝐸)‘𝑡)
109107, 108syl6eq 2655 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑟𝑇𝐸) → (𝑓𝑡) = ((𝑡𝑇𝐸)‘𝑡))
110109oveq1d 6538 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑟𝑇𝐸) → ((𝑓𝑡) · (𝑔𝑡)) = (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡)))
111110mpteq2dv 4663 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑟𝑇𝐸) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))))
112111eleq1d 2667 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑟𝑇𝐸) → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
113112imbi2d 328 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑟𝑇𝐸) → (((𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴)))
114763com12 1260 . . . . . . . . . . . . . . . . . 18 ((𝑓𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
1151143expib 1259 . . . . . . . . . . . . . . . . 17 (𝑓𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴))
116113, 115vtoclga 3240 . . . . . . . . . . . . . . . 16 ((𝑟𝑇𝐸) ∈ 𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
117106, 116sylbir 223 . . . . . . . . . . . . . . 15 ((𝑡𝑇𝐸) ∈ 𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
1181173impib 1253 . . . . . . . . . . . . . 14 (((𝑡𝑇𝐸) ∈ 𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴)
1191183com13 1261 . . . . . . . . . . . . 13 ((𝑔𝐴𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴)
1201193expib 1259 . . . . . . . . . . . 12 (𝑔𝐴 → ((𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
121105, 120vtoclga 3240 . . . . . . . . . . 11 ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴))
1221213impib 1253 . . . . . . . . . 10 (((𝑋‘(𝑚 + 1)) ∈ 𝐴𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
12391, 92, 100, 122syl3anc 1317 . . . . . . . . 9 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
12490, 123eqeltrrd 2684 . . . . . . . 8 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
125124ad2antll 760 . . . . . . 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 799 . . . . . . 7 (((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → 𝜑)
127 simpl 471 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℕ0)
128 simprl 789 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝜑)
1291ad2antrl 759 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ)
130129nnnn0d 11194 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ0)
131 nn0re 11144 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
132131adantr 479 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℝ)
133 peano2nn0 11176 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
134133nn0red 11195 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ)
135134adantr 479 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ∈ ℝ)
1361nnred 10878 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℝ)
137136ad2antrl 759 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℝ)
138 lep1 10707 . . . . . . . . . . . . 13 (𝑚 ∈ ℝ → 𝑚 ≤ (𝑚 + 1))
139127, 131, 1383syl 18 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ≤ (𝑚 + 1))
140 elfzle2 12167 . . . . . . . . . . . . 13 ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ≤ 𝑁)
141140ad2antll 760 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ≤ 𝑁)
142132, 135, 137, 139, 141letrd 10041 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚𝑁)
143 elfz2nn0 12251 . . . . . . . . . . 11 (𝑚 ∈ (0...𝑁) ↔ (𝑚 ∈ ℕ0𝑁 ∈ ℕ0𝑚𝑁))
144127, 130, 142, 143syl3anbrc 1238 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ (0...𝑁))
145127, 128, 144jca32 555 . . . . . . . . 9 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ0 ∧ (𝜑𝑚 ∈ (0...𝑁))))
146145adantl 480 . . . . . . . 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 pm3.31 459 . . . . . . . . 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 479 . . . . . . . 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 6084 . . . . . . . . . . . 12 (𝑟 = 𝑡 → ((𝑋‘(𝑚 + 1))‘𝑟) = ((𝑋‘(𝑚 + 1))‘𝑡))
151150oveq2d 6539 . . . . . . . . . . 11 (𝑟 = 𝑡 → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
152151cbvmptv 4668 . . . . . . . . . 10 (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
153152eleq1i 2674 . . . . . . . . 9 ((𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
154 fveq1 6083 . . . . . . . . . . . . . . 15 (𝑔 = (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔𝑡) = ((𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡))
155152fveq1i 6085 . . . . . . . . . . . . . . 15 ((𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡) = ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)
156154, 155syl6eq 2655 . . . . . . . . . . . . . 14 (𝑔 = (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔𝑡) = ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))
157156oveq2d 6539 . . . . . . . . . . . . 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 4663 . . . . . . . . . . . 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 2667 . . . . . . . . . . 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 328 . . . . . . . . . 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 6084 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑡 → ((𝑋𝑖)‘𝑟) = ((𝑋𝑖)‘𝑡))
