Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem17 Structured version   Visualization version   GIF version

Theorem stoweidlem17 40706
 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 11514 . 2 (𝜑𝑁 ∈ ℕ0)
3 nn0uz 11886 . . . . 5 0 = (ℤ‘0)
42, 3syl6eleq 2837 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
5 eluzfz2 12513 . . . 4 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
64, 5syl 17 . . 3 (𝜑𝑁 ∈ (0...𝑁))
76ancli 575 . 2 (𝜑 → (𝜑𝑁 ∈ (0...𝑁)))
8 eleq1 2815 . . . . 5 (𝑛 = 0 → (𝑛 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
98anbi2d 742 . . . 4 (𝑛 = 0 → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
10 oveq2 6809 . . . . . . 7 (𝑛 = 0 → (0...𝑛) = (0...0))
1110sumeq1d 14601 . . . . . 6 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡)))
1211mpteq2dv 4885 . . . . 5 (𝑛 = 0 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))))
1312eleq1d 2812 . . . 4 (𝑛 = 0 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
149, 13imbi12d 333 . . 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 2815 . . . . 5 (𝑛 = 𝑚 → (𝑛 ∈ (0...𝑁) ↔ 𝑚 ∈ (0...𝑁)))
1615anbi2d 742 . . . 4 (𝑛 = 𝑚 → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑𝑚 ∈ (0...𝑁))))
17 oveq2 6809 . . . . . . 7 (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
1817sumeq1d 14601 . . . . . 6 (𝑛 = 𝑚 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
1918mpteq2dv 4885 . . . . 5 (𝑛 = 𝑚 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))))
2019eleq1d 2812 . . . 4 (𝑛 = 𝑚 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
2116, 20imbi12d 333 . . 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 2815 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑛 ∈ (0...𝑁) ↔ (𝑚 + 1) ∈ (0...𝑁)))
2322anbi2d 742 . . . 4 (𝑛 = (𝑚 + 1) → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))))
24 oveq2 6809 . . . . . . 7 (𝑛 = (𝑚 + 1) → (0...𝑛) = (0...(𝑚 + 1)))
2524sumeq1d 14601 . . . . . 6 (𝑛 = (𝑚 + 1) → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡)))
2625mpteq2dv 4885 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))))
2726eleq1d 2812 . . . 4 (𝑛 = (𝑚 + 1) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
2823, 27imbi12d 333 . . 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 2815 . . . . 5 (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁)))
3029anbi2d 742 . . . 4 (𝑛 = 𝑁 → ((𝜑𝑛 ∈ (0...𝑁)) ↔ (𝜑𝑁 ∈ (0...𝑁))))
31 oveq2 6809 . . . . . . 7 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
3231sumeq1d 14601 . . . . . 6 (𝑛 = 𝑁 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡)))
3332mpteq2dv 4885 . . . . 5 (𝑛 = 𝑁 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))))
3433eleq1d 2812 . . . 4 (𝑛 = 𝑁 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴))
3530, 34imbi12d 333 . . 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 11551 . . . . . . . . 9 0 ∈ ℤ
37 fzsn 12547 . . . . . . . . 9 (0 ∈ ℤ → (0...0) = {0})
3836, 37ax-mp 5 . . . . . . . 8 (0...0) = {0}
3938sumeq1i 14598 . . . . . . 7 Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡))
4039mpteq2i 4881 . . . . . 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 472 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 𝐸 ∈ ℝ)
4443recnd 10231 . . . . . . . . 9 ((𝜑𝑡𝑇) → 𝐸 ∈ ℂ)
45 stoweidlem17.3 . . . . . . . . . . . . 13 (𝜑𝑋:(0...𝑁)⟶𝐴)
46 nnz 11562 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
47 nngt0 11212 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → 0 < 𝑁)
48 0re 10203 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℝ
49 nnre 11190 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
50 ltle 10289 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁))
5148, 49, 50sylancr 698 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → (0 < 𝑁 → 0 ≤ 𝑁))
5247, 51mpd 15 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → 0 ≤ 𝑁)
5346, 52jca 555 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
541, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
5536eluz1i 11858 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘0) ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
5654, 55sylibr 224 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ (ℤ‘0))
57 eluzfz1 12512 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
5856, 57syl 17 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ (0...𝑁))
5945, 58ffvelrnd 6511 . . . . . . . . . . . 12 (𝜑 → (𝑋‘0) ∈ 𝐴)
60 feq1 6175 . . . . . . . . . . . . . 14 (𝑓 = (𝑋‘0) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘0):𝑇⟶ℝ))
6160imbi2d 329 . . . . . . . . . . . . 13 (𝑓 = (𝑋‘0) → ((𝜑𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘0):𝑇⟶ℝ)))
62 stoweidlem17.8 . . . . . . . . . . . . . 14 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
6362expcom 450 . . . . . . . . . . . . 13 (𝑓𝐴 → (𝜑𝑓:𝑇⟶ℝ))
6461, 63vtoclga 3400 . . . . . . . . . . . 12 ((𝑋‘0) ∈ 𝐴 → (𝜑 → (𝑋‘0):𝑇⟶ℝ))
