stoweidlem17 - Mathbox for Glauco Siliprandi

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

Theorem stoweidlem17 43448

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 12223	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
3		nn0uz 12549	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2849	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
5		eluzfz2 13193	. . . 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 628	. . . 4 ⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
10		oveq2 7263	. . . . . . 7 ⊢ (𝑛 = 0 → (0...𝑛) = (0...0))
11	10	sumeq1d 15341	. . . . . 6 ⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
12	11	mpteq2dv 5172	. . . . 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 628	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑚 ∈ (0...𝑁))))
17		oveq2 7263	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
18	17	sumeq1d 15341	. . . . . 6 ⊢ (𝑛 = 𝑚 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
19	18	mpteq2dv 5172	. . . . 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 628	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))))
24		oveq2 7263	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (0...𝑛) = (0...(𝑚 + 1)))
25	24	sumeq1d 15341	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)))
26	25	mpteq2dv 5172	. . . . 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 628	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑁 ∈ (0...𝑁))))
31		oveq2 7263	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
32	31	sumeq1d 15341	. . . . . 6 ⊢ (𝑛 = 𝑁 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
33	32	mpteq2dv 5172	. . . . 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 12260	. . . . . . . . 9 ⊢ 0 ∈ ℤ
37		fzsn 13227	. . . . . . . . 9 ⊢ (0 ∈ ℤ → (0...0) = {0})
38	36, 37	ax-mp 5	. . . . . . . 8 ⊢ (0...0) = {0}
39	38	sumeq1i 15338	. . . . . . 7 ⊢ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡))
40	39	mpteq2i 5175	. . . . . 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 10934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℂ)
45		stoweidlem17.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝐴)
46		nnz 12272	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
47		nngt0 11934	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
48		0re 10908	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
49		nnre 11910	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
50		ltle 10994	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁))
51	48, 49, 50	sylancr 586	. . . . . . . . . . . . . . . . . 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 12519	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘0) ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
56	54, 55	sylibr 233	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
57		eluzfz1 13192	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑁))
58	56, 57	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑁))
59	45, 58	ffvelrnd 6944	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋‘0) ∈ 𝐴)
60		feq1 6565	. . . . . . . . . . . . . 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 3503	. . . . . . . . . . . 12 ⊢ ((𝑋‘0) ∈ 𝐴 → (𝜑 → (𝑋‘0):𝑇⟶ℝ))
65	59, 64	mpcom 38	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋‘0):𝑇⟶ℝ)
66	65	ffvelrnda 6943	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑋‘0)‘𝑡) ∈ ℝ)
67	66	recnd 10934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑋‘0)‘𝑡) ∈ ℂ)
68	44, 67	mulcld 10926	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ)
69		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → (𝑋‘𝑖) = (𝑋‘0))
70	69	fveq1d 6758	. . . . . . . . . 10 ⊢ (𝑖 = 0 → ((𝑋‘𝑖)‘𝑡) = ((𝑋‘0)‘𝑡))
71	70	oveq2d 7271	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
72	71	sumsn 15386	. . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
73	36, 68, 72	sylancr 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡)))
74	41, 73	mpteq2da 5168	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))))
75	40, 74	syl5eq 2791	. . . . 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 43433	. . . . 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 480	. . 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 5183	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ 𝑇 ↦ 𝐸) = (𝑡 ∈ 𝑇 ↦ 𝐸)
83	82	eqcomi 2747	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑇 ↦ 𝐸) = (𝑟 ∈ 𝑇 ↦ 𝐸)
84		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
85	83, 81, 84, 43	fvmptd3 6880	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) = 𝐸)
86	85	oveq1d 7270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
87	41, 86	mpteq2da 5168	. . . . . . . . . 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	ffvelrnda 6943	. . . . . . . . . 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 5172	. . . . . . . . . . . . . . 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 3503	. . . . . . . . . . . 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 6755	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑔‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
100	99	oveq2d 7271	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)))
101	100	mpteq2dv 5172	. . . . . . . . . . . . . 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 6755	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑟 ∈ 𝑇 ↦ 𝐸)‘𝑡))
106	82	fveq1i 6757	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸)‘𝑡) = ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡)
107	105, 106	eqtrdi 2795	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡))
108	107	oveq1d 7270	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → ((𝑓‘𝑡) · (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡)))
109	108	mpteq2dv 5172	. . . . . . . . . . . . . . . . . . 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 1121	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
113	112	3expib 1120	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
114	111, 113	vtoclga 3503	. . . . . . . . . . . . . . . 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 1114	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
117	116	3com13 1122	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ 𝐴 ∧ 𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
118	117	3expib 1120	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
119	103, 118	vtoclga 3503	. . . . . . . . . . 11 ⊢ ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴))
120	119	3impib 1114	. . . . . . . . . 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 1369	. . . . . . . . 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 725	. . . . . . 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 777	. . . . . . 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 767	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝜑)
127	1	ad2antrl 724	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ)
128	127	nnnn0d 12223	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ₀)
129		nn0re 12172	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
130	129	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℝ)
131		peano2nn0 12203	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ₀)
132	131	nn0red 12224	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℝ)
133	132	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ∈ ℝ)
134	1	nnred 11918	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℝ)
135	134	ad2antrl 724	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℝ)
136		lep1 11746	. . . . . . . . . . . . 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 13189	. . . . . . . . . . . . 13 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ≤ 𝑁)
