stoweidlem17 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem17 46596

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 12544	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
3		nn0uz 12879	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2874	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
5		eluzfz2 13539	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
7	6	ancli 556	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝑁 ∈ (0...𝑁)))
8		eleq1 2852	. . . . 5 ⊢ (𝑛 = 0 → (𝑛 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
9	8	anbi2d 639	. . . 4 ⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
10		oveq2 7406	. . . . . . 7 ⊢ (𝑛 = 0 → (0...𝑛) = (0...0))
11	10	sumeq1d 15729	. . . . . 6 ⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
12	11	mpteq2dv 5196	. . . . 5 ⊢ (𝑛 = 0 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))))
13	12	eleq1d 2849	. . . 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 346	. . 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 2852	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ (0...𝑁) ↔ 𝑚 ∈ (0...𝑁)))
16	15	anbi2d 639	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑚 ∈ (0...𝑁))))
17		oveq2 7406	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
18	17	sumeq1d 15729	. . . . . 6 ⊢ (𝑛 = 𝑚 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
19	18	mpteq2dv 5196	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))))
20	19	eleq1d 2849	. . . 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 346	. . 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 2852	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 ∈ (0...𝑁) ↔ (𝑚 + 1) ∈ (0...𝑁)))
23	22	anbi2d 639	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))))
24		oveq2 7406	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (0...𝑛) = (0...(𝑚 + 1)))
25	24	sumeq1d 15729	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)))
26	25	mpteq2dv 5196	. . . . 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 2849	. . . 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 346	. . 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 2852	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁)))
30	29	anbi2d 639	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑁 ∈ (0...𝑁))))
31		oveq2 7406	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
32	31	sumeq1d 15729	. . . . . 6 ⊢ (𝑛 = 𝑁 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
33	32	mpteq2dv 5196	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))))
34	33	eleq1d 2849	. . . 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 346	. . 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 12581	. . . . . . . . 9 ⊢ 0 ∈ ℤ
37		fzsn 13573	. . . . . . . . 9 ⊢ (0 ∈ ℤ → (0...0) = {0})
38	36, 37	ax-mp 5	. . . . . . . 8 ⊢ (0...0) = {0}
39	38	sumeq1i 15726	. . . . . . 7 ⊢ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡))
40	39	mpteq2i 5198	. . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ)
44	43	recnd 11212	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℂ)
45		stoweidlem17.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝐴)
46		nnz 12591	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
47		nngt0 12246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
48		0re 11185	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
49		nnre 12219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
50		ltle 11273	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁))
51	48, 49, 50	sylancr 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (0 < 𝑁 → 0 ≤ 𝑁))
52	47, 51	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 0 ≤ 𝑁)
53	46, 52	jca 519	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
54	1, 53	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
55	36	eluz1i 12849	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘0) ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
56	54, 55	sylibr 236	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
57		eluzfz1 13538	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑁))
58	56, 57	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑁))
59	45, 58	ffvelcdmd 7068	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋‘0) ∈ 𝐴)
60		feq1 6671	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑋‘0) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘0):𝑇⟶ℝ))
61	60	imbi2d 342	. . . . . . . . . . . . 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 417	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝐴 → (𝜑 → 𝑓:𝑇⟶ℝ))
64	61, 63	vtoclga 3543	. . . . . . . . . . . 12 ⊢ ((𝑋‘0) ∈ 𝐴 → (𝜑 → (𝑋‘0):𝑇⟶ℝ))
65	59, 64	mpcom 38	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋‘0):𝑇⟶ℝ)
66	65	ffvelcdmda 7067	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑋‘0)‘𝑡) ∈ ℝ)
67	66	recnd 11212	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑋‘0)‘𝑡) ∈ ℂ)
68	44, 67	mulcld 11204	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ)
69		fveq2 6869	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → (𝑋‘𝑖) = (𝑋‘0))
70	69	fveq1d 6871	. . . . . . . . . 10 ⊢ (𝑖 = 0 → ((𝑋‘𝑖)‘𝑡) = ((𝑋‘0)‘𝑡))
71	70	oveq2d 7414	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
72	71	sumsn 15775	. . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
73	36, 68, 72	sylancr 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
74	41, 73	mpteq2da 5194	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))))
75	40, 74	eqtrid 2811	. . . . 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 46581	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))) ∈ 𝐴)
