fmuldfeqlem1 - Mathbox for Glauco Siliprandi

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Theorem fmuldfeqlem1 45503

Description: induction step for the proof of fmuldfeq 45504. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
fmuldfeqlem1.1	⊢ Ⅎ𝑓𝜑
fmuldfeqlem1.2	⊢ Ⅎ𝑔𝜑
fmuldfeqlem1.3	⊢ Ⅎ𝑡𝑌
fmuldfeqlem1.5	⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
fmuldfeqlem1.6	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
fmuldfeqlem1.7	⊢ (𝜑 → 𝑇 ∈ V)
fmuldfeqlem1.8	⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
fmuldfeqlem1.9	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)
fmuldfeqlem1.10	⊢ (𝜑 → 𝑁 ∈ (1...𝑀))
fmuldfeqlem1.11	⊢ (𝜑 → (𝑁 + 1) ∈ (1...𝑀))
fmuldfeqlem1.12	⊢ (𝜑 → ((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑁))
fmuldfeqlem1.13	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)

**Assertion**
Ref	Expression
fmuldfeqlem1	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘(𝑁 + 1))‘𝑡) = (seq1( · , (𝐹‘𝑡))‘(𝑁 + 1)))

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

Proof of Theorem fmuldfeqlem1 Dummy variables ℎ 𝑙 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7481	. . . . . . . 8 ⊢ (1...𝑀) ∈ V
2	1	mptex 7260	. . . . . . 7 ⊢ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V
3		fmuldfeqlem1.6	. . . . . . . 8 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
4	3	fvmpt2 7040	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
5	2, 4	mpan2 690	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
6		fveq2 6920	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑈‘𝑖) = (𝑈‘𝑗))
7	6	fveq1d 6922	. . . . . . 7 ⊢ (𝑖 = 𝑗 → ((𝑈‘𝑖)‘𝑡) = ((𝑈‘𝑗)‘𝑡))
8	7	cbvmptv 5279	. . . . . 6 ⊢ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) = (𝑗 ∈ (1...𝑀) ↦ ((𝑈‘𝑗)‘𝑡))
9	5, 8	eqtrdi 2796	. . . . 5 ⊢ (𝑡 ∈ 𝑇 → (𝐹‘𝑡) = (𝑗 ∈ (1...𝑀) ↦ ((𝑈‘𝑗)‘𝑡)))
10	9	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = (𝑗 ∈ (1...𝑀) ↦ ((𝑈‘𝑗)‘𝑡)))
11		fveq2 6920	. . . . . 6 ⊢ (𝑗 = (𝑁 + 1) → (𝑈‘𝑗) = (𝑈‘(𝑁 + 1)))
12	11	fveq1d 6922	. . . . 5 ⊢ (𝑗 = (𝑁 + 1) → ((𝑈‘𝑗)‘𝑡) = ((𝑈‘(𝑁 + 1))‘𝑡))
13	12	adantl 481	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑗 = (𝑁 + 1)) → ((𝑈‘𝑗)‘𝑡) = ((𝑈‘(𝑁 + 1))‘𝑡))
14		fmuldfeqlem1.11	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (1...𝑀))
15	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑁 + 1) ∈ (1...𝑀))
16		fmuldfeqlem1.8	. . . . . . 7 ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
17	16, 14	ffvelcdmd 7119	. . . . . 6 ⊢ (𝜑 → (𝑈‘(𝑁 + 1)) ∈ 𝑌)
18	17	ancli 548	. . . . . 6 ⊢ (𝜑 → (𝜑 ∧ (𝑈‘(𝑁 + 1)) ∈ 𝑌))
19		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑓(𝑈‘(𝑁 + 1))
20		fmuldfeqlem1.1	. . . . . . . . 9 ⊢ Ⅎ𝑓𝜑
21		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑓(𝑈‘(𝑁 + 1)) ∈ 𝑌
22	20, 21	nfan 1898	. . . . . . . 8 ⊢ Ⅎ𝑓(𝜑 ∧ (𝑈‘(𝑁 + 1)) ∈ 𝑌)
23		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑓(𝑈‘(𝑁 + 1)):𝑇⟶ℝ
24	22, 23	nfim 1895	. . . . . . 7 ⊢ Ⅎ𝑓((𝜑 ∧ (𝑈‘(𝑁 + 1)) ∈ 𝑌) → (𝑈‘(𝑁 + 1)):𝑇⟶ℝ)
25		eleq1 2832	. . . . . . . . 9 ⊢ (𝑓 = (𝑈‘(𝑁 + 1)) → (𝑓 ∈ 𝑌 ↔ (𝑈‘(𝑁 + 1)) ∈ 𝑌))