6559, 64mpcom 38 . . . . . . . . . . 11 (𝜑 → (𝑋‘0):𝑇⟶ℝ)
6665ffvelrnda 6510 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝑋‘0)‘𝑡) ∈ ℝ)
6766recnd 10231 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑋‘0)‘𝑡) ∈ ℂ)
6844, 67mulcld 10223 . . . . . . . 8 ((𝜑𝑡𝑇) → (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ)
69 fveq2 6340 . . . . . . . . . . 11 (𝑖 = 0 → (𝑋𝑖) = (𝑋‘0))
7069fveq1d 6342 . . . . . . . . . 10 (𝑖 = 0 → ((𝑋𝑖)‘𝑡) = ((𝑋‘0)‘𝑡))
7170oveq2d 6817 . . . . . . . . 9 (𝑖 = 0 → (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
7271sumsn 14645 . . . . . . . 8 ((0 ∈ ℤ ∧ (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
7336, 68, 72sylancr 698 . . . . . . 7 ((𝜑𝑡𝑇) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
7441, 73mpteq2da 4883 . . . . . 6 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ {0} (𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))))
7540, 74syl5eq 2794 . . . . 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 40691 . . . . 5 (𝜑 → (𝑡𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))) ∈ 𝐴)
7975, 78eqeltrd 2827 . . . 4 (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
8079adantr 472 . . 3 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
81 eqidd 2749 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑡𝐸 = 𝐸)
8281cbvmptv 4890 . . . . . . . . . . . . . . 15 (𝑟𝑇𝐸) = (𝑡𝑇𝐸)
8382eqcomi 2757 . . . . . . . . . . . . . 14 (𝑡𝑇𝐸) = (𝑟𝑇𝐸)
8483a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝑡𝑇𝐸) = (𝑟𝑇𝐸))
85 eqidd 2749 . . . . . . . . . . . . 13 (((𝜑𝑡𝑇) ∧ 𝑟 = 𝑡) → 𝐸 = 𝐸)
86 simpr 479 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → 𝑡𝑇)
8784, 85, 86, 43fvmptd 6438 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → ((𝑡𝑇𝐸)‘𝑡) = 𝐸)
8887oveq1d 6816 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
8941, 88mpteq2da 4883 . . . . . . . . . 10 (𝜑 → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
9089adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
9145ffvelrnda 6510 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑋‘(𝑚 + 1)) ∈ 𝐴)
92 simpl 474 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝜑)
93 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐸𝑥 = 𝐸)
9493mpteq2dv 4885 . . . . . . . . . . . . . . 15 (𝑥 = 𝐸 → (𝑡𝑇𝑥) = (𝑡𝑇𝐸))
9594eleq1d 2812 . . . . . . . . . . . . . 14 (𝑥 = 𝐸 → ((𝑡𝑇𝑥) ∈ 𝐴 ↔ (𝑡𝑇𝐸) ∈ 𝐴))
9695imbi2d 329 . . . . . . . . . . . . 13 (𝑥 = 𝐸 → ((𝜑 → (𝑡𝑇𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)))
9777expcom 450 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → (𝜑 → (𝑡𝑇𝑥) ∈ 𝐴))
9896, 97vtoclga 3400 . . . . . . . . . . . 12 (𝐸 ∈ ℝ → (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴))
9942, 98mpcom 38 . . . . . . . . . . 11 (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)
10099adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇𝐸) ∈ 𝐴)
101 fveq1 6339 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑔𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
102101oveq2d 6817 . . . . . . . . . . . . . . 15 (𝑔 = (𝑋‘(𝑚 + 1)) → (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡)) = (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)))
103102mpteq2dv 4885 . . . . . . . . . . . . . 14 (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))))
104103eleq1d 2812 . . . . . . . . . . . . 13 (𝑔 = (𝑋‘(𝑚 + 1)) → ((𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴))
105104imbi2d 329 . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . 16 ((𝑟𝑇𝐸) ∈ 𝐴 ↔ (𝑡𝑇𝐸) ∈ 𝐴)
107 fveq1 6339 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = (𝑟𝑇𝐸) → (𝑓𝑡) = ((𝑟𝑇𝐸)‘𝑡))
10882fveq1i 6341 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑟𝑇𝐸)‘𝑡) = ((𝑡𝑇𝐸)‘𝑡)
109107, 108syl6eq 2798 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑟𝑇𝐸) → (𝑓𝑡) = ((𝑡𝑇𝐸)‘𝑡))
110109oveq1d 6816 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑟𝑇𝐸) → ((𝑓𝑡) · (𝑔𝑡)) = (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡)))
111110mpteq2dv 4885 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑟𝑇𝐸) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))))
112111eleq1d 2812 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑟𝑇𝐸) → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
113112imbi2d 329 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑟𝑇𝐸) → (((𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴)))
114763com12 1117 . . . . . . . . . . . . . . . . . 18 ((𝑓𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
1151143expib 1116 . . . . . . . . . . . . . . . . 17 (𝑓𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴))
116113, 115vtoclga 3400 . . . . . . . . . . . . . . . 16 ((𝑟𝑇𝐸) ∈ 𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
117106, 116sylbir 225 . . . . . . . . . . . . . . 15 ((𝑡𝑇𝐸) ∈ 𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
