stoweidlem17 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem17 45972

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	⊢ Ⅎ𝑡𝜑
stoweidlem17.2	⊢ (𝜑 → 𝑁 ∈ ℕ)
stoweidlem17.3	⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝐴)
stoweidlem17.4	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem17.5	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem17.6	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
stoweidlem17.7	⊢ (𝜑 → 𝐸 ∈ ℝ)
stoweidlem17.8	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)

**Assertion**
Ref	Expression
stoweidlem17	⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘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.

Step	Hyp	Ref	Expression
1		stoweidlem17.2	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	nnnn0d 12584	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
3		nn0uz 12917	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2848	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
5		eluzfz2 13568	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
7	6	ancli 548	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝑁 ∈ (0...𝑁)))
8		eleq1 2826	. . . . 5 ⊢ (𝑛 = 0 → (𝑛 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
9	8	anbi2d 630	. . . 4 ⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
10		oveq2 7438	. . . . . . 7 ⊢ (𝑛 = 0 → (0...𝑛) = (0...0))
11	10	sumeq1d 15732	. . . . . 6 ⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
12	11	mpteq2dv 5249	. . . . 5 ⊢ (𝑛 = 0 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))))
13	12	eleq1d 2823	. . . 4 ⊢ (𝑛 = 0 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))
14	9, 13	imbi12d 344	. . 3 ⊢ (𝑛 = 0 → (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)))
15		eleq1 2826	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ (0...𝑁) ↔ 𝑚 ∈ (0...𝑁)))
16	15	anbi2d 630	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑚 ∈ (0...𝑁))))
17		oveq2 7438	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
18	17	sumeq1d 15732	. . . . . 6 ⊢ (𝑛 = 𝑚 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
19	18	mpteq2dv 5249	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))))
20	19	eleq1d 2823	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))
21	16, 20	imbi12d 344	. . 3 ⊢ (𝑛 = 𝑚 → (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)))
22		eleq1 2826	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 ∈ (0...𝑁) ↔ (𝑚 + 1) ∈ (0...𝑁)))
23	22	anbi2d 630	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))))
24		oveq2 7438	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (0...𝑛) = (0...(𝑚 + 1)))
25	24	sumeq1d 15732	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)))
26	25	mpteq2dv 5249	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))))
27	26	eleq1d 2823	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))
28	23, 27	imbi12d 344	. . 3 ⊢ (𝑛 = (𝑚 + 1) → (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)))
29		eleq1 2826	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁)))
30	29	anbi2d 630	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑁 ∈ (0...𝑁))))
31		oveq2 7438	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
32	31	sumeq1d 15732	. . . . . 6 ⊢ (𝑛 = 𝑁 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
33	32	mpteq2dv 5249	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))))
34	33	eleq1d 2823	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))
35	30, 34	imbi12d 344	. . 3 ⊢ (𝑛 = 𝑁 → (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑁 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)))
36		0z 12621	. . . . . . . . 9 ⊢ 0 ∈ ℤ
37		fzsn 13602	. . . . . . . . 9 ⊢ (0 ∈ ℤ → (0...0) = {0})
38	36, 37	ax-mp 5	. . . . . . . 8 ⊢ (0...0) = {0}
39	38	sumeq1i 15729	. . . . . . 7 ⊢ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡))
40	39	mpteq2i 5252	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡)))
41		stoweidlem17.1	. . . . . . 7 ⊢ Ⅎ𝑡𝜑
42		stoweidlem17.7	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ)
43	42	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ)
44	43	recnd 11286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℂ)
45		stoweidlem17.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝐴)
46		nnz 12631	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
47		nngt0 12294	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
48		0re 11260	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
49		nnre 12270	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
50		ltle 11346	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁))
51	48, 49, 50	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (0 < 𝑁 → 0 ≤ 𝑁))
52	47, 51	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 0 ≤ 𝑁)
53	46, 52	jca 511	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
54	1, 53	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
55	36	eluz1i 12883	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘0) ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
56	54, 55	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
57		eluzfz1 13567	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑁))
58	56, 57	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑁))
59	45, 58	ffvelcdmd 7104	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋‘0) ∈ 𝐴)
60		feq1 6716	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑋‘0) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘0):𝑇⟶ℝ))
61	60	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑋‘0) → ((𝜑 → 𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘0):𝑇⟶ℝ)))
62		stoweidlem17.8	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
63	62	expcom 413	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝐴 → (𝜑 → 𝑓:𝑇⟶ℝ))
64	61, 63	vtoclga 3576	. . . . . . . . . . . 12 ⊢ ((𝑋‘0) ∈ 𝐴 → (𝜑 → (𝑋‘0):𝑇⟶ℝ))
65	59, 64	mpcom 38	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋‘0):𝑇⟶ℝ)
66	65	ffvelcdmda 7103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑋‘0)‘𝑡) ∈ ℝ)
67	66	recnd 11286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑋‘0)‘𝑡) ∈ ℂ)
68	44, 67	mulcld 11278	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ)
69		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → (𝑋‘𝑖) = (𝑋‘0))