26	25	anbi2d 629	. . . . . . . 8 ⊢ (𝑓 = (𝑈‘(𝑁 + 1)) → ((𝜑 ∧ 𝑓 ∈ 𝑌) ↔ (𝜑 ∧ (𝑈‘(𝑁 + 1)) ∈ 𝑌)))
27		feq1 6728	. . . . . . . 8 ⊢ (𝑓 = (𝑈‘(𝑁 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘(𝑁 + 1)):𝑇⟶ℝ))
28	26, 27	imbi12d 344	. . . . . . 7 ⊢ (𝑓 = (𝑈‘(𝑁 + 1)) → (((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘(𝑁 + 1)) ∈ 𝑌) → (𝑈‘(𝑁 + 1)):𝑇⟶ℝ)))
29		fmuldfeqlem1.13	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)
30	19, 24, 28, 29	vtoclgf 3581	. . . . . 6 ⊢ ((𝑈‘(𝑁 + 1)) ∈ 𝑌 → ((𝜑 ∧ (𝑈‘(𝑁 + 1)) ∈ 𝑌) → (𝑈‘(𝑁 + 1)):𝑇⟶ℝ))
31	17, 18, 30	sylc 65	. . . . 5 ⊢ (𝜑 → (𝑈‘(𝑁 + 1)):𝑇⟶ℝ)
32	31	ffvelcdmda 7118	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑈‘(𝑁 + 1))‘𝑡) ∈ ℝ)
33	10, 13, 15, 32	fvmptd 7036	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)‘(𝑁 + 1)) = ((𝑈‘(𝑁 + 1))‘𝑡))
34	33	oveq2d 7464	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1( · , (𝐹‘𝑡))‘𝑁) · ((𝐹‘𝑡)‘(𝑁 + 1))) = ((seq1( · , (𝐹‘𝑡))‘𝑁) · ((𝑈‘(𝑁 + 1))‘𝑡)))
35		fmuldfeqlem1.10	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (1...𝑀))
36		elfzuz 13580	. . . . 5 ⊢ (𝑁 ∈ (1...𝑀) → 𝑁 ∈ (ℤ_≥‘1))
37	35, 36	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
38		seqp1 14067	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘1) → (seq1( · , (𝐹‘𝑡))‘(𝑁 + 1)) = ((seq1( · , (𝐹‘𝑡))‘𝑁) · ((𝐹‘𝑡)‘(𝑁 + 1))))
39	37, 38	syl 17	. . 3 ⊢ (𝜑 → (seq1( · , (𝐹‘𝑡))‘(𝑁 + 1)) = ((seq1( · , (𝐹‘𝑡))‘𝑁) · ((𝐹‘𝑡)‘(𝑁 + 1))))
40	39	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (seq1( · , (𝐹‘𝑡))‘(𝑁 + 1)) = ((seq1( · , (𝐹‘𝑡))‘𝑁) · ((𝐹‘𝑡)‘(𝑁 + 1))))
41		seqp1 14067	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘1) → (seq1(𝑃, 𝑈)‘(𝑁 + 1)) = ((seq1(𝑃, 𝑈)‘𝑁)𝑃(𝑈‘(𝑁 + 1))))
42	37, 41	syl 17	. . . . 5 ⊢ (𝜑 → (seq1(𝑃, 𝑈)‘(𝑁 + 1)) = ((seq1(𝑃, 𝑈)‘𝑁)𝑃(𝑈‘(𝑁 + 1))))
43		fmuldfeqlem1.5	. . . . . . . 8 ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
44		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎℎ(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
45		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑙(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
46		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑓(𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡)))
47		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑔(𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡)))
48		fveq1 6919	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘𝑡) = (ℎ‘𝑡))
49		fveq1 6919	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑙 → (𝑔‘𝑡) = (𝑙‘𝑡))
50	48, 49	oveqan12d 7467	. . . . . . . . . 10 ⊢ ((𝑓 = ℎ ∧ 𝑔 = 𝑙) → ((𝑓‘𝑡) · (𝑔‘𝑡)) = ((ℎ‘𝑡) · (𝑙‘𝑡)))
51	50	mpteq2dv 5268	. . . . . . . . 9 ⊢ ((𝑓 = ℎ ∧ 𝑔 = 𝑙) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))))
52	44, 45, 46, 47, 51	cbvmpo 7544	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) = (ℎ ∈ 𝑌, 𝑙 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))))
53	43, 52	eqtri 2768	. . . . . . 7 ⊢ 𝑃 = (ℎ ∈ 𝑌, 𝑙 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))))