1181173impib 1108 . . . . . . . . . . . . . 14 (((𝑡𝑇𝐸) ∈ 𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴)
1191183com13 1118 . . . . . . . . . . . . 13 ((𝑔𝐴𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴)
1201193expib 1116 . . . . . . . . . . . 12 (𝑔𝐴 → ((𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · (𝑔𝑡))) ∈ 𝐴))
121105, 120vtoclga 3400 . . . . . . . . . . 11 ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴))
1221213impib 1108 . . . . . . . . . 10 (((𝑋‘(𝑚 + 1)) ∈ 𝐴𝜑 ∧ (𝑡𝑇𝐸) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
12391, 92, 100, 122syl3anc 1463 . . . . . . . . 9 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ (((𝑡𝑇𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
12490, 123eqeltrrd 2828 . . . . . . . 8 ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
125124ad2antll 767 . . . . . . 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 823 . . . . . . 7 (((𝑚 ∈ ℕ0 → ((𝜑𝑚 ∈ (0...𝑁)) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → 𝜑)
127 simpl 474 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℕ0)
128 simprl 811 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝜑)
1291ad2antrl 766 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ)
130129nnnn0d 11514 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ0)
131 nn0re 11464 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
132131adantr 472 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℝ)
133 peano2nn0 11496 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
134133nn0red 11515 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ)
135134adantr 472 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ∈ ℝ)
1361nnred 11198 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℝ)
137136ad2antrl 766 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℝ)
138 lep1 11025 . . . . . . . . . . . . 13 (𝑚 ∈ ℝ → 𝑚 ≤ (𝑚 + 1))
139127, 131, 1383syl 18 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ≤ (𝑚 + 1))
140 elfzle2 12509 . . . . . . . . . . . . 13 ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ≤ 𝑁)
141140ad2antll 767 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ≤ 𝑁)
142132, 135, 137, 139, 141letrd 10357 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚𝑁)
143 elfz2nn0 12595 . . . . . . . . . . 11 (𝑚 ∈ (0...𝑁) ↔ (𝑚 ∈ ℕ0𝑁 ∈ ℕ0𝑚𝑁))
144127, 130, 142, 143syl3anbrc 1407 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ (0...𝑁))
145127, 128, 144jca32 559 . . . . . . . . 9 ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ0 ∧ (𝜑𝑚 ∈ (0...𝑁))))
146145adantl 473 . . . . . . . 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 pm3.31 460 . . . . . . . . 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 472 . . . . . . . 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 6340 . . . . . . . . . . . 12 (𝑟 = 𝑡 → ((𝑋‘(𝑚 + 1))‘𝑟) = ((𝑋‘(𝑚 + 1))‘𝑡))
151150oveq2d 6817 . . . . . . . . . . 11 (𝑟 = 𝑡 → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
152151cbvmptv 4890 . . . . . . . . . 10 (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
153152eleq1i 2818 . . . . . . . . 9 ((𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
154 fveq1 6339 . . . . . . . . . . . . . . 15 (𝑔 = (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔𝑡) = ((𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡))
155152fveq1i 6341 . . . . . . . . . . . . . . 15 ((𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡) = ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)
156154, 155syl6eq 2798 . . . . . . . . . . . . . 14 (𝑔 = (𝑟𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔𝑡) = ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))
157156oveq2d 6817 . . . . . . . . . . . . 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 4885 . . . . . . . . . . . 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 2812 . . . . . . . . . . 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 329 . . . . . . . . . 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 6340 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑡 → ((𝑋𝑖)‘𝑟) = ((𝑋𝑖)‘𝑡))
162161oveq2d 6817 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑡 → (𝐸 · ((𝑋𝑖)‘𝑟)) = (𝐸 · ((𝑋𝑖)‘𝑡)))
163162sumeq2sdv 14605 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑡 → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
164163cbvmptv 4890 . . . . . . . . . . . . . . 15 (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
165164eleq1i 2818 . . . . . . . . . . . . . 14 ((𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
166 fveq1 6339 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → (𝑓𝑡) = ((𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟)))‘𝑡))
167164fveq1i 6341 . . . . . . . . . . . . . . . . . . . 20 ((𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟)))‘𝑡) = ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡)