162161oveq2d 6539 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑡 → (𝐸 · ((𝑋𝑖)‘𝑟)) = (𝐸 · ((𝑋𝑖)‘𝑡)))
163162sumeq2sdv 14224 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑡 → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
164163cbvmptv 4668 . . . . . . . . . . . . . . 15 (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
165164eleq1i 2674 . . . . . . . . . . . . . 14 ((𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
166 fveq1 6083 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → (𝑓𝑡) = ((𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟)))‘𝑡))
167164fveq1i 6085 . . . . . . . . . . . . . . . . . . . 20 ((𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟)))‘𝑡) = ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡)
168166, 167syl6eq 2655 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → (𝑓𝑡) = ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡))
169168oveq1d 6538 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → ((𝑓𝑡) + (𝑔𝑡)) = (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡)))
170169mpteq2dv 4663 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))))
171170eleq1d 2667 . . . . . . . . . . . . . . . 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 328 . . . . . . . . . . . . . . 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 1260 . . . . . . . . . . . . . . . 16 ((𝑓𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
1751743expib 1259 . . . . . . . . . . . . . . 15 (𝑓𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴))
176172, 175vtoclga 3240 . . . . . . . . . . . . . 14 ((𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) ∈ 𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴))
177165, 176sylbir 223 . . . . . . . . . . . . 13 ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴))
1781773impib 1253 . . . . . . . . . . . 12 (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴)
1791783com13 1261 . . . . . . . . . . 11 ((𝑔𝐴𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴)
1801793expib 1259 . . . . . . . . . 10 (𝑔𝐴 → ((𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴))
181160, 180vtoclga 3240 . . . . . . . . 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 223 . . . . . . . 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 1253 . . . . . . 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 1317 . . . . . 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 1034 . . . . . . . . 9 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ↔ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))))
186185biimpri 216 . . . . . . . 8 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
187186adantl 480 . . . . . . 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 1828 . . . . . . . . . 10 𝑡 𝑚 ∈ ℕ0
189 nfv 1828 . . . . . . . . . 10 𝑡(𝑚 + 1) ∈ (0...𝑁)
190188, 41, 189nf3an 1817 . . . . . . . . 9 𝑡(𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))
191 simpr 475 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 𝑡𝑇)
192 fzfid 12585 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (0...𝑚) ∈ Fin)
193423ad2ant2 1075 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝐸 ∈ ℝ)
194193adantr 479 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 𝐸 ∈ ℝ)
195194adantr 479 . . . . . . . . . . . . . 14 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → 𝐸 ∈ ℝ)
196 fzelp1 12214 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0...𝑚) → 𝑖 ∈ (0...(𝑚 + 1)))
197196anim2i 590 . . . . . . . . . . . . . . . 16 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))))
198 an32 834 . . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . . . 15 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ∧ 𝑡𝑇))
200453ad2ant2 1075 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑋:(0...𝑁)⟶𝐴)
201200adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑋:(0...𝑁)⟶𝐴)
202 elfzuz3 12161 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 + 1) ∈ (0...𝑁) → 𝑁 ∈ (ℤ‘(𝑚 + 1)))
203 fzss2 12203 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘(𝑚 + 1)) → (0...(𝑚 + 1)) ⊆ (0...𝑁))
204202, 203syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 + 1) ∈ (0...𝑁) → (0...(𝑚 + 1)) ⊆ (0...𝑁))
205204sselda 3563 . . . . . . . . . . . . . . . . . . 19 (((𝑚 + 1) ∈ (0...𝑁) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
2062053ad2antl3 1217 . . . . . . . . . . . . . . . . . 18 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
207201, 206ffvelrnd 6249 . . . . . . . . . . . . . . . . 17 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋𝑖) ∈ 𝐴)
208 simpl2 1057 . . . . . . . . . . . . . . . . 17 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝜑)
209 feq1 5921 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑋𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑋𝑖):𝑇⟶ℝ))
210209imbi2d 328 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑋𝑖) → ((𝜑𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋𝑖):𝑇⟶ℝ)))
211210, 63vtoclga 3240 . . . . . . . . . . . . . . . . 17 ((𝑋𝑖) ∈ 𝐴 → (𝜑 → (𝑋𝑖):𝑇⟶ℝ))
212207, 208, 211sylc 62 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋𝑖):𝑇⟶ℝ)