54	53	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑃 = (ℎ ∈ 𝑌, 𝑙 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡)))))
55		nfcv 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡1
56		fmuldfeqlem1.3	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡𝑌
57		nfmpt1 5274	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
58	56, 56, 57	nfmpo 7532	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
59	43, 58	nfcxfr 2906	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝑃
60		nfcv 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝑈
61	55, 59, 60	nfseq 14062	. . . . . . . . . . 11 ⊢ Ⅎ𝑡seq1(𝑃, 𝑈)
62		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑁
63	61, 62	nffv 6930	. . . . . . . . . 10 ⊢ Ⅎ𝑡(seq1(𝑃, 𝑈)‘𝑁)
64	63	nfeq2 2926	. . . . . . . . 9 ⊢ Ⅎ𝑡 ℎ = (seq1(𝑃, 𝑈)‘𝑁)
65		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑙 = (𝑈‘(𝑁 + 1))
66	64, 65	nfan 1898	. . . . . . . 8 ⊢ Ⅎ𝑡(ℎ = (seq1(𝑃, 𝑈)‘𝑁) ∧ 𝑙 = (𝑈‘(𝑁 + 1)))
67		fveq1 6919	. . . . . . . . . 10 ⊢ (ℎ = (seq1(𝑃, 𝑈)‘𝑁) → (ℎ‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑁)‘𝑡))
68	67	ad2antrr 725	. . . . . . . . 9 ⊢ (((ℎ = (seq1(𝑃, 𝑈)‘𝑁) ∧ 𝑙 = (𝑈‘(𝑁 + 1))) ∧ 𝑡 ∈ 𝑇) → (ℎ‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑁)‘𝑡))
69		fveq1 6919	. . . . . . . . . 10 ⊢ (𝑙 = (𝑈‘(𝑁 + 1)) → (𝑙‘𝑡) = ((𝑈‘(𝑁 + 1))‘𝑡))
70	69	ad2antlr 726	. . . . . . . . 9 ⊢ (((ℎ = (seq1(𝑃, 𝑈)‘𝑁) ∧ 𝑙 = (𝑈‘(𝑁 + 1))) ∧ 𝑡 ∈ 𝑇) → (𝑙‘𝑡) = ((𝑈‘(𝑁 + 1))‘𝑡))
71	68, 70	oveq12d 7466	. . . . . . . 8 ⊢ (((ℎ = (seq1(𝑃, 𝑈)‘𝑁) ∧ 𝑙 = (𝑈‘(𝑁 + 1))) ∧ 𝑡 ∈ 𝑇) → ((ℎ‘𝑡) · (𝑙‘𝑡)) = (((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) · ((𝑈‘(𝑁 + 1))‘𝑡)))
72	66, 71	mpteq2da 5264	. . . . . . 7 ⊢ ((ℎ = (seq1(𝑃, 𝑈)‘𝑁) ∧ 𝑙 = (𝑈‘(𝑁 + 1))) → (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) · ((𝑈‘(𝑁 + 1))‘𝑡))))
73	72	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (ℎ = (seq1(𝑃, 𝑈)‘𝑁) ∧ 𝑙 = (𝑈‘(𝑁 + 1)))) → (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) · ((𝑈‘(𝑁 + 1))‘𝑡))))
74		eqid 2740	. . . . . . 7 ⊢ (seq1(𝑃, 𝑈)‘𝑁) = (seq1(𝑃, 𝑈)‘𝑁)
75		3simpc 1150	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑙 ∈ 𝑌) → (ℎ ∈ 𝑌 ∧ 𝑙 ∈ 𝑌))
76		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑓ℎ
77		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑔ℎ
78		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑔𝑙
79		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑓 ℎ ∈ 𝑌
80		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑓 𝑔 ∈ 𝑌
81	20, 79, 80	nf3an 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑓(𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑔 ∈ 𝑌)
82		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑓(𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌
83	81, 82	nfim 1895	. . . . . . . . 9 ⊢ Ⅎ𝑓((𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)
84		fmuldfeqlem1.2	. . . . . . . . . . 11 ⊢ Ⅎ𝑔𝜑
85		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑔 ℎ ∈ 𝑌
86		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑔 𝑙 ∈ 𝑌
87	84, 85, 86	nf3an 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑔(𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑙 ∈ 𝑌)