70	69	fveq1d 6908	. . . . . . . . . 10 ⊢ (𝑖 = 0 → ((𝑋‘𝑖)‘𝑡) = ((𝑋‘0)‘𝑡))
71	70	oveq2d 7446	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
72	71	sumsn 15778	. . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
73	36, 68, 72	sylancr 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
74	41, 73	mpteq2da 5245	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))))
75	40, 74	eqtrid 2786	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))))
76		stoweidlem17.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
77		stoweidlem17.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
78	41, 76, 77, 62, 42, 59	stoweidlem2 45957	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))) ∈ 𝐴)
79	75, 78	eqeltrd 2838	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)
80	79	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)
81		eqidd 2735	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑡 → 𝐸 = 𝐸)
82	81	cbvmptv 5260	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ 𝑇 ↦ 𝐸) = (𝑡 ∈ 𝑇 ↦ 𝐸)
83	82	eqcomi 2743	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑇 ↦ 𝐸) = (𝑟 ∈ 𝑇 ↦ 𝐸)
84		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
85	83, 81, 84, 43	fvmptd3 7038	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) = 𝐸)
86	85	oveq1d 7445	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
87	41, 86	mpteq2da 5245	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
88	87	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
89	45	ffvelcdmda 7103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑋‘(𝑚 + 1)) ∈ 𝐴)
90		simpl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝜑)
91		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐸 → 𝑥 = 𝐸)
92	91	mpteq2dv 5249	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐸 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝐸))
93	92	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐸 → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))
94	93	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐸 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)))
95	77	expcom 413	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴))
96	94, 95	vtoclga 3576	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))
97	42, 96	mpcom 38	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
98	97	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
99		fveq1 6905	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑔‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
100	99	oveq2d 7446	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)))
101	100	mpteq2dv 5249	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))))
102	101	eleq1d 2823	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → ((𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴))
103	102	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)))
104	82	eleq1i 2829	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
105		fveq1 6905	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑟 ∈ 𝑇 ↦ 𝐸)‘𝑡))
106	82	fveq1i 6907	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸)‘𝑡) = ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡)
107	105, 106	eqtrdi 2790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡))
108	107	oveq1d 7445	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → ((𝑓‘𝑡) · (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡)))
109	108	mpteq2dv 5249	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))))
110	109	eleq1d 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
111	110	imbi2d 340	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)))
112	76	3com12 1122	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
113	112	3expib 1121	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
114	111, 113	vtoclga 3576	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
115	104, 114	sylbir 235	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
116	115	3impib 1115	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
117	116	3com13 1123	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ 𝐴 ∧ 𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
118	117	3expib 1121	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
119	103, 118	vtoclga 3576	. . . . . . . . . . 11 ⊢ ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴))
120	119	3impib 1115	. . . . . . . . . 10 ⊢ (((𝑋‘(𝑚 + 1)) ∈ 𝐴 ∧ 𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
121	89, 90, 98, 120	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
122	88, 121	eqeltrrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
123	122	ad2antll 729	. . . . . . 7 ⊢ (((𝑚 ∈ ℕ₀ → ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
124		simprrl 781	. . . . . . 7 ⊢ (((𝑚 ∈ ℕ₀ → ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → 𝜑)
125		simpl 482	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℕ₀)
126		simprl 771	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝜑)
127	1	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ)
128	127	nnnn0d 12584	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ₀)
129		nn0re 12532	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
130	129	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℝ)
131		peano2nn0 12563	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ₀)
132	131	nn0red 12585	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℝ)
133	132	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ∈ ℝ)
134	1	nnred 12278	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℝ)
135	134	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℝ)
136		lep1 12105	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℝ → 𝑚 ≤ (𝑚 + 1))
137	125, 129, 136	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ≤ (𝑚 + 1))
138		elfzle2 13564	. . . . . . . . . . . . 13 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ≤ 𝑁)
139	138	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ≤ 𝑁)
140	130, 133, 135, 137, 139	letrd 11415	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ≤ 𝑁)
141		elfz2nn0 13654	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...𝑁) ↔ (𝑚 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑚 ≤ 𝑁))
142	125, 128, 140, 141	syl3anbrc 1342	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ (0...𝑁))
