fmuldfeq - Mathbox for Glauco Siliprandi

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

Theorem fmuldfeq 41427

Description: X and Z are two equivalent definitions of the finite product of real functions. Y is a set of real functions from a common domain T, Y is closed under function multiplication and U is a finite sequence of functions in Y. M is the number of functions multiplied together. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
fmuldfeq.1	⊢ Ⅎ𝑖𝜑
fmuldfeq.2	⊢ Ⅎ𝑡𝑌
fmuldfeq.3	⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
fmuldfeq.4	⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
fmuldfeq.5	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
fmuldfeq.6	⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
fmuldfeq.7	⊢ (𝜑 → 𝑇 ∈ V)
fmuldfeq.8	⊢ (𝜑 → 𝑀 ∈ ℕ)
fmuldfeq.9	⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
fmuldfeq.10	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)
fmuldfeq.11	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)

**Assertion**
Ref	Expression
fmuldfeq	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡))

Distinct variable groups: 𝑡,𝑇 𝑓,𝑔,𝑡,𝑇 𝑓,𝑖,𝑡,𝑇 𝑓,𝐹,𝑔 𝑓,𝑀,𝑔 𝑈,𝑓,𝑔,𝑡 𝑓,𝑌,𝑔 𝜑,𝑓,𝑔 𝑖,𝑀 𝑈,𝑖 Allowed substitution hints: 𝜑(𝑡,𝑖) 𝑃(𝑡,𝑓,𝑔,𝑖) 𝐹(𝑡,𝑖) 𝑀(𝑡) 𝑋(𝑡,𝑓,𝑔,𝑖) 𝑌(𝑡,𝑖) 𝑍(𝑡,𝑓,𝑔,𝑖)

Proof of Theorem fmuldfeq Dummy variables 𝑘 𝑏 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmuldfeq.8	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2	1	nnge1d 11539	. . . . 5 ⊢ (𝜑 → 1 ≤ 𝑀)
3	2	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ≤ 𝑀)
4		nnre 11499	. . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
5		leid 10589	. . . . . 6 ⊢ (𝑀 ∈ ℝ → 𝑀 ≤ 𝑀)
6	1, 4, 5	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑀 ≤ 𝑀)
8	1	nnzd 11940	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑀 ∈ ℤ)
10		1zzd 11867	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℤ)
11		elfz 12752	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (1...𝑀) ↔ (1 ≤ 𝑀 ∧ 𝑀 ≤ 𝑀)))
12	9, 10, 9, 11	syl3anc 1364	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑀 ∈ (1...𝑀) ↔ (1 ≤ 𝑀 ∧ 𝑀 ≤ 𝑀)))
13	3, 7, 12	mpbir2and 709	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑀 ∈ (1...𝑀))
14	1	3ad2ant1 1126	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑀 ∈ (1...𝑀)) → 𝑀 ∈ ℕ)
15		eleq1 2872	. . . . . . 7 ⊢ (𝑚 = 1 → (𝑚 ∈ (1...𝑀) ↔ 1 ∈ (1...𝑀)))
16	15	3anbi3d 1434	. . . . . 6 ⊢ (𝑚 = 1 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑚 ∈ (1...𝑀)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 1 ∈ (1...𝑀))))
17		fveq2 6545	. . . . . . . 8 ⊢ (𝑚 = 1 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘1))
18	17	fveq1d 6547	. . . . . . 7 ⊢ (𝑚 = 1 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘1)‘𝑡))
19		fveq2 6545	. . . . . . 7 ⊢ (𝑚 = 1 → (seq1( · , (𝐹‘𝑡))‘𝑚) = (seq1( · , (𝐹‘𝑡))‘1))
20	18, 19	eqeq12d 2812	. . . . . 6 ⊢ (𝑚 = 1 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘1)))
21	16, 20	imbi12d 346	. . . . 5 ⊢ (𝑚 = 1 → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑚 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑚)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 1 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘1))))
22		eleq1 2872	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑀) ↔ 𝑛 ∈ (1...𝑀)))
23	22	3anbi3d 1434	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑚 ∈ (1...𝑀)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀))))
24		fveq2 6545	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘𝑛))
25	24	fveq1d 6547	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡))
26		fveq2 6545	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (seq1( · , (𝐹‘𝑡))‘𝑚) = (seq1( · , (𝐹‘𝑡))‘𝑛))
27	25, 26	eqeq12d 2812	. . . . . 6 ⊢ (𝑚 = 𝑛 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)))
28	23, 27	imbi12d 346	. . . . 5 ⊢ (𝑚 = 𝑛 → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑚 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑚)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))))
29		eleq1 2872	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 ∈ (1...𝑀) ↔ (𝑛 + 1) ∈ (1...𝑀)))
30	29	3anbi3d 1434	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑚 ∈ (1...𝑀)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))))
31		fveq2 6545	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘(𝑛 + 1)))