88		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑔(𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))) ∈ 𝑌
89	87, 88	nfim 1895	. . . . . . . . 9 ⊢ Ⅎ𝑔((𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑙 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))) ∈ 𝑌)
90		eleq1 2832	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓 ∈ 𝑌 ↔ ℎ ∈ 𝑌))
91	90	3anbi2d 1441	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ↔ (𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑔 ∈ 𝑌)))
92	48	oveq1d 7463	. . . . . . . . . . . 12 ⊢ (𝑓 = ℎ → ((𝑓‘𝑡) · (𝑔‘𝑡)) = ((ℎ‘𝑡) · (𝑔‘𝑡)))
93	92	mpteq2dv 5268	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑔‘𝑡))))
94	93	eleq1d 2829	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌 ↔ (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌))
95	91, 94	imbi12d 344	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌) ↔ ((𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)))
96		eleq1 2832	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑙 → (𝑔 ∈ 𝑌 ↔ 𝑙 ∈ 𝑌))
97	96	3anbi3d 1442	. . . . . . . . . 10 ⊢ (𝑔 = 𝑙 → ((𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ↔ (𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑙 ∈ 𝑌)))
98	49	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑙 → ((ℎ‘𝑡) · (𝑔‘𝑡)) = ((ℎ‘𝑡) · (𝑙‘𝑡)))
99	98	mpteq2dv 5268	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑙 → (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))))
100	99	eleq1d 2829	. . . . . . . . . 10 ⊢ (𝑔 = 𝑙 → ((𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌 ↔ (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))) ∈ 𝑌))
101	97, 100	imbi12d 344	. . . . . . . . 9 ⊢ (𝑔 = 𝑙 → (((𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌) ↔ ((𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑙 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))) ∈ 𝑌)))
102		fmuldfeqlem1.9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)
103	76, 77, 78, 83, 89, 95, 101, 102	vtocl2gf 3584	. . . . . . . 8 ⊢ ((ℎ ∈ 𝑌 ∧ 𝑙 ∈ 𝑌) → ((𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑙 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))) ∈ 𝑌))
104	75, 103	mpcom 38	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ 𝑌 ∧ 𝑙 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) · (𝑙‘𝑡))) ∈ 𝑌)
105		fmuldfeqlem1.7	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ V)
106	53, 74, 35, 16, 104, 105	fmulcl 45502	. . . . . 6 ⊢ (𝜑 → (seq1(𝑃, 𝑈)‘𝑁) ∈ 𝑌)
107		mptexg 7258	. . . . . . 7 ⊢ (𝑇 ∈ V → (𝑡 ∈ 𝑇 ↦ (((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) · ((𝑈‘(𝑁 + 1))‘𝑡))) ∈ V)
108	105, 107	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) · ((𝑈‘(𝑁 + 1))‘𝑡))) ∈ V)
109	54, 73, 106, 17, 108	ovmpod 7602	. . . . 5 ⊢ (𝜑 → ((seq1(𝑃, 𝑈)‘𝑁)𝑃(𝑈‘(𝑁 + 1))) = (𝑡 ∈ 𝑇 ↦ (((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) · ((𝑈‘(𝑁 + 1))‘𝑡))))
110	42, 109	eqtrd 2780	. . . 4 ⊢ (𝜑 → (seq1(𝑃, 𝑈)‘(𝑁 + 1)) = (𝑡 ∈ 𝑇 ↦ (((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) · ((𝑈‘(𝑁 + 1))‘𝑡))))
