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Theorem fmul01 38647
 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 6103 . . . . . 6 (𝑘 = 𝐿 → (𝐴𝑘) = (𝐴𝐿))
32breq2d 4595 . . . . 5 (𝑘 = 𝐿 → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴𝐿)))
42breq1d 4593 . . . . 5 (𝑘 = 𝐿 → ((𝐴𝑘) ≤ 1 ↔ (𝐴𝐿) ≤ 1))
53, 4anbi12d 743 . . . 4 (𝑘 = 𝐿 → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1)))
65imbi2d 329 . . 3 (𝑘 = 𝐿 → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1))))
7 fveq2 6103 . . . . . 6 (𝑘 = 𝑗 → (𝐴𝑘) = (𝐴𝑗))
87breq2d 4595 . . . . 5 (𝑘 = 𝑗 → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴𝑗)))
97breq1d 4593 . . . . 5 (𝑘 = 𝑗 → ((𝐴𝑘) ≤ 1 ↔ (𝐴𝑗) ≤ 1))
108, 9anbi12d 743 . . . 4 (𝑘 = 𝑗 → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)))
1110imbi2d 329 . . 3 (𝑘 = 𝑗 → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))))
12 fveq2 6103 . . . . . 6 (𝑘 = (𝑗 + 1) → (𝐴𝑘) = (𝐴‘(𝑗 + 1)))
1312breq2d 4595 . . . . 5 (𝑘 = (𝑗 + 1) → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴‘(𝑗 + 1))))
1412breq1d 4593 . . . . 5 (𝑘 = (𝑗 + 1) → ((𝐴𝑘) ≤ 1 ↔ (𝐴‘(𝑗 + 1)) ≤ 1))
1513, 14anbi12d 743 . . . 4 (𝑘 = (𝑗 + 1) → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1)))
1615imbi2d 329 . . 3 (𝑘 = (𝑗 + 1) → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
17 fveq2 6103 . . . . . 6 (𝑘 = 𝐾 → (𝐴𝑘) = (𝐴𝐾))
1817breq2d 4595 . . . . 5 (𝑘 = 𝐾 → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴𝐾)))
1917breq1d 4593 . . . . 5 (𝑘 = 𝐾 → ((𝐴𝑘) ≤ 1 ↔ (𝐴𝐾) ≤ 1))
2018, 19anbi12d 743 . . . 4 (𝑘 = 𝐾 → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1)))
2120imbi2d 329 . . 3 (𝑘 = 𝐾 → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1))))
22 fmul01.4 . . . . . . . . . 10 (𝜑𝐿 ∈ ℤ)
2322zred 11358 . . . . . . . . 9 (𝜑𝐿 ∈ ℝ)
2423leidd 10473 . . . . . . . 8 (𝜑𝐿𝐿)
25 fmul01.5 . . . . . . . . 9 (𝜑𝑀 ∈ (ℤ𝐿))
26 eluzelz 11573 . . . . . . . . . . 11 (𝑀 ∈ (ℤ𝐿) → 𝑀 ∈ ℤ)
2725, 26syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ ℤ)
28 eluz 11577 . . . . . . . . . 10 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (ℤ𝐿) ↔ 𝐿𝑀))
2922, 27, 28syl2anc 691 . . . . . . . . 9 (𝜑 → (𝑀 ∈ (ℤ𝐿) ↔ 𝐿𝑀))
3025, 29mpbid 221 . . . . . . . 8 (𝜑𝐿𝑀)
31 elfz 12203 . . . . . . . . 9 ((𝐿 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ∈ (𝐿...𝑀) ↔ (𝐿𝐿𝐿𝑀)))
3222, 22, 27, 31syl3anc 1318 . . . . . . . 8 (𝜑 → (𝐿 ∈ (𝐿...𝑀) ↔ (𝐿𝐿𝐿𝑀)))
3324, 30, 32mpbir2and 959 . . . . . . 7 (𝜑𝐿 ∈ (𝐿...𝑀))
3433ancli 572 . . . . . . 7 (𝜑 → (𝜑𝐿 ∈ (𝐿...𝑀)))
35 fmul01.2 . . . . . . . . . 10 𝑖𝜑
36 nfv 1830 . . . . . . . . . 10 𝑖 𝐿 ∈ (𝐿...𝑀)
3735, 36nfan 1816 . . . . . . . . 9 𝑖(𝜑𝐿 ∈ (𝐿...𝑀))
38 nfcv 2751 . . . . . . . . . 10 𝑖0
39 nfcv 2751 . . . . . . . . . 10 𝑖
40 fmul01.1 . . . . . . . . . . 11 𝑖𝐵
41 nfcv 2751 . . . . . . . . . . 11 𝑖𝐿
4240, 41nffv 6110 . . . . . . . . . 10 𝑖(𝐵𝐿)
4338, 39, 42nfbr 4629 . . . . . . . . 9 𝑖0 ≤ (𝐵𝐿)
4437, 43nfim 1813 . . . . . . . 8 𝑖((𝜑𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝐿))
45 eleq1 2676 . . . . . . . . . 10 (𝑖 = 𝐿 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝐿 ∈ (𝐿...𝑀)))
4645anbi2d 736 . . . . . . . . 9 (𝑖 = 𝐿 → ((𝜑𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑𝐿 ∈ (𝐿...𝑀))))
47 fveq2 6103 . . . . . . . . . 10 (𝑖 = 𝐿 → (𝐵𝑖) = (𝐵𝐿))
4847breq2d 4595 . . . . . . . . 9 (𝑖 = 𝐿 → (0 ≤ (𝐵𝑖) ↔ 0 ≤ (𝐵𝐿)))
4946, 48imbi12d 333 . . . . . . . 8 (𝑖 = 𝐿 → (((𝜑𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝑖)) ↔ ((𝜑𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝐿))))