32	31	fveq1d 6547	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡))
33		fveq2 6545	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (seq1( · , (𝐹‘𝑡))‘𝑚) = (seq1( · , (𝐹‘𝑡))‘(𝑛 + 1)))
34	32, 33	eqeq12d 2812	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹‘𝑡))‘(𝑛 + 1))))
35	30, 34	imbi12d 346	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑚 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑚)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹‘𝑡))‘(𝑛 + 1)))))
36		eleq1 2872	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚 ∈ (1...𝑀) ↔ 𝑀 ∈ (1...𝑀)))
37	36	3anbi3d 1434	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑚 ∈ (1...𝑀)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑀 ∈ (1...𝑀))))
38		fveq2 6545	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘𝑀))
39	38	fveq1d 6547	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡))
40		fveq2 6545	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (seq1( · , (𝐹‘𝑡))‘𝑚) = (seq1( · , (𝐹‘𝑡))‘𝑀))
41	39, 40	eqeq12d 2812	. . . . . 6 ⊢ (𝑚 = 𝑀 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀)))
42	37, 41	imbi12d 346	. . . . 5 ⊢ (𝑚 = 𝑀 → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑚 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑚)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))))
43		1z 11866	. . . . . . . 8 ⊢ 1 ∈ ℤ
44		seq1 13236	. . . . . . . 8 ⊢ (1 ∈ ℤ → (seq1( · , (𝐹‘𝑡))‘1) = ((𝐹‘𝑡)‘1))
45	43, 44	ax-mp 5	. . . . . . 7 ⊢ (seq1( · , (𝐹‘𝑡))‘1) = ((𝐹‘𝑡)‘1)
46		1le1 11122	. . . . . . . . . . . . 13 ⊢ 1 ≤ 1
47	46	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 1)
48		1zzd 11867	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℤ)
49		elfz 12752	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (1 ∈ (1...𝑀) ↔ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
50	48, 48, 8, 49	syl3anc 1364	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 ∈ (1...𝑀) ↔ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
51	47, 2, 50	mpbir2and 709	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ (1...𝑀))
52		nfv 1896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑡 ∈ 𝑇
53		fmuldfeq.5	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
54		nfcv 2951	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖𝑇
55		nfmpt1 5065	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
56	54, 55	nfmpt 5064	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖(𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
57	53, 56	nfcxfr 2949	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖𝐹
58		nfcv 2951	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖𝑡
59	57, 58	nffv 6555	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖(𝐹‘𝑡)
60		nfcv 2951	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖1
61	59, 60	nffv 6555	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘1)
62		nffvmpt1 6556	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘1)
63	61, 62	nfeq 2962	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘1)
64	52, 63	nfim 1882	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑡 ∈ 𝑇 → ((𝐹‘𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘1))
65		fveq2 6545	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 1 → ((𝐹‘𝑡)‘𝑖) = ((𝐹‘𝑡)‘1))
66		fveq2 6545	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 1 → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘1))
67	65, 66	eqeq12d 2812	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (((𝐹‘𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖) ↔ ((𝐹‘𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘1)))
68	67	imbi2d 342	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → ((𝑡 ∈ 𝑇 → ((𝐹‘𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖)) ↔ (𝑡 ∈ 𝑇 → ((𝐹‘𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘1))))
69		ovex 7055	. . . . . . . . . . . . . . 15 ⊢ (1...𝑀) ∈ V
70	69	mptex 6859	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V
71	53	fvmpt2 6652	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
72	70, 71	mpan2 687	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑇 → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
73	72	fveq1d 6547	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝑇 → ((𝐹‘𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖))
74	64, 68, 73	vtoclg1f 3512	. . . . . . . . . . 11 ⊢ (1 ∈ (1...𝑀) → (𝑡 ∈ 𝑇 → ((𝐹‘𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘1)))