213212fnvinran 37995 . . . . . . . . . . . . . . 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 9922 . . . . . . . . . . . . 13 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℝ)
216192, 215fsumrecl 14254 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℝ)
217 eqid 2605 . . . . . . . . . . . . 13 (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
218217fvmpt2 6181 . . . . . . . . . . . 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 690 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
220219oveq1d 6538 . . . . . . . . . 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 1052 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
222221adantr 479 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
223 feq1 5921 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑋‘(𝑚 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘(𝑚 + 1)):𝑇⟶ℝ))
224223imbi2d 328 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑋‘(𝑚 + 1)) → ((𝜑𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)))
225224, 63vtoclga 3240 . . . . . . . . . . . . . . . 16 ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ))
22691, 92, 225sylc 62 . . . . . . . . . . . . . . 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 6249 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑋‘(𝑚 + 1))‘𝑡) ∈ ℝ)
229194, 228remulcld 9922 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ)
230 eqid 2605 . . . . . . . . . . . . 13 (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
231230fvmpt2 6181 . . . . . . . . . . . 12 ((𝑡𝑇 ∧ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ) → ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
232191, 229, 231syl2anc 690 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
233232oveq2d 6539 . . . . . . . . . 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 12160 . . . . . . . . . . . . . 14 ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ∈ (ℤ‘0))
2352343ad2ant3 1076 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑚 + 1) ∈ (ℤ‘0))
236235adantr 479 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (𝑚 + 1) ∈ (ℤ‘0))
237194adantr 479 . . . . . . . . . . . . 13 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝐸 ∈ ℝ)
238213an32s 841 . . . . . . . . . . . . 13 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → ((𝑋𝑖)‘𝑡) ∈ ℝ)
239 remulcl 9873 . . . . . . . . . . . . . 14 ((𝐸 ∈ ℝ ∧ ((𝑋𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℝ)
240239recnd 9920 . . . . . . . . . . . . 13 ((𝐸 ∈ ℝ ∧ ((𝑋𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℂ)
241237, 238, 240syl2anc 690 . . . . . . . . . . . 12 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℂ)
242 fveq2 6084 . . . . . . . . . . . . . 14 (𝑖 = (𝑚 + 1) → (𝑋𝑖) = (𝑋‘(𝑚 + 1)))
243242fveq1d 6086 . . . . . . . . . . . . 13 (𝑖 = (𝑚 + 1) → ((𝑋𝑖)‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
244243oveq2d 6539 . . . . . . . . . . . 12 (𝑖 = (𝑚 + 1) → (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
245236, 241, 244fsumm1 14266 . . . . . . . . . . 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 11145 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
2472463ad2ant1 1074 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑚 ∈ ℂ)
248247adantr 479 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 𝑚 ∈ ℂ)
249 1cnd 9908 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 1 ∈ ℂ)
250248, 249pncand 10240 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑚 + 1) − 1) = 𝑚)
251250oveq2d 6539 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (0...((𝑚 + 1) − 1)) = (0...𝑚))
252251sumeq1d 14221 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → Σ𝑖 ∈ (0...((𝑚 + 1) − 1))(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
253252oveq1d 6538 . . . . . . . . . . 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 2639 . . . . . . . . . 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 2650 . . . . . . . . 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 4661 . . . . . . . 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 2667 . . . . . . 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 245 . . . . 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 628 . . . 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 106 . . 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 11300 . 2 (𝑁 ∈ ℕ0 → ((𝜑𝑁 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
2632, 7, 262sylc 62 1 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wnf 1698  wcel 1975  wss 3535  {csn 4120   class class class wbr 4573  cmpt 4633  wf 5782  cfv 5786  (class class class)co 6523  cc 9786  cr 9787  0cc0 9788  1c1 9789   + caddc 9791   · cmul 9793   < clt 9926  cle 9927  cmin 10113  cn 10863  0cn0 11135  cz 11206  cuz 11515  ...cfz 12148  Σcsu 14206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-sup 8204  df-oi 8271  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-3 10923  df-n0 11136  df-z 11207  df-uz 11516  df-rp 11661  df-fz 12149  df-fzo 12286  df-seq 12615  df-exp 12674  df-hash 12931  df-cj 13629  df-re 13630  df-im 13631  df-sqrt 13765  df-abs 13766  df-clim 14009  df-sum 14207
This theorem is referenced by:  stoweidlem60  38753
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