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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmul01lt1 | Structured version Visualization version GIF version |
Description: Given a finite multiplication of values between 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
fmul01lt1.1 | ⊢ Ⅎ𝑖𝐵 |
fmul01lt1.2 | ⊢ Ⅎ𝑖𝜑 |
fmul01lt1.3 | ⊢ Ⅎ𝑗𝐴 |
fmul01lt1.4 | ⊢ 𝐴 = seq1( · , 𝐵) |
fmul01lt1.5 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
fmul01lt1.6 | ⊢ (𝜑 → 𝐵:(1...𝑀)⟶ℝ) |
fmul01lt1.7 | ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ (𝐵‘𝑖)) |
fmul01lt1.8 | ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ≤ 1) |
fmul01lt1.9 | ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
fmul01lt1.10 | ⊢ (𝜑 → ∃𝑗 ∈ (1...𝑀)(𝐵‘𝑗) < 𝐸) |
Ref | Expression |
---|---|
fmul01lt1 | ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmul01lt1.10 | . 2 ⊢ (𝜑 → ∃𝑗 ∈ (1...𝑀)(𝐵‘𝑗) < 𝐸) | |
2 | nfv 1909 | . . 3 ⊢ Ⅎ𝑗𝜑 | |
3 | fmul01lt1.3 | . . . . 5 ⊢ Ⅎ𝑗𝐴 | |
4 | nfcv 2891 | . . . . 5 ⊢ Ⅎ𝑗𝑀 | |
5 | 3, 4 | nffv 6910 | . . . 4 ⊢ Ⅎ𝑗(𝐴‘𝑀) |
6 | nfcv 2891 | . . . 4 ⊢ Ⅎ𝑗 < | |
7 | nfcv 2891 | . . . 4 ⊢ Ⅎ𝑗𝐸 | |
8 | 5, 6, 7 | nfbr 5199 | . . 3 ⊢ Ⅎ𝑗(𝐴‘𝑀) < 𝐸 |
9 | fmul01lt1.1 | . . . . 5 ⊢ Ⅎ𝑖𝐵 | |
10 | fmul01lt1.2 | . . . . . 6 ⊢ Ⅎ𝑖𝜑 | |
11 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑖 𝑗 ∈ (1...𝑀) | |
12 | nfcv 2891 | . . . . . . . 8 ⊢ Ⅎ𝑖𝑗 | |
13 | 9, 12 | nffv 6910 | . . . . . . 7 ⊢ Ⅎ𝑖(𝐵‘𝑗) |
14 | nfcv 2891 | . . . . . . 7 ⊢ Ⅎ𝑖 < | |
15 | nfcv 2891 | . . . . . . 7 ⊢ Ⅎ𝑖𝐸 | |
16 | 13, 14, 15 | nfbr 5199 | . . . . . 6 ⊢ Ⅎ𝑖(𝐵‘𝑗) < 𝐸 |
17 | 10, 11, 16 | nf3an 1896 | . . . . 5 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) |
18 | fmul01lt1.4 | . . . . 5 ⊢ 𝐴 = seq1( · , 𝐵) | |
19 | 1zzd 12640 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 1 ∈ ℤ) | |
20 | fmul01lt1.5 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
21 | elnnuz 12913 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ≥‘1)) | |
22 | 20, 21 | sylib 217 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1)) |
23 | 22 | 3ad2ant1 1130 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝑀 ∈ (ℤ≥‘1)) |
24 | fmul01lt1.6 | . . . . . . 7 ⊢ (𝜑 → 𝐵:(1...𝑀)⟶ℝ) | |
25 | 24 | ffvelcdmda 7097 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
26 | 25 | 3ad2antl1 1182 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
27 | fmul01lt1.7 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ (𝐵‘𝑖)) | |
28 | 27 | 3ad2antl1 1182 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ (𝐵‘𝑖)) |
29 | fmul01lt1.8 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ≤ 1) | |
30 | 29 | 3ad2antl1 1182 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ≤ 1) |
31 | fmul01lt1.9 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
32 | 31 | 3ad2ant1 1130 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝐸 ∈ ℝ+) |
33 | simp2 1134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝑗 ∈ (1...𝑀)) | |
34 | simp3 1135 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → (𝐵‘𝑗) < 𝐸) | |
35 | 9, 17, 18, 19, 23, 26, 28, 30, 32, 33, 34 | fmul01lt1lem2 45143 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → (𝐴‘𝑀) < 𝐸) |
36 | 35 | 3exp 1116 | . . 3 ⊢ (𝜑 → (𝑗 ∈ (1...𝑀) → ((𝐵‘𝑗) < 𝐸 → (𝐴‘𝑀) < 𝐸))) |
37 | 2, 8, 36 | rexlimd 3253 | . 2 ⊢ (𝜑 → (∃𝑗 ∈ (1...𝑀)(𝐵‘𝑗) < 𝐸 → (𝐴‘𝑀) < 𝐸)) |
38 | 1, 37 | mpd 15 | 1 ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2875 ∃wrex 3059 class class class wbr 5152 ⟶wf 6549 ‘cfv 6553 (class class class)co 7423 ℝcr 11153 0cc0 11154 1c1 11155 · cmul 11159 < clt 11294 ≤ cle 11295 ℕcn 12259 ℤ≥cuz 12869 ℝ+crp 13023 ...cfz 13533 seqcseq 14016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-n0 12520 df-z 12606 df-uz 12870 df-rp 13024 df-fz 13534 df-fzo 13677 df-seq 14017 |
This theorem is referenced by: stoweidlem48 45606 |
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