79	75, 78	eqeltrd 2864	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)
80	79	adantr 484	. . 3 ⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)
81		eqidd 2765	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑡 → 𝐸 = 𝐸)
82	81	cbvmptv 5206	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ 𝑇 ↦ 𝐸) = (𝑡 ∈ 𝑇 ↦ 𝐸)
83	82	eqcomi 2773	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑇 ↦ 𝐸) = (𝑟 ∈ 𝑇 ↦ 𝐸)
84		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
85	83, 81, 84, 43	fvmptd3 7001	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) = 𝐸)
86	85	oveq1d 7413	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
87	41, 86	mpteq2da 5194	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
88	87	adantr 484	. . . . . . . . 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 7067	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑋‘(𝑚 + 1)) ∈ 𝐴)
90		simpl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝜑)
91		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐸 → 𝑥 = 𝐸)
92	91	mpteq2dv 5196	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐸 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝐸))
93	92	eleq1d 2849	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐸 → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))
94	93	imbi2d 342	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐸 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)))
95	77	expcom 417	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴))
96	94, 95	vtoclga 3543	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))
97	42, 96	mpcom 38	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
98	97	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
99		fveq1 6868	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑔‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
100	99	oveq2d 7414	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)))
101	100	mpteq2dv 5196	. . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . 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 342	. . . . . . . . . . . 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 2855	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
105		fveq1 6868	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑟 ∈ 𝑇 ↦ 𝐸)‘𝑡))
106	82	fveq1i 6870	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸)‘𝑡) = ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡)
107	105, 106	eqtrdi 2815	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡))
108	107	oveq1d 7413	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → ((𝑓‘𝑡) · (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡)))
109	108	mpteq2dv 5196	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))))
110	109	eleq1d 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
111	110	imbi2d 342	. . . . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
113	112	3expib 1136	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
114	111, 113	vtoclga 3543	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
115	104, 114	sylbir 237	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
116	115	3impib 1130	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
117	116	3com13 1138	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ 𝐴 ∧ 𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
118	117	3expib 1136	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
119	103, 118	vtoclga 3543	. . . . . . . . . . 11 ⊢ ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴))
120	119	3impib 1130	. . . . . . . . . 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 1392	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
122	88, 121	eqeltrrd 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
123	122	ad2antll 739	. . . . . . 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 790	. . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℕ₀)
126		simprl 780	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝜑)
127	1	ad2antrl 738	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ)
128	127	nnnn0d 12544	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ₀)
129		nn0re 12492	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
130	129	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℝ)
131		peano2nn0 12523	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ₀)
132	131	nn0red 12545	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℝ)
133	132	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ∈ ℝ)
134	1	nnred 12227	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℝ)
135	134	ad2antrl 738	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℝ)
136		lep1 12034	. . . . . . . . . . . . 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 13535	. . . . . . . . . . . . 13 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ≤ 𝑁)
139	138	ad2antll 739	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ≤ 𝑁)
140	130, 133, 135, 137, 139	letrd 11342	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ≤ 𝑁)
141		elfz2nn0 13625	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...𝑁) ↔ (𝑚 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑚 ≤ 𝑁))
142	125, 128, 140, 141	syl3anbrc 1358	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ (0...𝑁))
143	125, 126, 142	jca32 523	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ 𝑚 ∈ (0...𝑁))))
144	143	adantl 485	. . . . . . . 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		pm3.31 453	. . . . . . . . 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 484	. . . . . . . 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 6869	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑡 → ((𝑋‘(𝑚 + 1))‘𝑟) = ((𝑋‘(𝑚 + 1))‘𝑡))