75	51, 74	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → ((𝐹‘𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘1)))
76	75	imp 407	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘1))
77		eqid 2797	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
78		fveq2 6545	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → (𝑈‘𝑖) = (𝑈‘1))
79	78	fveq1d 6547	. . . . . . . . . 10 ⊢ (𝑖 = 1 → ((𝑈‘𝑖)‘𝑡) = ((𝑈‘1)‘𝑡))
80	51	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ (1...𝑀))
81		fmuldfeq.9	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
82	81, 51	ffvelrnd 6724	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈‘1) ∈ 𝑌)
83	82	ancli 549	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝜑 ∧ (𝑈‘1) ∈ 𝑌))
84		eleq1 2872	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘1) → (𝑓 ∈ 𝑌 ↔ (𝑈‘1) ∈ 𝑌))
85	84	anbi2d 628	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑈‘1) → ((𝜑 ∧ 𝑓 ∈ 𝑌) ↔ (𝜑 ∧ (𝑈‘1) ∈ 𝑌)))
86		feq1 6370	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑈‘1) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘1):𝑇⟶ℝ))
87	85, 86	imbi12d 346	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑈‘1) → (((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘1) ∈ 𝑌) → (𝑈‘1):𝑇⟶ℝ)))
88		fmuldfeq.10	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)
89	88	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝑌 → ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ))
90	87, 89	vtoclga 3519	. . . . . . . . . . . 12 ⊢ ((𝑈‘1) ∈ 𝑌 → ((𝜑 ∧ (𝑈‘1) ∈ 𝑌) → (𝑈‘1):𝑇⟶ℝ))
91	82, 83, 90	sylc 65	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈‘1):𝑇⟶ℝ)
92	91	ffvelrnda 6723	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑈‘1)‘𝑡) ∈ ℝ)
93	77, 79, 80, 92	fvmptd3 6664	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘1) = ((𝑈‘1)‘𝑡))
94	76, 93	eqtrd 2833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)‘1) = ((𝑈‘1)‘𝑡))
95		seq1 13236	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → (seq1(𝑃, 𝑈)‘1) = (𝑈‘1))
96	43, 95	ax-mp 5	. . . . . . . . 9 ⊢ (seq1(𝑃, 𝑈)‘1) = (𝑈‘1)
97	96	fveq1i 6546	. . . . . . . 8 ⊢ ((seq1(𝑃, 𝑈)‘1)‘𝑡) = ((𝑈‘1)‘𝑡)
98	94, 97	syl6eqr 2851	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)‘1) = ((seq1(𝑃, 𝑈)‘1)‘𝑡))
99	45, 98	syl5req 2846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘1))
100	99	3adant3 1125	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 1 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘1))
101		simp31 1202	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)) ∧ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝜑)
102		simp1 1129	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)) ∧ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑛 ∈ ℕ)
103		simp33 1204	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)) ∧ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → (𝑛 + 1) ∈ (1...𝑀))
104	102, 103	jca 512	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)) ∧ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → (𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑀)))
105		elnnuz 12135	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
106	105	biimpi 217	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘1))
107	106	anim1i 614	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑀)) → (𝑛 ∈ (ℤ_≥‘1) ∧ (𝑛 + 1) ∈ (1...𝑀)))
108		peano2fzr 12774	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ_≥‘1) ∧ (𝑛 + 1) ∈ (1...𝑀)) → 𝑛 ∈ (1...𝑀))
109	104, 107, 108	3syl 18	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)) ∧ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑛 ∈ (1...𝑀))
110		simp32 1203	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)) ∧ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑡 ∈ 𝑇)
111		simp2 1130	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)) ∧ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)))
112	101, 110, 109, 111	mp3and 1456	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)) ∧ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))
113	109, 103, 112	3jca 1121	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)) ∧ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)))
114		nfv 1896	. . . . . . . . 9 ⊢ Ⅎ𝑓𝜑
115		nfv 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑓 𝑛 ∈ (1...𝑀)
116		nfv 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑓(𝑛 + 1) ∈ (1...𝑀)