50 fmul01.8 . . . . . . . 8 ((𝜑𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝑖))
5144, 49, 50vtoclg1f 3238 . . . . . . 7 (𝐿 ∈ (𝐿...𝑀) → ((𝜑𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝐿)))
5233, 34, 51sylc 63 . . . . . 6 (𝜑 → 0 ≤ (𝐵𝐿))
53 fmul01.3 . . . . . . . 8 𝐴 = seq𝐿( · , 𝐵)
5453fveq1i 6104 . . . . . . 7 (𝐴𝐿) = (seq𝐿( · , 𝐵)‘𝐿)
55 seq1 12676 . . . . . . . 8 (𝐿 ∈ ℤ → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵𝐿))
5622, 55syl 17 . . . . . . 7 (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵𝐿))
5754, 56syl5eq 2656 . . . . . 6 (𝜑 → (𝐴𝐿) = (𝐵𝐿))
5852, 57breqtrrd 4611 . . . . 5 (𝜑 → 0 ≤ (𝐴𝐿))
59 nfcv 2751 . . . . . . . . . 10 𝑖1
6042, 39, 59nfbr 4629 . . . . . . . . 9 𝑖(𝐵𝐿) ≤ 1
6137, 60nfim 1813 . . . . . . . 8 𝑖((𝜑𝐿 ∈ (𝐿...𝑀)) → (𝐵𝐿) ≤ 1)
6247breq1d 4593 . . . . . . . . 9 (𝑖 = 𝐿 → ((𝐵𝑖) ≤ 1 ↔ (𝐵𝐿) ≤ 1))
6346, 62imbi12d 333 . . . . . . . 8 (𝑖 = 𝐿 → (((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ≤ 1) ↔ ((𝜑𝐿 ∈ (𝐿...𝑀)) → (𝐵𝐿) ≤ 1)))
64 fmul01.9 . . . . . . . 8 ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ≤ 1)
6561, 63, 64vtoclg1f 3238 . . . . . . 7 (𝐿 ∈ (𝐿...𝑀) → ((𝜑𝐿 ∈ (𝐿...𝑀)) → (𝐵𝐿) ≤ 1))
6633, 34, 65sylc 63 . . . . . 6 (𝜑 → (𝐵𝐿) ≤ 1)
6757, 66eqbrtrd 4605 . . . . 5 (𝜑 → (𝐴𝐿) ≤ 1)
6858, 67jca 553 . . . 4 (𝜑 → (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1))
6968a1i 11 . . 3 (𝑀 ∈ (ℤ𝐿) → (𝜑 → (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1)))
70 elfzouz 12343 . . . . . . . . . 10 (𝑗 ∈ (𝐿..^𝑀) → 𝑗 ∈ (ℤ𝐿))
71703ad2ant1 1075 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (ℤ𝐿))
72 simpl3 1059 . . . . . . . . . 10 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝜑)
73 elfzouz2 12353 . . . . . . . . . . . . 13 (𝑗 ∈ (𝐿..^𝑀) → 𝑀 ∈ (ℤ𝑗))
74 fzss2 12252 . . . . . . . . . . . . 13 (𝑀 ∈ (ℤ𝑗) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
7573, 74syl 17 . . . . . . . . . . . 12 (𝑗 ∈ (𝐿..^𝑀) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
76753ad2ant1 1075 . . . . . . . . . . 11 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
7776sselda 3568 . . . . . . . . . 10 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝑘 ∈ (𝐿...𝑀))
78 nfv 1830 . . . . . . . . . . . . 13 𝑖 𝑘 ∈ (𝐿...𝑀)
7935, 78nfan 1816 . . . . . . . . . . . 12 𝑖(𝜑𝑘 ∈ (𝐿...𝑀))
80 nfcv 2751 . . . . . . . . . . . . . 14 𝑖𝑘
8140, 80nffv 6110 . . . . . . . . . . . . 13 𝑖(𝐵𝑘)
8281nfel1 2765 . . . . . . . . . . . 12 𝑖(𝐵𝑘) ∈ ℝ
8379, 82nfim 1813 . . . . . . . . . . 11 𝑖((𝜑𝑘 ∈ (𝐿...𝑀)) → (𝐵𝑘) ∈ ℝ)
84 eleq1 2676 . . . . . . . . . . . . 13 (𝑖 = 𝑘 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑘 ∈ (𝐿...𝑀)))
8584anbi2d 736 . . . . . . . . . . . 12 (𝑖 = 𝑘 → ((𝜑𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑𝑘 ∈ (𝐿...𝑀))))
86 fveq2 6103 . . . . . . . . . . . . 13 (𝑖 = 𝑘 → (𝐵𝑖) = (𝐵𝑘))
8786eleq1d 2672 . . . . . . . . . . . 12 (𝑖 = 𝑘 → ((𝐵𝑖) ∈ ℝ ↔ (𝐵𝑘) ∈ ℝ))
8885, 87imbi12d 333 . . . . . . . . . . 11 (𝑖 = 𝑘 → (((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ∈ ℝ) ↔ ((𝜑𝑘 ∈ (𝐿...𝑀)) → (𝐵𝑘) ∈ ℝ)))
89 fmul01.7 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ∈ ℝ)
9083, 88, 89chvar 2250 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝐿...𝑀)) → (𝐵𝑘) ∈ ℝ)
9172, 77, 90syl2anc 691 . . . . . . . . 9 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → (𝐵𝑘) ∈ ℝ)
92 remulcl 9900 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑘 · 𝑙) ∈ ℝ)