139	138	ad2antll 725	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ≤ 𝑁)
140	130, 133, 135, 137, 139	letrd 11062	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ≤ 𝑁)
141		elfz2nn0 13276	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...𝑁) ↔ (𝑚 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑚 ≤ 𝑁))
142	125, 128, 140, 141	syl3anbrc 1341	. . . . . . . . . 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 6756	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑡 → ((𝑋‘(𝑚 + 1))‘𝑟) = ((𝑋‘(𝑚 + 1))‘𝑡))
149	148	oveq2d 7271	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑡 → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
150	149	cbvmptv 5183	. . . . . . . . . 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 6755	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔‘𝑡) = ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡))
153	150	fveq1i 6757	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡) = ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)
154	152, 153	eqtrdi 2795	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔‘𝑡) = ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))
155	154	oveq2d 7271	. . . . . . . . . . . . 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 5172	. . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝑡 → ((𝑋‘𝑖)‘𝑟) = ((𝑋‘𝑖)‘𝑡))
160	159	oveq2d 7271	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑡 → (𝐸 · ((𝑋‘𝑖)‘𝑟)) = (𝐸 · ((𝑋‘𝑖)‘𝑡)))
161	160	sumeq2sdv 15344	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑡 → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
162	161	cbvmptv 5183	. . . . . . . . . . . . . . 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 6755	. . . . . . . . . . . . . . . . . . . 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 6757	. . . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . . . 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 7270	. . . . . . . . . . . . . . . . . 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 5172	. . . . . . . . . . . . . . . . 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 1121	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
173	172	3expib 1120	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴))
174	170, 173	vtoclga 3503	. . . . . . . . . . . . . 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 1114	. . . . . . . . . . . 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 1122	. . . . . . . . . . 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 1120	. . . . . . . . . 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 3503	. . . . . . . . 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 1114	. . . . . . 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 1369	. . . . . 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 1093	. . . . . . . . 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 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 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ₀
187		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑚 + 1) ∈ (0...𝑁)
188	186, 41, 187	nf3an 1905	. . . . . . . . 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 13621	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (0...𝑚) ∈ Fin)
191	42	3ad2ant2 1132	. . . . . . . . . . . . . . . 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 13237	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0...𝑚) → 𝑖 ∈ (0...(𝑚 + 1)))
195	194	anim2i 616	. . . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . . . . 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 13182	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘(𝑚 + 1)))
201		fzss2 13225	. . . . . . . . . . . . . . . . . . . . 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 3917	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 + 1) ∈ (0...𝑁) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
204	203	3ad2antl3 1185	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁))
205	199, 204	ffvelrnd 6944	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋‘𝑖) ∈ 𝐴)
206		simpl2 1190	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝜑)
207		feq1 6565	. . . . . . . . . . . . . . . . . . 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 3503	. . . . . . . . . . . . . . . . 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 6943	. . . . . . . . . . . . . . 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 10936	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
214	190, 213	fsumrecl 15374	. . . . . . . . . . . 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 6868	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 7270	. . . . . . . . . 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 1148	. . . . . . . . . . . . . . . 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 6565	. . . . . . . . . . . . . . . . . 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 3503	. . . . . . . . . . . . . . . 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 6944	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑋‘(𝑚 + 1))‘𝑡) ∈ ℝ)
227	192, 226	remulcld 10936	. . . . . . . . . . . 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 6868	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 7271	. . . . . . . . . 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 13181	. . . . . . . . . . . . . 14 ⊢ ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ∈ (ℤ_≥‘0))
233	232	3ad2ant3 1133	. . . . . . . . . . . . 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 648	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ)
237		remulcl 10887	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ ℝ ∧ ((𝑋‘𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
238	237	recnd 10934	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℝ ∧ ((𝑋‘𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℂ)
239	235, 236, 238	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℂ)
240		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑚 + 1) → (𝑋‘𝑖) = (𝑋‘(𝑚 + 1)))
241	240	fveq1d 6758	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑚 + 1) → ((𝑋‘𝑖)‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡))
242	241	oveq2d 7271	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑚 + 1) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))
243	234, 239, 242	fsumm1 15391	. . . . . . . . . . 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 12173	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
245	244	3ad2ant1 1131	. . . . . . . . . . . . . . . 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 10901	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℂ)
248	246, 247	pncand 11263	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑚 + 1) − 1) = 𝑚)
249	248	oveq2d 7271	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (0...((𝑚 + 1) − 1)) = (0...𝑚))
250	249	sumeq1d 15341	. . . . . . . . . . . 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 7270	. . . . . . . . . . 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 5168	. . . . . . . 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 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 12345	. 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 395 ∧ w3a 1085 = wceq 1539 Ⅎwnf 1787 ∈ wcel 2108 ⊆ wss 3883 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ...cfz 13168 Σcsu 15325

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326

This theorem is referenced by: stoweidlem60 43491

W3C validator