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 betweeen 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 1922 | . . 3 ⊢ Ⅎ𝑗𝜑 | |
3 | fmul01lt1.3 | . . . . 5 ⊢ Ⅎ𝑗𝐴 | |
4 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑗𝑀 | |
5 | 3, 4 | nffv 6727 | . . . 4 ⊢ Ⅎ𝑗(𝐴‘𝑀) |
6 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑗 < | |
7 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑗𝐸 | |
8 | 5, 6, 7 | nfbr 5100 | . . 3 ⊢ Ⅎ𝑗(𝐴‘𝑀) < 𝐸 |
9 | fmul01lt1.1 | . . . . 5 ⊢ Ⅎ𝑖𝐵 | |
10 | fmul01lt1.2 | . . . . . 6 ⊢ Ⅎ𝑖𝜑 | |
11 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑖 𝑗 ∈ (1...𝑀) | |
12 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑖𝑗 | |
13 | 9, 12 | nffv 6727 | . . . . . . 7 ⊢ Ⅎ𝑖(𝐵‘𝑗) |
14 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑖 < | |
15 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑖𝐸 | |
16 | 13, 14, 15 | nfbr 5100 | . . . . . 6 ⊢ Ⅎ𝑖(𝐵‘𝑗) < 𝐸 |
17 | 10, 11, 16 | nf3an 1909 | . . . . 5 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) |
18 | fmul01lt1.4 | . . . . 5 ⊢ 𝐴 = seq1( · , 𝐵) | |
19 | 1zzd 12208 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 1 ∈ ℤ) | |
20 | fmul01lt1.5 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
21 | elnnuz 12478 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ≥‘1)) | |
22 | 20, 21 | sylib 221 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1)) |
23 | 22 | 3ad2ant1 1135 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝑀 ∈ (ℤ≥‘1)) |
24 | fmul01lt1.6 | . . . . . . 7 ⊢ (𝜑 → 𝐵:(1...𝑀)⟶ℝ) | |
25 | 24 | ffvelrnda 6904 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
26 | 25 | 3ad2antl1 1187 | . . . . 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 1187 | . . . . 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 1187 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ≤ 1) |
31 | fmul01lt1.9 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
32 | 31 | 3ad2ant1 1135 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝐸 ∈ ℝ+) |
33 | simp2 1139 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝑗 ∈ (1...𝑀)) | |
34 | simp3 1140 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → (𝐵‘𝑗) < 𝐸) | |
35 | 9, 17, 18, 19, 23, 26, 28, 30, 32, 33, 34 | fmul01lt1lem2 42801 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → (𝐴‘𝑀) < 𝐸) |
36 | 35 | 3exp 1121 | . . 3 ⊢ (𝜑 → (𝑗 ∈ (1...𝑀) → ((𝐵‘𝑗) < 𝐸 → (𝐴‘𝑀) < 𝐸))) |
37 | 2, 8, 36 | rexlimd 3236 | . 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 399 ∧ w3a 1089 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2884 ∃wrex 3062 class class class wbr 5053 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 0cc0 10729 1c1 10730 · cmul 10734 < clt 10867 ≤ cle 10868 ℕcn 11830 ℤ≥cuz 12438 ℝ+crp 12586 ...cfz 13095 seqcseq 13574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-seq 13575 |
This theorem is referenced by: stoweidlem48 43264 |
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