117		nfcv 2951	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓1
118		fmuldfeq.3	. . . . . . . . . . . . . . 15 ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
119		nfmpo1 7099	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑓(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
120	118, 119	nfcxfr 2949	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓𝑃
121		nfcv 2951	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓𝑈
122	117, 120, 121	nfseq 13233	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑓seq1(𝑃, 𝑈)
123		nfcv 2951	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑓𝑛
124	122, 123	nffv 6555	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓(seq1(𝑃, 𝑈)‘𝑛)
125		nfcv 2951	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓𝑡
126	124, 125	nffv 6555	. . . . . . . . . . 11 ⊢ Ⅎ𝑓((seq1(𝑃, 𝑈)‘𝑛)‘𝑡)
127		nfcv 2951	. . . . . . . . . . 11 ⊢ Ⅎ𝑓(seq1( · , (𝐹‘𝑡))‘𝑛)
128	126, 127	nfeq 2962	. . . . . . . . . 10 ⊢ Ⅎ𝑓((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)
129	115, 116, 128	nf3an 1887	. . . . . . . . 9 ⊢ Ⅎ𝑓(𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))
130	114, 129	nfan 1885	. . . . . . . 8 ⊢ Ⅎ𝑓(𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)))
131		nfv 1896	. . . . . . . . 9 ⊢ Ⅎ𝑔𝜑
132		nfv 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑔 𝑛 ∈ (1...𝑀)
133		nfv 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑔(𝑛 + 1) ∈ (1...𝑀)
134		nfcv 2951	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑔1
135		nfmpo2 7100	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑔(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
136	118, 135	nfcxfr 2949	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑔𝑃
137		nfcv 2951	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑔𝑈
138	134, 136, 137	nfseq 13233	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑔seq1(𝑃, 𝑈)
139		nfcv 2951	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑔𝑛
140	138, 139	nffv 6555	. . . . . . . . . . . 12 ⊢ Ⅎ𝑔(seq1(𝑃, 𝑈)‘𝑛)
141		nfcv 2951	. . . . . . . . . . . 12 ⊢ Ⅎ𝑔𝑡
142	140, 141	nffv 6555	. . . . . . . . . . 11 ⊢ Ⅎ𝑔((seq1(𝑃, 𝑈)‘𝑛)‘𝑡)
143		nfcv 2951	. . . . . . . . . . 11 ⊢ Ⅎ𝑔(seq1( · , (𝐹‘𝑡))‘𝑛)
144	142, 143	nfeq 2962	. . . . . . . . . 10 ⊢ Ⅎ𝑔((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)
145	132, 133, 144	nf3an 1887	. . . . . . . . 9 ⊢ Ⅎ𝑔(𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))
146	131, 145	nfan 1885	. . . . . . . 8 ⊢ Ⅎ𝑔(𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)))
147		fmuldfeq.2	. . . . . . . 8 ⊢ Ⅎ𝑡𝑌
148		fmuldfeq.7	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ V)
149	148	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))) → 𝑇 ∈ V)
150	81	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))) → 𝑈:(1...𝑀)⟶𝑌)
151		fmuldfeq.11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)
152	151	3adant1r 1170	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))) ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)
153		simpr1 1187	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))) → 𝑛 ∈ (1...𝑀))
154		simpr2 1188	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))) → (𝑛 + 1) ∈ (1...𝑀))
155		simpr3 1189	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))
156	88	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))) ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)
157	130, 146, 147, 118, 53, 149, 150, 152, 153, 154, 155, 156	fmuldfeqlem1 41426	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛))) ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹‘𝑡))‘(𝑛 + 1)))
158	101, 113, 110, 157	syl21anc 834	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)) ∧ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹‘𝑡))‘(𝑛 + 1)))
159	158	3exp 1112	. . . . 5 ⊢ (𝑛 ∈ ℕ → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑛)) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹‘𝑡))‘(𝑛 + 1)))))
160	21, 28, 35, 42, 100, 159	nnind 11510	. . . 4 ⊢ (𝑀 ∈ ℕ → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀)))
161	14, 160	mpcom 38	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
162	13, 161	mpd3an3 1454	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
163		fmuldfeq.4	. . . 4 ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
164	163	fveq1i 6546	. . 3 ⊢ (𝑋‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡)
