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Theorem fmul01 40730
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 6448 . . . . . 6 (𝑘 = 𝐿 → (𝐴𝑘) = (𝐴𝐿))
32breq2d 4900 . . . . 5 (𝑘 = 𝐿 → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴𝐿)))
42breq1d 4898 . . . . 5 (𝑘 = 𝐿 → ((𝐴𝑘) ≤ 1 ↔ (𝐴𝐿) ≤ 1))
53, 4anbi12d 624 . . . 4 (𝑘 = 𝐿 → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1)))
65imbi2d 332 . . 3 (𝑘 = 𝐿 → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1))))
7 fveq2 6448 . . . . . 6 (𝑘 = 𝑗 → (𝐴𝑘) = (𝐴𝑗))
87breq2d 4900 . . . . 5 (𝑘 = 𝑗 → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴𝑗)))
97breq1d 4898 . . . . 5 (𝑘 = 𝑗 → ((𝐴𝑘) ≤ 1 ↔ (𝐴𝑗) ≤ 1))
108, 9anbi12d 624 . . . 4 (𝑘 = 𝑗 → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)))
1110imbi2d 332 . . 3 (𝑘 = 𝑗 → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))))
12 fveq2 6448 . . . . . 6 (𝑘 = (𝑗 + 1) → (𝐴𝑘) = (𝐴‘(𝑗 + 1)))
1312breq2d 4900 . . . . 5 (𝑘 = (𝑗 + 1) → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴‘(𝑗 + 1))))
1412breq1d 4898 . . . . 5 (𝑘 = (𝑗 + 1) → ((𝐴𝑘) ≤ 1 ↔ (𝐴‘(𝑗 + 1)) ≤ 1))
1513, 14anbi12d 624 . . . 4 (𝑘 = (𝑗 + 1) → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1)))
1615imbi2d 332 . . 3 (𝑘 = (𝑗 + 1) → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
17 fveq2 6448 . . . . . 6 (𝑘 = 𝐾 → (𝐴𝑘) = (𝐴𝐾))
1817breq2d 4900 . . . . 5 (𝑘 = 𝐾 → (0 ≤ (𝐴𝑘) ↔ 0 ≤ (𝐴𝐾)))
1917breq1d 4898 . . . . 5 (𝑘 = 𝐾 → ((𝐴𝑘) ≤ 1 ↔ (𝐴𝐾) ≤ 1))
2018, 19anbi12d 624 . . . 4 (𝑘 = 𝐾 → ((0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1) ↔ (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1)))
2120imbi2d 332 . . 3 (𝑘 = 𝐾 → ((𝜑 → (0 ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ 1)) ↔ (𝜑 → (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1))))
22 fmul01.4 . . . . . . . . . 10 (𝜑𝐿 ∈ ℤ)
2322zred 11838 . . . . . . . . 9 (𝜑𝐿 ∈ ℝ)
2423leidd 10943 . . . . . . . 8 (𝜑𝐿𝐿)
25 fmul01.5 . . . . . . . . 9 (𝜑𝑀 ∈ (ℤ𝐿))
26 eluzelz 12006 . . . . . . . . . . 11 (𝑀 ∈ (ℤ𝐿) → 𝑀 ∈ ℤ)
2725, 26syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ ℤ)
28 eluz 12010 . . . . . . . . . 10 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (ℤ𝐿) ↔ 𝐿𝑀))
2922, 27, 28syl2anc 579 . . . . . . . . 9 (𝜑 → (𝑀 ∈ (ℤ𝐿) ↔ 𝐿𝑀))
3025, 29mpbid 224 . . . . . . . 8 (𝜑𝐿𝑀)
31 elfz 12653 . . . . . . . . 9 ((𝐿 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ∈ (𝐿...𝑀) ↔ (𝐿𝐿𝐿𝑀)))
3222, 22, 27, 31syl3anc 1439 . . . . . . . 8 (𝜑 → (𝐿 ∈ (𝐿...𝑀) ↔ (𝐿𝐿𝐿𝑀)))
3324, 30, 32mpbir2and 703 . . . . . . 7 (𝜑𝐿 ∈ (𝐿...𝑀))
3433ancli 544 . . . . . . 7 (𝜑 → (𝜑𝐿 ∈ (𝐿...𝑀)))
35 fmul01.2 . . . . . . . . . 10 𝑖𝜑
36 nfv 1957 . . . . . . . . . 10 𝑖 𝐿 ∈ (𝐿...𝑀)
3735, 36nfan 1946 . . . . . . . . 9 𝑖(𝜑𝐿 ∈ (𝐿...𝑀))
38 nfcv 2934 . . . . . . . . . 10 𝑖0
39 nfcv 2934 . . . . . . . . . 10 𝑖
40 fmul01.1 . . . . . . . . . . 11 𝑖𝐵
41 nfcv 2934 . . . . . . . . . . 11 𝑖𝐿
4240, 41nffv 6458 . . . . . . . . . 10 𝑖(𝐵𝐿)
4338, 39, 42nfbr 4935 . . . . . . . . 9 𝑖0 ≤ (𝐵𝐿)
4437, 43nfim 1943 . . . . . . . 8 𝑖((𝜑𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝐿))
45 eleq1 2847 . . . . . . . . . 10 (𝑖 = 𝐿 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝐿 ∈ (𝐿...𝑀)))
4645anbi2d 622 . . . . . . . . 9 (𝑖 = 𝐿 → ((𝜑𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑𝐿 ∈ (𝐿...𝑀))))
47 fveq2 6448 . . . . . . . . . 10 (𝑖 = 𝐿 → (𝐵𝑖) = (𝐵𝐿))
4847breq2d 4900 . . . . . . . . 9 (𝑖 = 𝐿 → (0 ≤ (𝐵𝑖) ↔ 0 ≤ (𝐵𝐿)))