149	148	oveq2d 7414	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑡 → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
150	149	cbvmptv 5206	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
151	150	eleq1i 2855	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
152		fveq1 6868	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔‘𝑡) = ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡))
153	150	fveq1i 6870	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡) = ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)
154	152, 153	eqtrdi 2815	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔‘𝑡) = ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))
155	154	oveq2d 7414	. . . . . . . . . . . . 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 5196	. . . . . . . . . . . 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 2849	. . . . . . . . . . 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 342	. . . . . . . . . 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 6869	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝑡 → ((𝑋‘𝑖)‘𝑟) = ((𝑋‘𝑖)‘𝑡))
160	159	oveq2d 7414	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑡 → (𝐸 · ((𝑋‘𝑖)‘𝑟)) = (𝐸 · ((𝑋‘𝑖)‘𝑡)))
161	160	sumeq2sdv 15732	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑡 → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
162	161	cbvmptv 5206	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
163	162	eleq1i 2855	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)
164		fveq1 6868	. . . . . . . . . . . . . . . . . . . 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 6870	. . . . . . . . . . . . . . . . . . . 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 2815	. . . . . . . . . . . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . . 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 5196	. . . . . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . . . . 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 342	. . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
173	172	3expib 1136	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴))
174	170, 173	vtoclga 3543	. . . . . . . . . . . . . 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 237	. . . . . . . . . . . . 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 1130	. . . . . . . . . . . 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 1138	. . . . . . . . . . 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 1136	. . . . . . . . . 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 3543	. . . . . . . . 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 237	. . . . . . . 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 1130	. . . . . . 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 1392	. . . . . 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 1107	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ↔ (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))))
184	183	bilanri 510	. . . . . . 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)))
185		nfv 1936	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ₀
186		nfv 1936	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑚 + 1) ∈ (0...𝑁)
187	185, 41, 186	nf3an 1923	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))
188		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
189		fzfid 13988	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (0...𝑚) ∈ Fin)
190	42	3ad2ant2 1148	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝐸 ∈ ℝ)
191	190	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ)
192	191	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → 𝐸 ∈ ℝ)
193		fzelp1 13583	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0...𝑚) → 𝑖 ∈ (0...(𝑚 + 1)))
194	193	anim2i 626	. . . . . . . . . . . . . . . 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))))
195		an32 656	. . . . . . . . . . . . . . . 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))
196	194, 195	sylib 220	. . . . . . . . . . . . . . 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))
197	45	3ad2ant2 1148	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑋:(0...𝑁)⟶𝐴)
198	197	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑋:(0...𝑁)⟶𝐴)
199		elfzuz3 13528	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘(𝑚 + 1)))
200		fzss2 13571	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑚 + 1)) → (0...(𝑚 + 1)) ⊆ (0...𝑁))
201	199, 200	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → (0...(𝑚 + 1)) ⊆ (0...𝑁))
202	201	sselda 3938	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 + 1) ∈ (0...𝑁) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
203	202	3ad2antl3 1202	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
204	198, 203	ffvelcdmd 7068	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋‘𝑖) ∈ 𝐴)
205		simpl2 1207	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝜑)
206		feq1 6671	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑋‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘𝑖):𝑇⟶ℝ))
207	206	imbi2d 342	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑋‘𝑖) → ((𝜑 → 𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘𝑖):𝑇⟶ℝ)))
208	207, 63	vtoclga 3543	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋‘𝑖) ∈ 𝐴 → (𝜑 → (𝑋‘𝑖):𝑇⟶ℝ))
209	204, 205, 208	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋‘𝑖):𝑇⟶ℝ)
210	209	ffvelcdmda 7067	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ∧ 𝑡 ∈ 𝑇) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ)