165	164	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡))
166		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
167		elnnuz 12135	. . . . . 6 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
168	1, 167	sylib 219	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
169	168	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑀 ∈ (ℤ_≥‘1))
170		fmuldfeq.1	. . . . . . . 8 ⊢ Ⅎ𝑖𝜑
171	170, 52	nfan 1885	. . . . . . 7 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑡 ∈ 𝑇)
172		nfv 1896	. . . . . . 7 ⊢ Ⅎ𝑖 𝑘 ∈ (1...𝑀)
173	171, 172	nfan 1885	. . . . . 6 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑘 ∈ (1...𝑀))
174		nfcv 2951	. . . . . . . 8 ⊢ Ⅎ𝑖𝑘
175	59, 174	nffv 6555	. . . . . . 7 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑘)
176	175	nfel1 2965	. . . . . 6 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑘) ∈ ℝ
177	173, 176	nfim 1882	. . . . 5 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑘) ∈ ℝ)
178		eleq1 2872	. . . . . . 7 ⊢ (𝑖 = 𝑘 → (𝑖 ∈ (1...𝑀) ↔ 𝑘 ∈ (1...𝑀)))
179	178	anbi2d 628	. . . . . 6 ⊢ (𝑖 = 𝑘 → (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑘 ∈ (1...𝑀))))
180		fveq2 6545	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ((𝐹‘𝑡)‘𝑖) = ((𝐹‘𝑡)‘𝑘))
181	180	eleq1d 2869	. . . . . 6 ⊢ (𝑖 = 𝑘 → (((𝐹‘𝑡)‘𝑖) ∈ ℝ ↔ ((𝐹‘𝑡)‘𝑘) ∈ ℝ))
182	179, 181	imbi12d 346	. . . . 5 ⊢ (𝑖 = 𝑘 → ((((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) ∈ ℝ) ↔ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑘) ∈ ℝ)))
183	73	ad2antlr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖))
184		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑀))
185	81	ffvelrnda 6723	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌)
186		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
187	186, 185	jca 512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌))
188		eleq1 2872	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓 ∈ 𝑌 ↔ (𝑈‘𝑖) ∈ 𝑌))
189	188	anbi2d 628	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑈‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝑌) ↔ (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌)))
190		feq1 6370	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘𝑖):𝑇⟶ℝ))
191	189, 190	imbi12d 346	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑈‘𝑖) → (((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌) → (𝑈‘𝑖):𝑇⟶ℝ)))
192	191, 89	vtoclga 3519	. . . . . . . . . . 11 ⊢ ((𝑈‘𝑖) ∈ 𝑌 → ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌) → (𝑈‘𝑖):𝑇⟶ℝ))
193	185, 187, 192	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
194	193	adantlr 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
195		simplr 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡 ∈ 𝑇)
196	194, 195	ffvelrnd 6724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
197	77	fvmpt2 6652	. . . . . . . 8 ⊢ ((𝑖 ∈ (1...𝑀) ∧ ((𝑈‘𝑖)‘𝑡) ∈ ℝ) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖) = ((𝑈‘𝑖)‘𝑡))
198	184, 196, 197	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖) = ((𝑈‘𝑖)‘𝑡))
199	198, 196	eqeltrd 2885	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖) ∈ ℝ)
200	183, 199	eqeltrd 2885	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) ∈ ℝ)
201	177, 182, 200	chvar 2371	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑘) ∈ ℝ)
202		remulcl 10475	. . . . 5 ⊢ ((𝑘 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑘 · 𝑏) ∈ ℝ)
203	202	adantl 482	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑘 · 𝑏) ∈ ℝ)
204	169, 201, 203	seqcl 13244	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ)
205		fmuldfeq.6	. . . 4 ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
206	205	fvmpt2 6652	. . 3 ⊢ ((𝑡 ∈ 𝑇 ∧ (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
207	166, 204, 206	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
208	162, 165, 207	3eqtr4d 2843	1 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1525 Ⅎwnf 1769 ∈ wcel 2083 Ⅎwnfc 2935 Vcvv 3440 class class class wbr 4968 ↦ cmpt 5047 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 ∈ cmpo 7025 ℝcr 10389 1c1 10391 + caddc 10393 · cmul 10395 ≤ cle 10529 ℕcn 11492 ℤcz 11835 ℤ_≥cuz 12097 ...cfz 12746 seqcseq 13223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-n0 11752 df-z 11836 df-uz 12098 df-fz 12747 df-seq 13224

This theorem is referenced by: stoweidlem42 41891 stoweidlem48 41897

W3C validator