4946, 48imbi12d 336 . . . . . . . 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 3466 . . . . . . 7 (𝐿 ∈ (𝐿...𝑀) → ((𝜑𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝐿)))
5233, 34, 51sylc 65 . . . . . 6 (𝜑 → 0 ≤ (𝐵𝐿))
53 fmul01.3 . . . . . . . 8 𝐴 = seq𝐿( · , 𝐵)
5453fveq1i 6449 . . . . . . 7 (𝐴𝐿) = (seq𝐿( · , 𝐵)‘𝐿)
55 seq1 13136 . . . . . . . 8 (𝐿 ∈ ℤ → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵𝐿))
5622, 55syl 17 . . . . . . 7 (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵𝐿))
5754, 56syl5eq 2826 . . . . . 6 (𝜑 → (𝐴𝐿) = (𝐵𝐿))
5852, 57breqtrrd 4916 . . . . 5 (𝜑 → 0 ≤ (𝐴𝐿))
59 nfcv 2934 . . . . . . . . . 10 𝑖1
6042, 39, 59nfbr 4935 . . . . . . . . 9 𝑖(𝐵𝐿) ≤ 1
6137, 60nfim 1943 . . . . . . . 8 𝑖((𝜑𝐿 ∈ (𝐿...𝑀)) → (𝐵𝐿) ≤ 1)
6247breq1d 4898 . . . . . . . . 9 (𝑖 = 𝐿 → ((𝐵𝑖) ≤ 1 ↔ (𝐵𝐿) ≤ 1))
6346, 62imbi12d 336 . . . . . . . 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 3466 . . . . . . 7 (𝐿 ∈ (𝐿...𝑀) → ((𝜑𝐿 ∈ (𝐿...𝑀)) → (𝐵𝐿) ≤ 1))
6633, 34, 65sylc 65 . . . . . 6 (𝜑 → (𝐵𝐿) ≤ 1)
6757, 66eqbrtrd 4910 . . . . 5 (𝜑 → (𝐴𝐿) ≤ 1)
6858, 67jca 507 . . . 4 (𝜑 → (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1))
6968a1i 11 . . 3 (𝑀 ∈ (ℤ𝐿) → (𝜑 → (0 ≤ (𝐴𝐿) ∧ (𝐴𝐿) ≤ 1)))
70 elfzouz 12797 . . . . . . . . . 10 (𝑗 ∈ (𝐿..^𝑀) → 𝑗 ∈ (ℤ𝐿))
71703ad2ant1 1124 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (ℤ𝐿))
72 simpl3 1203 . . . . . . . . . 10 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝜑)
73 elfzouz2 12807 . . . . . . . . . . . . 13 (𝑗 ∈ (𝐿..^𝑀) → 𝑀 ∈ (ℤ𝑗))
74 fzss2 12702 . . . . . . . . . . . . 13 (𝑀 ∈ (ℤ𝑗) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
7573, 74syl 17 . . . . . . . . . . . 12 (𝑗 ∈ (𝐿..^𝑀) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
76753ad2ant1 1124 . . . . . . . . . . 11 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐿...𝑗) ⊆ (𝐿...𝑀))
7776sselda 3821 . . . . . . . . . 10 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → 𝑘 ∈ (𝐿...𝑀))
78 nfv 1957 . . . . . . . . . . . . 13 𝑖 𝑘 ∈ (𝐿...𝑀)
7935, 78nfan 1946 . . . . . . . . . . . 12 𝑖(𝜑𝑘 ∈ (𝐿...𝑀))
80 nfcv 2934 . . . . . . . . . . . . . 14 𝑖𝑘
8140, 80nffv 6458 . . . . . . . . . . . . 13 𝑖(𝐵𝑘)
8281nfel1 2948 . . . . . . . . . . . 12 𝑖(𝐵𝑘) ∈ ℝ
8379, 82nfim 1943 . . . . . . . . . . 11 𝑖((𝜑𝑘 ∈ (𝐿...𝑀)) → (𝐵𝑘) ∈ ℝ)
84 eleq1 2847 . . . . . . . . . . . . 13 (𝑖 = 𝑘 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑘 ∈ (𝐿...𝑀)))
8584anbi2d 622 . . . . . . . . . . . 12 (𝑖 = 𝑘 → ((𝜑𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑𝑘 ∈ (𝐿...𝑀))))
86 fveq2 6448 . . . . . . . . . . . . 13 (𝑖 = 𝑘 → (𝐵𝑖) = (𝐵𝑘))
8786eleq1d 2844 . . . . . . . . . . . 12 (𝑖 = 𝑘 → ((𝐵𝑖) ∈ ℝ ↔ (𝐵𝑘) ∈ ℝ))
8885, 87imbi12d 336 . . . . . . . . . . 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 2360 . . . . . . . . . 10 ((𝜑𝑘 ∈ (𝐿...𝑀)) → (𝐵𝑘) ∈ ℝ)
9172, 77, 90syl2anc 579 . . . . . . . . 9 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ 𝑘 ∈ (𝐿...𝑗)) → (𝐵𝑘) ∈ ℝ)
92 remulcl 10359 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → (𝑘 · 𝑙) ∈ ℝ)
9392adantl 475 . . . . . . . . 9 (((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) ∧ (𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ)) → (𝑘 · 𝑙) ∈ ℝ)
9471, 91, 93seqcl 13143 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℝ)
95 simp3 1129 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 𝜑)
96 fzofzp1 12888 . . . . . . . . . 10 (𝑗 ∈ (𝐿..^𝑀) → (𝑗 + 1) ∈ (𝐿...𝑀))