9392adantl 481 . . . . . . . . 9 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ (𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ)) → (𝑘 · 𝑙) ∈ ℝ)
9471, 91, 93seqcl 12683 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℝ)
95 simp3 1056 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 𝜑)
96 fzofzp1 12431 . . . . . . . . . 10 (𝑗 ∈ (𝐿..^𝑀) → (𝑗 + 1) ∈ (𝐿...𝑀))
97963ad2ant1 1075 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝑗 + 1) ∈ (𝐿...𝑀))
98 nfv 1830 . . . . . . . . . . . . 13 𝑖(𝑗 + 1) ∈ (𝐿...𝑀)
9935, 98nfan 1816 . . . . . . . . . . . 12 𝑖(𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))
100 nfcv 2751 . . . . . . . . . . . . . 14 𝑖(𝑗 + 1)
10140, 100nffv 6110 . . . . . . . . . . . . 13 𝑖(𝐵‘(𝑗 + 1))
102101nfel1 2765 . . . . . . . . . . . 12 𝑖(𝐵‘(𝑗 + 1)) ∈ ℝ
10399, 102nfim 1813 . . . . . . . . . . 11 𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
104 eleq1 2676 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 1) → (𝑖 ∈ (𝐿...𝑀) ↔ (𝑗 + 1) ∈ (𝐿...𝑀)))
105104anbi2d 736 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → ((𝜑𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))))
106 fveq2 6103 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 1) → (𝐵𝑖) = (𝐵‘(𝑗 + 1)))
107106eleq1d 2672 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → ((𝐵𝑖) ∈ ℝ ↔ (𝐵‘(𝑗 + 1)) ∈ ℝ))
108105, 107imbi12d 333 . . . . . . . . . . 11 (𝑖 = (𝑗 + 1) → (((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ∈ ℝ) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)))
109103, 108, 89vtoclg1f 3238 . . . . . . . . . 10 ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ))
110109anabsi7 856 . . . . . . . . 9 ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
11195, 97, 110syl2anc 691 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
112 pm3.35 609 . . . . . . . . . . . 12 ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))) → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))
113112ancoms 468 . . . . . . . . . . 11 (((𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))
114 simpl 472 . . . . . . . . . . 11 ((0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1) → 0 ≤ (𝐴𝑗))
115113, 114syl 17 . . . . . . . . . 10 (((𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴𝑗))
1161153adant1 1072 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴𝑗))
11753fveq1i 6104 . . . . . . . . 9 (𝐴𝑗) = (seq𝐿( · , 𝐵)‘𝑗)
118116, 117syl6breq 4624 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘𝑗))
119 simp1 1054 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (𝐿..^𝑀))
12096adantl 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → (𝑗 + 1) ∈ (𝐿...𝑀))
121 simpl 472 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → 𝜑)
122121, 120jca 553 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
12338, 39, 101nfbr 4629 . . . . . . . . . . . 12 𝑖0 ≤ (𝐵‘(𝑗 + 1))
12499, 123nfim 1813 . . . . . . . . . . 11 𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
125106breq2d 4595 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → (0 ≤ (𝐵𝑖) ↔ 0 ≤ (𝐵‘(𝑗 + 1))))
126105, 125imbi12d 333 . . . . . . . . . . 11 (𝑖 = (𝑗 + 1) → (((𝜑𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝑖)) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))))
127124, 126, 50vtoclg1f 3238 . . . . . . . . . 10 ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1))))
128120, 122, 127sylc 63 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
12995, 119, 128syl2anc 691 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐵‘(𝑗 + 1)))
