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Theorem fmul01 45595
Description: Multiplying a finite number of values in [ 0 , 1 ] , gives the final product itself a number in [ 0 , 1 ]. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
fmul01.1 𝑖𝐵
fmul01.2 𝑖𝜑
fmul01.3 𝐴 = seq𝐿( · , 𝐵)
fmul01.4 (𝜑𝐿 ∈ ℤ)
fmul01.5 (𝜑𝑀 ∈ (ℤ𝐿))
fmul01.6 (𝜑𝐾 ∈ (𝐿...𝑀))
fmul01.7 ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ∈ ℝ)
fmul01.8 ((𝜑𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝑖))
fmul01.9 ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ≤ 1)
Assertion
Ref Expression
fmul01 (𝜑 → (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1))
Distinct variable groups:   𝑖,𝐿   𝑖,𝑀
Allowed substitution hints:   𝜑(𝑖)   𝐴(𝑖)   𝐵(𝑖)   𝐾(𝑖)

Proof of Theorem fmul01
Dummy variables 𝑗 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmul01.6 . 2 (𝜑𝐾 ∈ (𝐿...𝑀))
2 fveq2 6906 . . . . . 6 (𝑘 = 𝐿 → (𝐴𝑘) = (𝐴𝐿))
32breq2d 5155 . . . . 5 (𝑘 = 𝐿 → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴𝐿)))
42breq1d 5153 . . . . 5 (𝑘 = 𝐿 → ((𝐴𝑘) ≤ 1 ↔ (𝐴𝐿) ≤ 1))
53, 4anbi12d 632 . . . 4 (𝑘 = 𝐿 → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1)))
65imbi2d 340 . . 3 (𝑘 = 𝐿 → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1))))
7 fveq2 6906 . . . . . 6 (𝑘 = 𝑗 → (𝐴𝑘) = (𝐴𝑗))
87breq2d 5155 . . . . 5 (𝑘 = 𝑗 → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴𝑗)))
97breq1d 5153 . . . . 5 (𝑘 = 𝑗 → ((𝐴𝑘) ≤ 1 ↔ (𝐴𝑗) ≤ 1))
108, 9anbi12d 632 . . . 4 (𝑘 = 𝑗 → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)))
1110imbi2d 340 . . 3 (𝑘 = 𝑗 → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))))
12 fveq2 6906 . . . . . 6 (𝑘 = (𝑗 + 1) → (𝐴𝑘) = (𝐴‘(𝑗 + 1)))
1312breq2d 5155 . . . . 5 (𝑘 = (𝑗 + 1) → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴‘(𝑗 + 1))))
1412breq1d 5153 . . . . 5 (𝑘 = (𝑗 + 1) → ((𝐴𝑘) ≤ 1 ↔ (𝐴‘(𝑗 + 1)) ≤ 1))
1513, 14anbi12d 632 . . . 4 (𝑘 = (𝑗 + 1) → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1)))
1615imbi2d 340 . . 3 (𝑘 = (𝑗 + 1) → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
17 fveq2 6906 . . . . . 6 (𝑘 = 𝐾 → (𝐴𝑘) = (𝐴𝐾))
1817breq2d 5155 . . . . 5 (𝑘 = 𝐾 → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴𝐾)))
1917breq1d 5153 . . . . 5 (𝑘 = 𝐾 → ((𝐴𝑘) ≤ 1 ↔ (𝐴𝐾) ≤ 1))
2018, 19anbi12d 632 . . . 4 (𝑘 = 𝐾 → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1)))
2120imbi2d 340 . . 3 (𝑘 = 𝐾 → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1))))
22 fmul01.4 . . . . . . . 8 (𝜑𝐿 ∈ ℤ)
23 fmul01.5 . . . . . . . . 9 (𝜑𝑀 ∈ (ℤ𝐿))
24 eluzelz 12888 . . . . . . . . 9 (𝑀 ∈ (ℤ𝐿) → 𝑀 ∈ ℤ)
2523, 24syl 17 . . . . . . . 8 (𝜑𝑀 ∈ ℤ)
2622zred 12722 . . . . . . . . 9 (𝜑𝐿 ∈ ℝ)
2726leidd 11829 . . . . . . . 8 (𝜑𝐿𝐿)
28 eluz 12892 . . . . . . . . . 10 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (ℤ𝐿) ↔ 𝐿𝑀))
2922, 25, 28syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑀 ∈ (ℤ𝐿) ↔ 𝐿𝑀))
3023, 29mpbid 232 . . . . . . . 8 (𝜑𝐿𝑀)
3122, 25, 22, 27, 30elfzd 13555 . . . . . . 7 (𝜑𝐿 ∈ (𝐿...𝑀))
3231ancli 548 . . . . . . 7 (𝜑 → (𝜑𝐿 ∈ (𝐿...𝑀)))
33 fmul01.2 . . . . . . . . . 10 𝑖𝜑
34 nfv 1914 . . . . . . . . . 10 𝑖 𝐿 ∈ (𝐿...𝑀)
3533, 34nfan 1899 . . . . . . . . 9 𝑖(𝜑𝐿 ∈ (𝐿...𝑀))
36 nfcv 2905 . . . . . . . . . 10 𝑖0
37 nfcv 2905 . . . . . . . . . 10 𝑖
38 fmul01.1 . . . . . . . . . . 11 𝑖𝐵
39 nfcv 2905 . . . . . . . . . . 11 𝑖𝐿
4038, 39nffv 6916 . . . . . . . . . 10 𝑖(𝐵𝐿)
4136, 37, 40nfbr 5190 . . . . . . . . 9 𝑖0 ≤ (𝐵𝐿)
4235, 41nfim 1896 . . . . . . . 8 𝑖((𝜑𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝐿))
43 eleq1 2829 . . . . . . . . . 10 (𝑖 = 𝐿 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝐿 ∈ (𝐿...𝑀)))