168166, 167syl6eq 2798 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → (𝑓𝑡) = ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡))
169168oveq1d 6816 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → ((𝑓𝑡) + (𝑔𝑡)) = (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡)))
170169mpteq2dv 4885 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑟𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑟))) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) = (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))))
171170eleq1d 2812 . . . . . . . . . . . . . . . 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 329 . . . . . . . . . . . . . . 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 1117 . . . . . . . . . . . . . . . 16 ((𝑓𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
1751743expib 1116 . . . . . . . . . . . . . . 15 (𝑓𝐴 → ((𝜑𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴))
176172, 175vtoclga 3400 . . . . . . . . . . . . . 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 1108 . . . . . . . . . . . 12 (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴𝜑𝑔𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴)
1791783com13 1118 . . . . . . . . . . 11 ((𝑔𝐴𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴)
1801793expib 1116 . . . . . . . . . 10 (𝑔𝐴 → ((𝜑 ∧ (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴) → (𝑡𝑇 ↦ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) + (𝑔𝑡))) ∈ 𝐴))
181160, 180vtoclga 3400 . . . . . . . . 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 1108 . . . . . . 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 1463 . . . . . 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 1081 . . . . . . . . 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 473 . . . . . . 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 1980 . . . . . . . . . 10 𝑡 𝑚 ∈ ℕ0
189 nfv 1980 . . . . . . . . . 10 𝑡(𝑚 + 1) ∈ (0...𝑁)
190188, 41, 189nf3an 1968 . . . . . . . . 9 𝑡(𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))
191 simpr 479 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 𝑡𝑇)
192 fzfid 12937 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (0...𝑚) ∈ Fin)
193423ad2ant2 1126 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝐸 ∈ ℝ)
194193adantr 472 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 𝐸 ∈ ℝ)
195194adantr 472 . . . . . . . . . . . . . 14 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → 𝐸 ∈ ℝ)
196 fzelp1 12557 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0...𝑚) → 𝑖 ∈ (0...(𝑚 + 1)))
197196anim2i 594 . . . . . . . . . . . . . . . 16 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))))
198 an32 874 . . . . . . . . . . . . . . . 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 1126 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑋:(0...𝑁)⟶𝐴)
201200adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑋:(0...𝑁)⟶𝐴)
202 elfzuz3 12503 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 + 1) ∈ (0...𝑁) → 𝑁 ∈ (ℤ‘(𝑚 + 1)))
203 fzss2 12545 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘(𝑚 + 1)) → (0...(𝑚 + 1)) ⊆ (0...𝑁))
204202, 203syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 + 1) ∈ (0...𝑁) → (0...(𝑚 + 1)) ⊆ (0...𝑁))
205204sselda 3732 . . . . . . . . . . . . . . . . . . 19 (((𝑚 + 1) ∈ (0...𝑁) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
2062053ad2antl3 1179 . . . . . . . . . . . . . . . . . 18 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
207201, 206ffvelrnd 6511 . . . . . . . . . . . . . . . . 17 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋𝑖) ∈ 𝐴)
208 simpl2 1206 . . . . . . . . . . . . . . . . 17 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝜑)
209 feq1 6175 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑋𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑋𝑖):𝑇⟶ℝ))
210209imbi2d 329 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑋𝑖) → ((𝜑𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋𝑖):𝑇⟶ℝ)))
211210, 63vtoclga 3400 . . . . . . . . . . . . . . . . 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 6510 . . . . . . . . . . . . . . 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 10233 . . . . . . . . . . . . 13 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℝ)
216192, 215fsumrecl 14635 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℝ)
217 eqid 2748 . . . . . . . . . . . . 13 (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡))) = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
218217fvmpt2 6441 . . . . . . . . . . . 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 696 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
220219oveq1d 6816 . . . . . . . . . 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 1144 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