143	125, 126, 142	jca32 515	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ 𝑚 ∈ (0...𝑁))))
144	143	adantl 481	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ₀ → ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ 𝑚 ∈ (0...𝑁))))
145		pm3.31 449	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ₀ → ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) → ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ 𝑚 ∈ (0...𝑁))) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))
146	145	adantr 480	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ₀ → ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ 𝑚 ∈ (0...𝑁))) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))
147	144, 146	mpd 15	. . . . . . 7 ⊢ (((𝑚 ∈ ℕ₀ → ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)
148		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑡 → ((𝑋‘(𝑚 + 1))‘𝑟) = ((𝑋‘(𝑚 + 1))‘𝑡))
149	148	oveq2d 7446	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑡 → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
150	149	cbvmptv 5260	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
151	150	eleq1i 2829	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
152		fveq1 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔‘𝑡) = ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡))
153	150	fveq1i 6907	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡) = ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)
154	152, 153	eqtrdi 2790	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔‘𝑡) = ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))
155	154	oveq2d 7446	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)))
156	155	mpteq2dv 5249	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))))
157	156	eleq1d 2823	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → ((𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴))
158	157	imbi2d 340	. . . . . . . . . 10 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴)))
159		fveq2 6906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝑡 → ((𝑋‘𝑖)‘𝑟) = ((𝑋‘𝑖)‘𝑡))
160	159	oveq2d 7446	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑡 → (𝐸 · ((𝑋‘𝑖)‘𝑟)) = (𝐸 · ((𝑋‘𝑖)‘𝑡)))
161	160	sumeq2sdv 15735	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑡 → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
162	161	cbvmptv 5260	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
163	162	eleq1i 2829	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)
164		fveq1 6905	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → (𝑓‘𝑡) = ((𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟)))‘𝑡))
165	162	fveq1i 6907	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟)))‘𝑡) = ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡)
166	164, 165	eqtrdi 2790	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → (𝑓‘𝑡) = ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡))
167	166	oveq1d 7445	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → ((𝑓‘𝑡) + (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡)))
168	167	mpteq2dv 5249	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))))
169	168	eleq1d 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴))
170	169	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → (((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)))
171		stoweidlem17.4	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
172	171	3com12 1122	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
173	172	3expib 1121	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴))
174	170, 173	vtoclga 3576	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴))
175	163, 174	sylbir 235	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴))
176	175	3impib 1115	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
177	176	3com13 1123	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ 𝐴 ∧ 𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
178	177	3expib 1121	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴))
179	158, 178	vtoclga 3576	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴))
180	151, 179	sylbir 235	. . . . . . . 8 ⊢ ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴))
181	180	3impib 1115	. . . . . . 7 ⊢ (((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴 ∧ 𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴)
182	123, 124, 147, 181	syl3anc 1370	. . . . . 6 ⊢ (((𝑚 ∈ ℕ₀ → ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴)
183		3anass 1094	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ↔ (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))))
184	183	biimpri 228	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
185	184	adantl 481	. . . . . . 7 ⊢ (((𝑚 ∈ ℕ₀ → ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
186		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ₀
187		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑚 + 1) ∈ (0...𝑁)
188	186, 41, 187	nf3an 1898	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))
189		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
190		fzfid 14010	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (0...𝑚) ∈ Fin)
191	42	3ad2ant2 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝐸 ∈ ℝ)
192	191	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ)
193	192	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → 𝐸 ∈ ℝ)
194		fzelp1 13612	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0...𝑚) → 𝑖 ∈ (0...(𝑚 + 1)))
195	194	anim2i 617	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))))
196		an32 646	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ↔ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ∧ 𝑡 ∈ 𝑇))
197	195, 196	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ∧ 𝑡 ∈ 𝑇))
198	45	3ad2ant2 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑋:(0...𝑁)⟶𝐴)
199	198	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑋:(0...𝑁)⟶𝐴)
200		elfzuz3 13557	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘(𝑚 + 1)))
201		fzss2 13600	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑚 + 1)) → (0...(𝑚 + 1)) ⊆ (0...𝑁))
202	200, 201	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → (0...(𝑚 + 1)) ⊆ (0...𝑁))