97963ad2ant1 1124 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝑗 + 1) ∈ (𝐿...𝑀))
98 nfv 1957 . . . . . . . . . . . . 13 𝑖(𝑗 + 1) ∈ (𝐿...𝑀)
9935, 98nfan 1946 . . . . . . . . . . . 12 𝑖(𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))
100 nfcv 2934 . . . . . . . . . . . . . 14 𝑖(𝑗 + 1)
10140, 100nffv 6458 . . . . . . . . . . . . 13 𝑖(𝐵‘(𝑗 + 1))
102101nfel1 2948 . . . . . . . . . . . 12 𝑖(𝐵‘(𝑗 + 1)) ∈ ℝ
10399, 102nfim 1943 . . . . . . . . . . 11 𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
104 eleq1 2847 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 1) → (𝑖 ∈ (𝐿...𝑀) ↔ (𝑗 + 1) ∈ (𝐿...𝑀)))
105104anbi2d 622 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → ((𝜑𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀))))
106 fveq2 6448 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 1) → (𝐵𝑖) = (𝐵‘(𝑗 + 1)))
107106eleq1d 2844 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → ((𝐵𝑖) ∈ ℝ ↔ (𝐵‘(𝑗 + 1)) ∈ ℝ))
108105, 107imbi12d 336 . . . . . . . . . . 11 (𝑖 = (𝑗 + 1) → (((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ∈ ℝ) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)))
109103, 108, 89vtoclg1f 3466 . . . . . . . . . 10 ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ))
110109anabsi7 661 . . . . . . . . 9 ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
11195, 97, 110syl2anc 579 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐵‘(𝑗 + 1)) ∈ ℝ)
112 pm3.35 793 . . . . . . . . . . . 12 ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))) → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))
113112ancoms 452 . . . . . . . . . . 11 (((𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))
114 simpl 476 . . . . . . . . . . 11 ((0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1) → 0 ≤ (𝐴𝑗))
115113, 114syl 17 . . . . . . . . . 10 (((𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴𝑗))
1161153adant1 1121 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴𝑗))
11753fveq1i 6449 . . . . . . . . 9 (𝐴𝑗) = (seq𝐿( · , 𝐵)‘𝑗)
118116, 117syl6breq 4929 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘𝑗))
119 simp1 1127 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 𝑗 ∈ (𝐿..^𝑀))
12096adantl 475 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → (𝑗 + 1) ∈ (𝐿...𝑀))
121 simpl 476 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → 𝜑)
122121, 120jca 507 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
12338, 39, 101nfbr 4935 . . . . . . . . . . . 12 𝑖0 ≤ (𝐵‘(𝑗 + 1))
12499, 123nfim 1943 . . . . . . . . . . 11 𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
125106breq2d 4900 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → (0 ≤ (𝐵𝑖) ↔ 0 ≤ (𝐵‘(𝑗 + 1))))
126105, 125imbi12d 336 . . . . . . . . . . 11 (𝑖 = (𝑗 + 1) → (((𝜑𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵𝑖)) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))))
127124, 126, 50vtoclg1f 3466 . . . . . . . . . 10 ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1))))
128120, 122, 127sylc 65 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝐿..^𝑀)) → 0 ≤ (𝐵‘(𝑗 + 1)))
12995, 119, 128syl2anc 579 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐵‘(𝑗 + 1)))
13094, 111, 118, 129mulge0d 10954 . . . . . . 7 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))))
131 seqp1 13138 . . . . . . . 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 4916 . . . . . 6 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (seq𝐿( · , 𝐵)‘(𝑗 + 1)))