222221adantr 472 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
223 feq1 6175 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑋‘(𝑚 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘(𝑚 + 1)):𝑇⟶ℝ))
224223imbi2d 329 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑋‘(𝑚 + 1)) → ((𝜑𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)))
225224, 63vtoclga 3400 . . . . . . . . . . . . . . . 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 6511 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑋‘(𝑚 + 1))‘𝑡) ∈ ℝ)
229194, 228remulcld 10233 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ)
230 eqid 2748 . . . . . . . . . . . . 13 (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
231230fvmpt2 6441 . . . . . . . . . . . 12 ((𝑡𝑇 ∧ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ) → ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
232191, 229, 231syl2anc 696 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑡𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
233232oveq2d 6817 . . . . . . . . . 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 12502 . . . . . . . . . . . . . 14 ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ∈ (ℤ‘0))
2352343ad2ant3 1127 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑚 + 1) ∈ (ℤ‘0))
236235adantr 472 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (𝑚 + 1) ∈ (ℤ‘0))
237194adantr 472 . . . . . . . . . . . . 13 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝐸 ∈ ℝ)
238213an32s 881 . . . . . . . . . . . . 13 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → ((𝑋𝑖)‘𝑡) ∈ ℝ)
239 remulcl 10184 . . . . . . . . . . . . . 14 ((𝐸 ∈ ℝ ∧ ((𝑋𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℝ)
240239recnd 10231 . . . . . . . . . . . . 13 ((𝐸 ∈ ℝ ∧ ((𝑋𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℂ)
241237, 238, 240syl2anc 696 . . . . . . . . . . . 12 ((((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝐸 · ((𝑋𝑖)‘𝑡)) ∈ ℂ)
242 fveq2 6340 . . . . . . . . . . . . . 14 (𝑖 = (𝑚 + 1) → (𝑋𝑖) = (𝑋‘(𝑚 + 1)))
243242fveq1d 6342 . . . . . . . . . . . . 13 (𝑖 = (𝑚 + 1) → ((𝑋𝑖)‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
244243oveq2d 6817 . . . . . . . . . . . 12 (𝑖 = (𝑚 + 1) → (𝐸 · ((𝑋𝑖)‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
245236, 241, 244fsumm1 14650 . . . . . . . . . . 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 11465 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
2472463ad2ant1 1125 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑚 ∈ ℂ)
248247adantr 472 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 𝑚 ∈ ℂ)
249 1cnd 10219 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → 1 ∈ ℂ)
250248, 249pncand 10556 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → ((𝑚 + 1) − 1) = 𝑚)
251250oveq2d 6817 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → (0...((𝑚 + 1) − 1)) = (0...𝑚))
252251sumeq1d 14601 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡𝑇) → Σ𝑖 ∈ (0...((𝑚 + 1) − 1))(𝐸 · ((𝑋𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋𝑖)‘𝑡)))
253252oveq1d 6816 . . . . . . . . . . 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 2782 . . . . . . . . . 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 2793 . . . . . . . . 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 4883 . . . . . . . 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 2812 . . . . . . 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 632 . . . 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 11635 . 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 383   ∧ w3a 1072   = wceq 1620  Ⅎwnf 1845   ∈ wcel 2127   ⊆ wss 3703  {csn 4309   class class class wbr 4792   ↦ cmpt 4869  ⟶wf 6033  ‘cfv 6037  (class class class)co 6801  ℂcc 10097  ℝcr 10098  0cc0 10099  1c1 10100   + caddc 10102   · cmul 10104   < clt 10237   ≤ cle 10238   − cmin 10429  ℕcn 11183  ℕ0cn0 11455  ℤcz 11540  ℤ≥cuz 11850  ...cfz 12490  Σcsu 14586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-inf2 8699  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176  ax-pre-sup 10177 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-se 5214  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-isom 6046  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7899  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8501  df-oi 8568  df-card 8926  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-div 10848  df-nn 11184  df-2 11242  df-3 11243  df-n0 11456  df-z 11541  df-uz 11851  df-rp 11997  df-fz 12491  df-fzo 12631  df-seq 12967  df-exp 13026  df-hash 13283  df-cj 14009  df-re 14010  df-im 14011  df-sqrt 14145  df-abs 14146  df-clim 14389  df-sum 14587 This theorem is referenced by:  stoweidlem60  40749
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