stoweidlem17 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem17 43558

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 12293	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
3		nn0uz 12620	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2849	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
5		eluzfz2 13264	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
7	6	ancli 549	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝑁 ∈ (0...𝑁)))
8		eleq1 2826	. . . . 5 ⊢ (𝑛 = 0 → (𝑛 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
9	8	anbi2d 629	. . . 4 ⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
10		oveq2 7283	. . . . . . 7 ⊢ (𝑛 = 0 → (0...𝑛) = (0...0))
11	10	sumeq1d 15413	. . . . . 6 ⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
12	11	mpteq2dv 5176	. . . . 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 345	. . 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 629	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑚 ∈ (0...𝑁))))
17		oveq2 7283	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
18	17	sumeq1d 15413	. . . . . 6 ⊢ (𝑛 = 𝑚 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
19	18	mpteq2dv 5176	. . . . 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 345	. . 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 629	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))))
24		oveq2 7283	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (0...𝑛) = (0...(𝑚 + 1)))
25	24	sumeq1d 15413	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)))
26	25	mpteq2dv 5176	. . . . 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 345	. . 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 629	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑁 ∈ (0...𝑁))))
31		oveq2 7283	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
32	31	sumeq1d 15413	. . . . . 6 ⊢ (𝑛 = 𝑁 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
33	32	mpteq2dv 5176	. . . . 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 345	. . 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 12330	. . . . . . . . 9 ⊢ 0 ∈ ℤ
37		fzsn 13298	. . . . . . . . 9 ⊢ (0 ∈ ℤ → (0...0) = {0})
38	36, 37	ax-mp 5	. . . . . . . 8 ⊢ (0...0) = {0}
39	38	sumeq1i 15410	. . . . . . 7 ⊢ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡))
40	39	mpteq2i 5179	. . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ)
44	43	recnd 11003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℂ)
45		stoweidlem17.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝐴)
46		nnz 12342	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
47		nngt0 12004	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
48		0re 10977	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
49		nnre 11980	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
50		ltle 11063	. . . . . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
54	1, 53	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
55	36	eluz1i 12590	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘0) ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
56	54, 55	sylibr 233	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
57		eluzfz1 13263	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑁))
58	56, 57	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑁))
59	45, 58	ffvelrnd 6962	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋‘0) ∈ 𝐴)
60		feq1 6581	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑋‘0) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘0):𝑇⟶ℝ))
61	60	imbi2d 341	. . . . . . . . . . . . 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 414	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝐴 → (𝜑 → 𝑓:𝑇⟶ℝ))
64	61, 63	vtoclga 3513	. . . . . . . . . . . 12 ⊢ ((𝑋‘0) ∈ 𝐴 → (𝜑 → (𝑋‘0):𝑇⟶ℝ))
65	59, 64	mpcom 38	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋‘0):𝑇⟶ℝ)
66	65	ffvelrnda 6961	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑋‘0)‘𝑡) ∈ ℝ)
67	66	recnd 11003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑋‘0)‘𝑡) ∈ ℂ)
68	44, 67	mulcld 10995	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ)
69		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → (𝑋‘𝑖) = (𝑋‘0))
70	69	fveq1d 6776	. . . . . . . . . 10 ⊢ (𝑖 = 0 → ((𝑋‘𝑖)‘𝑡) = ((𝑋‘0)‘𝑡))
71	70	oveq2d 7291	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
72	71	sumsn 15458	. . . . . . . 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 5172	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))))
75	40, 74	eqtrid 2790	. . . . 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 43543	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))) ∈ 𝐴)
79	75, 78	eqeltrd 2839	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)
80	79	adantr 481	. . 3 ⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)
81		eqidd 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑡 → 𝐸 = 𝐸)
82	81	cbvmptv 5187	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ 𝑇 ↦ 𝐸) = (𝑡 ∈ 𝑇 ↦ 𝐸)
83	82	eqcomi 2747	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑇 ↦ 𝐸) = (𝑟 ∈ 𝑇 ↦ 𝐸)
84		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
85	83, 81, 84, 43	fvmptd3 6898	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) = 𝐸)
86	85	oveq1d 7290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
87	41, 86	mpteq2da 5172	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))))
88	87	adantr 481	. . . . . . . . 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	ffvelrnda 6961	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑋‘(𝑚 + 1)) ∈ 𝐴)
90		simpl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝜑)
91		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐸 → 𝑥 = 𝐸)
92	91	mpteq2dv 5176	. . . . . . . . . . . . . . 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 341	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐸 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)))
95	77	expcom 414	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴))
96	94, 95	vtoclga 3513	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))
97	42, 96	mpcom 38	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
98	97	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
99		fveq1 6773	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑔‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
100	99	oveq2d 7291	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)))
101	100	mpteq2dv 5176	. . . . . . . . . . . . . 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 341	. . . . . . . . . . . 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 6773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑟 ∈ 𝑇 ↦ 𝐸)‘𝑡))
106	82	fveq1i 6775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸)‘𝑡) = ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡)
107	105, 106	eqtrdi 2794	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡))
108	107	oveq1d 7290	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → ((𝑓‘𝑡) · (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡)))
109	108	mpteq2dv 5176	. . . . . . . . . . . . . . . . . . 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 341	. . . . . . . . . . . . . . . . 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 3513	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
115	104, 114	sylbir 234	. . . . . . . . . . . . . . 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 3513	. . . . . . . . . . 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 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)
123	122	ad2antll 726	. . . . . . 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 778	. . . . . . 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 483	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℕ₀)
126		simprl 768	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝜑)
127	1	ad2antrl 725	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ)
128	127	nnnn0d 12293	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ₀)
129		nn0re 12242	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
130	129	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℝ)
131		peano2nn0 12273	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ₀)
132	131	nn0red 12294	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℝ)