203	202	sselda 3994	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 + 1) ∈ (0...𝑁) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
204	203	3ad2antl3 1186	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
205	199, 204	ffvelcdmd 7104	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋‘𝑖) ∈ 𝐴)
206		simpl2 1191	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝜑)
207		feq1 6716	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑋‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘𝑖):𝑇⟶ℝ))
208	207	imbi2d 340	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑋‘𝑖) → ((𝜑 → 𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘𝑖):𝑇⟶ℝ)))
209	208, 63	vtoclga 3576	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋‘𝑖) ∈ 𝐴 → (𝜑 → (𝑋‘𝑖):𝑇⟶ℝ))
210	205, 206, 209	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋‘𝑖):𝑇⟶ℝ)
211	210	ffvelcdmda 7103	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ∧ 𝑡 ∈ 𝑇) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ)
212	197, 211	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ)
213	193, 212	remulcld 11288	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
214	190, 213	fsumrecl 15766	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
215		eqid 2734	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
216	215	fvmpt2 7026	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑇 ∧ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
217	189, 214, 216	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
218	217	oveq1d 7445	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
219		3simpc 1149	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
220	219	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
221		feq1 6716	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑋‘(𝑚 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘(𝑚 + 1)):𝑇⟶ℝ))
222	221	imbi2d 340	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑋‘(𝑚 + 1)) → ((𝜑 → 𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)))
223	222, 63	vtoclga 3576	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ))
224	89, 90, 223	sylc 65	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)
225	220, 224	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)
226	225, 189	ffvelcdmd 7104	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑋‘(𝑚 + 1))‘𝑡) ∈ ℝ)
227	192, 226	remulcld 11288	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ)
228		eqid 2734	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
229	228	fvmpt2 7026	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑇 ∧ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
230	189, 227, 229	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
231	230	oveq2d 7446	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
232		elfzuz 13556	. . . . . . . . . . . . . 14 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ∈ (ℤ_≥‘0))
233	232	3ad2ant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑚 + 1) ∈ (ℤ_≥‘0))
234	233	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝑚 + 1) ∈ (ℤ_≥‘0))
235	192	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝐸 ∈ ℝ)
236	211	an32s 652	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ)
237		remulcl 11237	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ ℝ ∧ ((𝑋‘𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
238	237	recnd 11286	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℝ ∧ ((𝑋‘𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℂ)
239	235, 236, 238	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℂ)
240		fveq2 6906	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑚 + 1) → (𝑋‘𝑖) = (𝑋‘(𝑚 + 1)))
241	240	fveq1d 6908	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑚 + 1) → ((𝑋‘𝑖)‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
242	241	oveq2d 7446	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑚 + 1) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
243	234, 239, 242	fsumm1 15783	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) = (Σ𝑖 ∈ (0...((𝑚 + 1) − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
244		nn0cn 12533	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
245	244	3ad2ant1 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑚 ∈ ℂ)
246	245	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℂ)
247		1cnd 11253	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℂ)
248	246, 247	pncand 11618	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑚 + 1) − 1) = 𝑚)
249	248	oveq2d 7446	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (0...((𝑚 + 1) − 1)) = (0...𝑚))
250	249	sumeq1d 15732	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...((𝑚 + 1) − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
251	250	oveq1d 7445	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (Σ𝑖 ∈ (0...((𝑚 + 1) − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
252	243, 251	eqtrd 2774	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) = (Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
253	218, 231, 252	3eqtr4rd 2785	. . . . . . . . 9 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)))
254	188, 253	mpteq2da 5245	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))))
255	254	eleq1d 2823	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴))
256	185, 255	syl 17	. . . . . 6 ⊢ (((𝑚 ∈ ℕ₀ → ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴))
257	182, 256	mpbird 257	. . . . 5 ⊢ (((𝑚 ∈ ℕ₀ → ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)
258	257	exp32 420	. . . 4 ⊢ ((𝑚 ∈ ℕ₀ → ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) → (𝑚 ∈ ℕ₀ → ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)))
259	258	pm2.86i 110	. . 3 ⊢ (𝑚 ∈ ℕ₀ → (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)))
260	14, 21, 28, 35, 80, 259	nn0ind 12710	. 2 ⊢ (𝑁 ∈ ℕ₀ → ((𝜑 ∧ 𝑁 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))
261	2, 7, 260	sylc 65	1 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 ⊆ wss 3962 {csn 4630 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 ≤ cle 11293 − cmin 11489 ℕcn 12263 ℕ₀cn0 12523 ℤcz 12610 ℤ_≥cuz 12875 ...cfz 13543 Σcsu 15718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719

This theorem is referenced by: stoweidlem60 46015

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