211	196, 210	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ)
212	192, 211	remulcld 11214	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
213	189, 212	fsumrecl 15763	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
214		eqid 2764	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
215	214	fvmpt2 6989	. . . . . . . . . . . 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)))
216	188, 213, 215	syl2anc 593	. . . . . . . . . . 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)))
217	216	oveq1d 7413	. . . . . . . . . 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))))
218		3simpc 1164	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
219	218	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
220		feq1 6671	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑋‘(𝑚 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘(𝑚 + 1)):𝑇⟶ℝ))
221	220	imbi2d 342	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑋‘(𝑚 + 1)) → ((𝜑 → 𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)))
222	221, 63	vtoclga 3543	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ))
223	89, 90, 222	sylc 65	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)
224	219, 223	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)
225	224, 188	ffvelcdmd 7068	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑋‘(𝑚 + 1))‘𝑡) ∈ ℝ)
226	191, 225	remulcld 11214	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ)
227		eqid 2764	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
228	227	fvmpt2 6989	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑇 ∧ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
229	188, 226, 228	syl2anc 593	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
230	229	oveq2d 7414	. . . . . . . . . 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))))
231		elfzuz 13527	. . . . . . . . . . . . . 14 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ∈ (ℤ_≥‘0))
232	231	3ad2ant3 1149	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑚 + 1) ∈ (ℤ_≥‘0))
233	232	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝑚 + 1) ∈ (ℤ_≥‘0))
234	191	adantr 484	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝐸 ∈ ℝ)
235	210	an32s 662	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ)
236		remulcl 11160	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ ℝ ∧ ((𝑋‘𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
237	236	recnd 11212	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℝ ∧ ((𝑋‘𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℂ)
238	234, 235, 237	syl2anc 593	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℂ)
239		fveq2 6869	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑚 + 1) → (𝑋‘𝑖) = (𝑋‘(𝑚 + 1)))
240	239	fveq1d 6871	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑚 + 1) → ((𝑋‘𝑖)‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
241	240	oveq2d 7414	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑚 + 1) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
242	233, 238, 241	fsumm1 15780	. . . . . . . . . . 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))))
243		nn0cn 12493	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
244	243	3ad2ant1 1147	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑚 ∈ ℂ)
245	244	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℂ)
246		1cnd 11177	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℂ)
247	245, 246	pncand 11545	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑚 + 1) − 1) = 𝑚)
248	247	oveq2d 7414	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (0...((𝑚 + 1) − 1)) = (0...𝑚))
249	248	sumeq1d 15729	. . . . . . . . . . . 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)))
250	249	oveq1d 7413	. . . . . . . . . . 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))))
251	242, 250	eqtrd 2799	. . . . . . . . . 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))))
252	217, 230, 251	3eqtr4rd 2810	. . . . . . . . 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)))
253	187, 252	mpteq2da 5194	. . . . . . . 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))))
254	253	eleq1d 2849	. . . . . . 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))
255	184, 254	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))
256	182, 255	mpbird 259	. . . . 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)
257	256	exp32 424	. . . 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)))
258	257	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)))
259	14, 21, 28, 35, 80, 258	nn0ind 12670	. 2 ⊢ (𝑁 ∈ ℕ₀ → ((𝜑 ∧ 𝑁 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))
260	2, 7, 259	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 208 ∧ wa 399 ∧ w3a 1099 = wceq 1562 Ⅎwnf 1805 ∈ wcel 2144 ⊆ wss 3906 {csn 4584 class class class wbr 5102 ↦ cmpt 5183 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 < clt 11218 ≤ cle 11219 − cmin 11416 ℕcn 12212 ℕ₀cn0 12483 ℤcz 12570 ℤ_≥cuz 12841 ...cfz 13514 Σcsu 15715

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-inf2 9598 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-er 8680 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-sup 9390 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-n0 12484 df-z 12571 df-uz 12842 df-rp 12996 df-fz 13515 df-fzo 13662 df-seq 14017 df-exp 14077 df-hash 14346 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-clim 15517 df-sum 15716

This theorem is referenced by: stoweidlem60 46639

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