111	106	ancli 548	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∧ (seq1(𝑃, 𝑈)‘𝑁) ∈ 𝑌))
112		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑓1
113		nfmpo1 7530	. . . . . . . . . . 11 ⊢ Ⅎ𝑓(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
114	43, 113	nfcxfr 2906	. . . . . . . . . 10 ⊢ Ⅎ𝑓𝑃
115		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑓𝑈
116	112, 114, 115	nfseq 14062	. . . . . . . . 9 ⊢ Ⅎ𝑓seq1(𝑃, 𝑈)
117		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑓𝑁
118	116, 117	nffv 6930	. . . . . . . 8 ⊢ Ⅎ𝑓(seq1(𝑃, 𝑈)‘𝑁)
119	118	nfel1 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑓(seq1(𝑃, 𝑈)‘𝑁) ∈ 𝑌
120	20, 119	nfan 1898	. . . . . . . . 9 ⊢ Ⅎ𝑓(𝜑 ∧ (seq1(𝑃, 𝑈)‘𝑁) ∈ 𝑌)
121		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑓𝑇
122		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑓ℝ
123	118, 121, 122	nff 6743	. . . . . . . . 9 ⊢ Ⅎ𝑓(seq1(𝑃, 𝑈)‘𝑁):𝑇⟶ℝ
124	120, 123	nfim 1895	. . . . . . . 8 ⊢ Ⅎ𝑓((𝜑 ∧ (seq1(𝑃, 𝑈)‘𝑁) ∈ 𝑌) → (seq1(𝑃, 𝑈)‘𝑁):𝑇⟶ℝ)
125		eleq1 2832	. . . . . . . . . 10 ⊢ (𝑓 = (seq1(𝑃, 𝑈)‘𝑁) → (𝑓 ∈ 𝑌 ↔ (seq1(𝑃, 𝑈)‘𝑁) ∈ 𝑌))
126	125	anbi2d 629	. . . . . . . . 9 ⊢ (𝑓 = (seq1(𝑃, 𝑈)‘𝑁) → ((𝜑 ∧ 𝑓 ∈ 𝑌) ↔ (𝜑 ∧ (seq1(𝑃, 𝑈)‘𝑁) ∈ 𝑌)))
127		feq1 6728	. . . . . . . . 9 ⊢ (𝑓 = (seq1(𝑃, 𝑈)‘𝑁) → (𝑓:𝑇⟶ℝ ↔ (seq1(𝑃, 𝑈)‘𝑁):𝑇⟶ℝ))
128	126, 127	imbi12d 344	. . . . . . . 8 ⊢ (𝑓 = (seq1(𝑃, 𝑈)‘𝑁) → (((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (seq1(𝑃, 𝑈)‘𝑁) ∈ 𝑌) → (seq1(𝑃, 𝑈)‘𝑁):𝑇⟶ℝ)))
129	118, 124, 128, 29	vtoclgf 3581	. . . . . . 7 ⊢ ((seq1(𝑃, 𝑈)‘𝑁) ∈ 𝑌 → ((𝜑 ∧ (seq1(𝑃, 𝑈)‘𝑁) ∈ 𝑌) → (seq1(𝑃, 𝑈)‘𝑁):𝑇⟶ℝ))
130	106, 111, 129	sylc 65	. . . . . 6 ⊢ (𝜑 → (seq1(𝑃, 𝑈)‘𝑁):𝑇⟶ℝ)
131	130	ffvelcdmda 7118	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) ∈ ℝ)
132	131, 32	remulcld 11320	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) · ((𝑈‘(𝑁 + 1))‘𝑡)) ∈ ℝ)
133	110, 132	fvmpt2d 7042	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘(𝑁 + 1))‘𝑡) = (((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) · ((𝑈‘(𝑁 + 1))‘𝑡)))
134		fmuldfeqlem1.12	. . . . 5 ⊢ (𝜑 → ((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑁))
135	134	oveq1d 7463	. . . 4 ⊢ (𝜑 → (((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) · ((𝑈‘(𝑁 + 1))‘𝑡)) = ((seq1( · , (𝐹‘𝑡))‘𝑁) · ((𝑈‘(𝑁 + 1))‘𝑡)))
136	135	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) · ((𝑈‘(𝑁 + 1))‘𝑡)) = ((seq1( · , (𝐹‘𝑡))‘𝑁) · ((𝑈‘(𝑁 + 1))‘𝑡)))
137	133, 136	eqtrd 2780	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘(𝑁 + 1))‘𝑡) = ((seq1( · , (𝐹‘𝑡))‘𝑁) · ((𝑈‘(𝑁 + 1))‘𝑡)))
138	34, 40, 137	3eqtr4rd 2791	1 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘(𝑁 + 1))‘𝑡) = (seq1( · , (𝐹‘𝑡))‘(𝑁 + 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 Ⅎwnf 1781 ∈ wcel 2108 Ⅎwnfc 2893 Vcvv 3488 ↦ cmpt 5249 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 ℝcr 11183 1c1 11185 + caddc 11187 · cmul 11189 ℤ_≥cuz 12903 ...cfz 13567 seqcseq 14052

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-seq 14053

This theorem is referenced by: fmuldfeq 45504

W3C validator