4443anbi2d 630 . . . . . . . . 9 (𝑖 = 𝐿 → ((𝜑𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑𝐿 ∈ (𝐿...𝑀))))
45 fveq2 6906 . . . . . . . . . 10 (𝑖 = 𝐿 → (𝐵𝑖) = (𝐵𝐿))
4645breq2d 5155 . . . . . . . . 9 (𝑖 = 𝐿 → (0 ≤ (𝐵𝑖) ↔ 0 ≤ (𝐵𝐿)))
4744, 46imbi12d 344 . . . . . . . 8 (𝑖 = 𝐿 → (((𝜑𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝑖)) ↔ ((𝜑𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝐿))))
48 fmul01.8 . . . . . . . 8 ((𝜑𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝑖))
4942, 47, 48vtoclg1f 3570 . . . . . . 7 (𝐿 ∈ (𝐿...𝑀) → ((𝜑𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝐿)))
5031, 32, 49sylc 65 . . . . . 6 (𝜑 → 0 ≤ (𝐵𝐿))
51 fmul01.3 . . . . . . . 8 𝐴 = seq𝐿( · , 𝐵)
5251fveq1i 6907 . . . . . . 7 (𝐴𝐿) = (seq𝐿( · , 𝐵)‘𝐿)
53 seq1 14055 . . . . . . . 8 (𝐿 ∈ ℤ → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵𝐿))
5422, 53syl 17 . . . . . . 7 (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵𝐿))
5552, 54eqtrid 2789 . . . . . 6 (𝜑 → (𝐴𝐿) = (𝐵𝐿))
5650, 55breqtrrd 5171 . . . . 5 (𝜑 → 0 ≤ (𝐴𝐿))
57 nfcv 2905 . . . . . . . . . 10 𝑖1
5840, 37, 57nfbr 5190 . . . . . . . . 9 𝑖(𝐵𝐿) ≤ 1
5935, 58nfim 1896 . . . . . . . 8 𝑖((𝜑𝐿 ∈ (𝐿...𝑀)) → (𝐵𝐿) ≤ 1)
6045breq1d 5153 . . . . . . . . 9 (𝑖 = 𝐿 → ((𝐵𝑖) ≤ 1 ↔ (𝐵𝐿) ≤ 1))
6144, 60imbi12d 344 . . . . . . . 8 (𝑖 = 𝐿 → (((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ≤ 1) ↔ ((𝜑𝐿 ∈ (𝐿...𝑀)) → (𝐵𝐿) ≤ 1)))
62 fmul01.9 . . . . . . . 8 ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ≤ 1)
6359, 61, 62vtoclg1f 3570 . . . . . . 7 (𝐿 ∈ (𝐿...𝑀) → ((𝜑𝐿 ∈ (𝐿...𝑀)) → (𝐵𝐿) ≤ 1))
6431, 32, 63sylc 65 . . . . . 6 (𝜑 → (𝐵𝐿) ≤ 1)
6555, 64eqbrtrd 5165 . . . . 5 (𝜑 → (𝐴𝐿) ≤ 1)
6656, 65jca 511 . . . 4 (𝜑 → (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1))
6766a1i 11 . . 3 (𝑀 ∈ (ℤ𝐿) → (𝜑 → (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1)))
68 elfzouz 13703 . . . . . . . . . 10 (𝑗 ∈ (𝐿..^𝑀) → 𝑗 ∈ (ℤ𝐿))
69683ad2ant1 1134 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (ℤ𝐿))
70 simpl3 1194 . . . . . . . . . 10 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝜑)
71 elfzouz2 13714 . . . . . . . . . . . . 13 (𝑗 ∈ (𝐿..^𝑀) → 𝑀 ∈ (ℤ𝑗))
72 fzss2 13604 . . . . . . . . . . . . 13 (𝑀 ∈ (ℤ𝑗) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
7371, 72syl 17 . . . . . . . . . . . 12 (𝑗 ∈ (𝐿..^𝑀) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
74733ad2ant1 1134 . . . . . . . . . . 11 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
7574sselda 3983 . . . . . . . . . 10 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝑘 ∈ (𝐿...𝑀))
76 nfv 1914 . . . . . . . . . . . . 13 𝑖 𝑘 ∈ (𝐿...𝑀)
7733, 76nfan 1899 . . . . . . . . . . . 12 𝑖(𝜑𝑘 ∈ (𝐿...𝑀))
78 nfcv 2905 . . . . . . . . . . . . . 14 𝑖𝑘
7938, 78nffv 6916 . . . . . . . . . . . . 13 𝑖(𝐵𝑘)
8079nfel1 2922 . . . . . . . . . . . 12 𝑖(𝐵𝑘) ∈ ℝ
8177, 80nfim 1896 . . . . . . . . . . 11 𝑖((𝜑𝑘 ∈ (𝐿...𝑀)) → (𝐵𝑘) ∈ ℝ)
82 eleq1 2829 . . . . . . . . . . . . 13 (𝑖 = 𝑘 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑘 ∈ (𝐿...𝑀)))
8382anbi2d 630 . . . . . . . . . . . 12 (𝑖 = 𝑘 → ((𝜑𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑𝑘 ∈ (𝐿...𝑀))))
84 fveq2 6906 . . . . . . . . . . . . 13 (𝑖 = 𝑘 → (𝐵𝑖) = (𝐵𝑘))
8584eleq1d 2826 . . . . . . . . . . . 12 (𝑖 = 𝑘 → ((𝐵𝑖) ∈ ℝ ↔ (𝐵𝑘) ∈ ℝ))
8683, 85imbi12d 344 . . . . . . . . . . 11 (𝑖 = 𝑘 → (((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ∈ ℝ) ↔ ((𝜑𝑘 ∈ (𝐿...𝑀)) → (𝐵𝑘) ∈ ℝ)))
87 fmul01.7 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ∈ ℝ)
8881, 86, 87chvarfv 2240 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝐿...𝑀)) → (𝐵𝑘) ∈ ℝ)