133	132	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ∈ ℝ)
134	1	nnred 11988	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℝ)
135	134	ad2antrl 725	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℝ)
136		lep1 11816	. . . . . . . . . . . . 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 13260	. . . . . . . . . . . . 13 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ≤ 𝑁)
139	138	ad2antll 726	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ≤ 𝑁)
140	130, 133, 135, 137, 139	letrd 11132	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ≤ 𝑁)
141		elfz2nn0 13347	. . . . . . . . . . 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 516	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ 𝑚 ∈ (0...𝑁))))
144	143	adantl 482	. . . . . . . 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		pm3.31 450	. . . . . . . . 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 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))) → (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 6774	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑡 → ((𝑋‘(𝑚 + 1))‘𝑟) = ((𝑋‘(𝑚 + 1))‘𝑡))
149	148	oveq2d 7291	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑡 → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
150	149	cbvmptv 5187	. . . . . . . . . 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 6773	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔‘𝑡) = ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡))
153	150	fveq1i 6775	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡) = ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)
154	152, 153	eqtrdi 2794	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔‘𝑡) = ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))
155	154	oveq2d 7291	. . . . . . . . . . . . 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 5176	. . . . . . . . . . . 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 341	. . . . . . . . . 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 6774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝑡 → ((𝑋‘𝑖)‘𝑟) = ((𝑋‘𝑖)‘𝑡))
160	159	oveq2d 7291	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑡 → (𝐸 · ((𝑋‘𝑖)‘𝑟)) = (𝐸 · ((𝑋‘𝑖)‘𝑡)))
161	160	sumeq2sdv 15416	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑡 → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
162	161	cbvmptv 5187	. . . . . . . . . . . . . . 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 6773	. . . . . . . . . . . . . . . . . . . 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 6775	. . . . . . . . . . . . . . . . . . . 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 2794	. . . . . . . . . . . . . . . . . . 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 7290	. . . . . . . . . . . . . . . . . 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 5176	. . . . . . . . . . . . . . . . 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 341	. . . . . . . . . . . . . . 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 3513	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 3513	. . . . . . . . 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 234	. . . . . . . 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 227	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
185	184	adantl 482	. . . . . . 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 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ₀
187		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑚 + 1) ∈ (0...𝑁)
188	186, 41, 187	nf3an 1904	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))
189		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
190		fzfid 13693	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ)
193	192	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → 𝐸 ∈ ℝ)
194		fzelp1 13308	. . . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑋:(0...𝑁)⟶𝐴)
200		elfzuz3 13253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘(𝑚 + 1)))
201		fzss2 13296	. . . . . . . . . . . . . . . . . . . . 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 3921	. . . . . . . . . . . . . . . . . . 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	ffvelrnd 6962	. . . . . . . . . . . . . . . . 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 6581	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑋‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘𝑖):𝑇⟶ℝ))
208	207	imbi2d 341	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑋‘𝑖) → ((𝜑 → 𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘𝑖):𝑇⟶ℝ)))
209	208, 63	vtoclga 3513	. . . . . . . . . . . . . . . . 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	ffvelrnda 6961	. . . . . . . . . . . . . . 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 11005	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
214	190, 213	fsumrecl 15446	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
215		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
216	215	fvmpt2 6886	. . . . . . . . . . . 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 7290	. . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))
221		feq1 6581	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑋‘(𝑚 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘(𝑚 + 1)):𝑇⟶ℝ))
222	221	imbi2d 341	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑋‘(𝑚 + 1)) → ((𝜑 → 𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)))
223	222, 63	vtoclga 3513	. . . . . . . . . . . . . . . 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	ffvelrnd 6962	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑋‘(𝑚 + 1))‘𝑡) ∈ ℝ)
227	192, 226	remulcld 11005	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ)
228		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
229	228	fvmpt2 6886	. . . . . . . . . . . 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 7291	. . . . . . . . . 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 13252	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝑚 + 1) ∈ (ℤ_≥‘0))
235	192	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝐸 ∈ ℝ)
236	211	an32s 649	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ)
237		remulcl 10956	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ ℝ ∧ ((𝑋‘𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
238	237	recnd 11003	. . . . . . . . . . . . 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 6774	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑚 + 1) → (𝑋‘𝑖) = (𝑋‘(𝑚 + 1)))
241	240	fveq1d 6776	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑚 + 1) → ((𝑋‘𝑖)‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
242	241	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑚 + 1) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
243	234, 239, 242	fsumm1 15463	. . . . . . . . . . 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 12243	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
245	244	3ad2ant1 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑚 ∈ ℂ)
246	245	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℂ)
247		1cnd 10970	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℂ)
248	246, 247	pncand 11333	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑚 + 1) − 1) = 𝑚)
249	248	oveq2d 7291	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (0...((𝑚 + 1) − 1)) = (0...𝑚))
250	249	sumeq1d 15413	. . . . . . . . . . . 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 7290	. . . . . . . . . . 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 2778	. . . . . . . . . 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 2789	. . . . . . . . 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 5172	. . . . . . . 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 256	. . . . 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 421	. . . 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 12415	. 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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 ⊆ wss 3887 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 Σcsu 15397

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398

This theorem is referenced by: stoweidlem60 43601

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