13094, 111, 118, 129mulge0d 10483 . . . . . . 7 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
131 seqp1 12678 . . . . . . . 8 (𝑗 ∈ (ℤ𝐿) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) = ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
13271, 131syl 17 . . . . . . 7 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) = ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
133130, 132breqtrrd 4611 . . . . . 6 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘(𝑗 + 1)))
13453fveq1i 6104 . . . . . 6 (𝐴‘(𝑗 + 1)) = (seq𝐿( · , 𝐵)‘(𝑗 + 1))
135133, 134syl6breqr 4625 . . . . 5 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘(𝑗 + 1)))
13694, 111remulcld 9949 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ∈ ℝ)
137 1red 9934 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 1 ∈ ℝ)
13895, 97jca 553 . . . . . . . . . . 11 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
139101, 39, 59nfbr 4629 . . . . . . . . . . . . 13 𝑖(𝐵‘(𝑗 + 1)) ≤ 1
14099, 139nfim 1813 . . . . . . . . . . . 12 𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)
141106breq1d 4593 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 1) → ((𝐵𝑖) ≤ 1 ↔ (𝐵‘(𝑗 + 1)) ≤ 1))
142105, 141imbi12d 333 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → (((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ≤ 1) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)))
143140, 142, 64vtoclg1f 3238 . . . . . . . . . . 11 ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1))
14497, 138, 143sylc 63 . . . . . . . . . 10 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐵‘(𝑗 + 1)) ≤ 1)
145111, 137, 94, 118, 144lemul2ad 10843 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ ((seq𝐿( · , 𝐵)‘𝑗) · 1))
14694recnd 9947 . . . . . . . . . 10 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℂ)
147146mulid1d 9936 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · 1) = (seq𝐿( · , 𝐵)‘𝑗))
148145, 147breqtrd 4609 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ (seq𝐿( · , 𝐵)‘𝑗))
149 simp2 1055 . . . . . . . . . 10 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)))
150112simprd 478 . . . . . . . . . 10 ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))) → (𝐴𝑗) ≤ 1)
15195, 149, 150syl2anc 691 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐴𝑗) ≤ 1)
152117, 151syl5eqbrr 4619 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ≤ 1)
153136, 94, 137, 148, 152letrd 10073 . . . . . . 7 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ 1)
154132, 153eqbrtrd 4605 . . . . . 6 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) ≤ 1)
155134, 154syl5eqbr 4618 . . . . 5 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐴‘(𝑗 + 1)) ≤ 1)
156135, 155jca 553 . . . 4 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))
1571563exp 1256 . . 3 (𝑗 ∈ (𝐿..^𝑀) → ((𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) → (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
1586, 11, 16, 21, 69, 157fzind2 12448 . 2 (𝐾 ∈ (𝐿...𝑀) → (𝜑 → (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1)))
1591, 158mpcom 37 1 (𝜑 → (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   ≤ cle 9954  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334  seqcseq 12663 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664 This theorem is referenced by:  fmul01lt1lem1  38651  fmul01lt1lem2  38652
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