8970, 75, 88syl2anc 584 . . . . . . . . 9 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → (𝐵𝑘) ∈ ℝ)
90 remulcl 11240 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑘 · 𝑙) ∈ ℝ)
9190adantl 481 . . . . . . . . 9 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ (𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ)) → (𝑘 · 𝑙) ∈ ℝ)
9269, 89, 91seqcl 14063 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℝ)
93 simp3 1139 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 𝜑)
94 fzofzp1 13803 . . . . . . . . . 10 (𝑗 ∈ (𝐿..^𝑀) → (𝑗 + 1) ∈ (𝐿...𝑀))
95943ad2ant1 1134 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝑗 + 1) ∈ (𝐿...𝑀))
96 nfv 1914 . . . . . . . . . . . . 13 𝑖(𝑗 + 1) ∈ (𝐿...𝑀)
9733, 96nfan 1899 . . . . . . . . . . . 12 𝑖(𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))
98 nfcv 2905 . . . . . . . . . . . . . 14 𝑖(𝑗 + 1)
9938, 98nffv 6916 . . . . . . . . . . . . 13 𝑖(𝐵‘(𝑗 + 1))
10099nfel1 2922 . . . . . . . . . . . 12 𝑖(𝐵‘(𝑗 + 1)) ∈ ℝ
10197, 100nfim 1896 . . . . . . . . . . 11 𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
102 eleq1 2829 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 1) → (𝑖 ∈ (𝐿...𝑀) ↔ (𝑗 + 1) ∈ (𝐿...𝑀)))
103102anbi2d 630 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → ((𝜑𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))))
104 fveq2 6906 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 1) → (𝐵𝑖) = (𝐵‘(𝑗 + 1)))
105104eleq1d 2826 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → ((𝐵𝑖) ∈ ℝ ↔ (𝐵‘(𝑗 + 1)) ∈ ℝ))
106103, 105imbi12d 344 . . . . . . . . . . 11 (𝑖 = (𝑗 + 1) → (((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ∈ ℝ) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)))
107101, 106, 87vtoclg1f 3570 . . . . . . . . . 10 ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ))
108107anabsi7 671 . . . . . . . . 9 ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
10993, 95, 108syl2anc 584 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
110 pm3.35 803 . . . . . . . . . . . 12 ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))) → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))
111110ancoms 458 . . . . . . . . . . 11 (((𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))
112 simpl 482 . . . . . . . . . . 11 ((0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1) → 0 ≤ (𝐴𝑗))
113111, 112syl 17 . . . . . . . . . 10 (((𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴𝑗))
1141133adant1 1131 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴𝑗))
11551fveq1i 6907 . . . . . . . . 9 (𝐴𝑗) = (seq𝐿( · , 𝐵)‘𝑗)
116114, 115breqtrdi 5184 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘𝑗))
117 simp1 1137 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (𝐿..^𝑀))
11894adantl 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → (𝑗 + 1) ∈ (𝐿...𝑀))
119 simpl 482 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → 𝜑)
120119, 118jca 511 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
12136, 37, 99nfbr 5190 . . . . . . . . . . . 12 𝑖0 ≤ (𝐵‘(𝑗 + 1))
12297, 121nfim 1896 . . . . . . . . . . 11 𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
123104breq2d 5155 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → (0 ≤ (𝐵𝑖) ↔ 0 ≤ (𝐵‘(𝑗 + 1))))
124103, 123imbi12d 344 . . . . . . . . . . 11 (𝑖 = (𝑗 + 1) → (((𝜑𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝑖)) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))))
125122, 124, 48vtoclg1f 3570 . . . . . . . . . 10 ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1))))
126118, 120, 125sylc 65 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