13453fveq1i 6449 . . . . . 6 (𝐴‘(𝑗 + 1)) = (seq𝐿( · , 𝐵)‘(𝑗 + 1))
135133, 134syl6breqr 4930 . . . . 5 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 0 ≤ (𝐴‘(𝑗 + 1)))
13694, 111remulcld 10409 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ∈ ℝ)
137 1red 10379 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → 1 ∈ ℝ)
13895, 97jca 507 . . . . . . . . . . 11 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)))
139101, 39, 59nfbr 4935 . . . . . . . . . . . . 13 𝑖(𝐵‘(𝑗 + 1)) ≤ 1
14099, 139nfim 1943 . . . . . . . . . . . 12 𝑖((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)
141106breq1d 4898 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 1) → ((𝐵𝑖) ≤ 1 ↔ (𝐵‘(𝑗 + 1)) ≤ 1))
142105, 141imbi12d 336 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 1) → (((𝜑𝑖 ∈ (𝐿...𝑀)) → (𝐵𝑖) ≤ 1) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1)))
143140, 142, 64vtoclg1f 3466 . . . . . . . . . . 11 ((𝑗 + 1) ∈ (𝐿...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (𝐿...𝑀)) → (𝐵‘(𝑗 + 1)) ≤ 1))
14497, 138, 143sylc 65 . . . . . . . . . 10 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐵‘(𝑗 + 1)) ≤ 1)
145111, 137, 94, 118, 144lemul2ad 11320 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ ((seq𝐿( · , 𝐵)‘𝑗) · 1))
14694recnd 10407 . . . . . . . . . 10 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ∈ ℂ)
147146mulid1d 10396 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · 1) = (seq𝐿( · , 𝐵)‘𝑗))
148145, 147breqtrd 4914 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ (seq𝐿( · , 𝐵)‘𝑗))
149 simp2 1128 . . . . . . . . . 10 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)))
150112simprd 491 . . . . . . . . . 10 ((𝜑 ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1))) → (𝐴𝑗) ≤ 1)
15195, 149, 150syl2anc 579 . . . . . . . . 9 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐴𝑗) ≤ 1)
152117, 151syl5eqbrr 4924 . . . . . . . 8 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘𝑗) ≤ 1)
153136, 94, 137, 148, 152letrd 10535 . . . . . . 7 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → ((seq𝐿( · , 𝐵)‘𝑗) · (𝐵‘(𝑗 + 1))) ≤ 1)
154132, 153eqbrtrd 4910 . . . . . 6 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (seq𝐿( · , 𝐵)‘(𝑗 + 1)) ≤ 1)
155134, 154syl5eqbr 4923 . . . . 5 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (𝐴‘(𝑗 + 1)) ≤ 1)
156135, 155jca 507 . . . 4 ((𝑗 ∈ (𝐿..^𝑀) ∧ (𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) ∧ 𝜑) → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))
1571563exp 1109 . . 3 (𝑗 ∈ (𝐿..^𝑀) → ((𝜑 → (0 ≤ (𝐴𝑗) ∧ (𝐴𝑗) ≤ 1)) → (𝜑 → (0 ≤ (𝐴‘(𝑗 + 1)) ∧ (𝐴‘(𝑗 + 1)) ≤ 1))))
1586, 11, 16, 21, 69, 157fzind2 12909 . 2 (𝐾 ∈ (𝐿...𝑀) → (𝜑 → (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1)))
1591, 158mpcom 38 1 (𝜑 → (0 ≤ (𝐴𝐾) ∧ (𝐴𝐾) ≤ 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wnf 1827  wcel 2107  wnfc 2919  wss 3792   class class class wbr 4888  cfv 6137  (class class class)co 6924  cr 10273  0cc0 10274  1c1 10275   + caddc 10277   · cmul 10279  cle 10414  cz 11732  cuz 11996  ...cfz 12647  ..^cfzo 12788  seqcseq 13123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11379  df-n0 11647  df-z 11733  df-uz 11997  df-fz 12648  df-fzo 12789  df-seq 13124
This theorem is referenced by:  fmul01lt1lem1  40734  fmul01lt1lem2  40735
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