12793, 117, 126syl2anc 584 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐵‘(𝑗 + 1)))
12892, 109, 116, 127mulge0d 11840 . . . . . . 7 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
129 seqp1 14057 . . . . . . . 8 (𝑗 ∈ (ℤ𝐿) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) = ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
13069, 129syl 17 . . . . . . 7 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) = ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
131128, 130breqtrrd 5171 . . . . . 6 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘(𝑗 + 1)))
13251fveq1i 6907 . . . . . 6 (𝐴‘(𝑗 + 1)) = (seq𝐿( · , 𝐵)‘(𝑗 + 1))
133131, 132breqtrrdi 5185 . . . . 5 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘(𝑗 + 1)))
13492, 109remulcld 11291 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ∈ ℝ)
135 1red 11262 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 1 ∈ ℝ)
13693, 95jca 511 . . . . . . . . . . 11 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
13799, 37, 57nfbr 5190 . . . . . . . . . . . . 13 𝑖(𝐵‘(𝑗 + 1)) ≤ 1
13897, 137nfim 1896 . . . . . . . . . . . 12 𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)
139104breq1d 5153 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 1) → ((𝐵𝑖) ≤ 1 ↔ (𝐵‘(𝑗 + 1)) ≤ 1))
140103, 139imbi12d 344 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → (((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ≤ 1) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)))
141138, 140, 62vtoclg1f 3570 . . . . . . . . . . 11 ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1))
14295, 136, 141sylc 65 . . . . . . . . . 10 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐵‘(𝑗 + 1)) ≤ 1)
143109, 135, 92, 116, 142lemul2ad 12208 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ ((seq𝐿( · , 𝐵)‘𝑗) · 1))
14492recnd 11289 . . . . . . . . . 10 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℂ)
145144mulridd 11278 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · 1) = (seq𝐿( · , 𝐵)‘𝑗))
146143, 145breqtrd 5169 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ (seq𝐿( · , 𝐵)‘𝑗))
147 simp2 1138 . . . . . . . . . 10 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)))
148110simprd 495 . . . . . . . . . 10 ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))) → (𝐴𝑗) ≤ 1)
14993, 147, 148syl2anc 584 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐴𝑗) ≤ 1)
150115, 149eqbrtrrid 5179 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ≤ 1)
151134, 92, 135, 146, 150letrd 11418 . . . . . . 7 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ 1)
152130, 151eqbrtrd 5165 . . . . . 6 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) ≤ 1)
153132, 152eqbrtrid 5178 . . . . 5 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐴‘(𝑗 + 1)) ≤ 1)
154133, 153jca 511 . . . 4 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))
1551543exp 1120 . . 3 (𝑗 ∈ (𝐿..^𝑀) → ((𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) → (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
1566, 11, 16, 21, 67, 155fzind2 13824 . 2 (𝐾 ∈ (𝐿...𝑀) → (𝜑 → (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1)))
1571, 156mpcom 38 1 (𝜑 → (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wnf 1783  wcel 2108  wnfc 2890  wss 3951   class class class wbr 5143  cfv 6561  (class class class)co 7431  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  cle 11296  cz 12613  cuz 12878  ...cfz 13547  ..^cfzo 13694  seqcseq 14042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043
This theorem is referenced by:  fmul01lt1lem1  45599  fmul01lt1lem2  45600
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