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 1843 | . . 3 ⊢ Ⅎ𝑗𝜑 | |
3 | fmul01lt1.3 | . . . . 5 ⊢ Ⅎ𝑗𝐴 | |
4 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑗𝑀 | |
5 | 3, 4 | nffv 6198 | . . . 4 ⊢ Ⅎ𝑗(𝐴‘𝑀) |
6 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑗 < | |
7 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑗𝐸 | |
8 | 5, 6, 7 | nfbr 4699 | . . 3 ⊢ Ⅎ𝑗(𝐴‘𝑀) < 𝐸 |
9 | fmul01lt1.1 | . . . . 5 ⊢ Ⅎ𝑖𝐵 | |
10 | fmul01lt1.2 | . . . . . 6 ⊢ Ⅎ𝑖𝜑 | |
11 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑖 𝑗 ∈ (1...𝑀) | |
12 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑖𝑗 | |
13 | 9, 12 | nffv 6198 | . . . . . . 7 ⊢ Ⅎ𝑖(𝐵‘𝑗) |
14 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑖 < | |
15 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑖𝐸 | |
16 | 13, 14, 15 | nfbr 4699 | . . . . . 6 ⊢ Ⅎ𝑖(𝐵‘𝑗) < 𝐸 |
17 | 10, 11, 16 | nf3an 1831 | . . . . 5 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) |
18 | fmul01lt1.4 | . . . . 5 ⊢ 𝐴 = seq1( · , 𝐵) | |
19 | 1zzd 11408 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 1 ∈ ℤ) | |
20 | fmul01lt1.5 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
21 | elnnuz 11724 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ≥‘1)) | |
22 | 20, 21 | sylib 208 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1)) |
23 | 22 | 3ad2ant1 1082 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝑀 ∈ (ℤ≥‘1)) |
24 | fmul01lt1.6 | . . . . . . 7 ⊢ (𝜑 → 𝐵:(1...𝑀)⟶ℝ) | |
25 | 24 | ffvelrnda 6359 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
26 | 25 | 3ad2antl1 1223 | . . . . 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 1223 | . . . . 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 1223 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ≤ 1) |
31 | fmul01lt1.9 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
32 | 31 | 3ad2ant1 1082 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝐸 ∈ ℝ+) |
33 | simp2 1062 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝑗 ∈ (1...𝑀)) | |
34 | simp3 1063 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → (𝐵‘𝑗) < 𝐸) | |
35 | 9, 17, 18, 19, 23, 26, 28, 30, 32, 33, 34 | fmul01lt1lem2 39817 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → (𝐴‘𝑀) < 𝐸) |
36 | 35 | 3exp 1264 | . . 3 ⊢ (𝜑 → (𝑗 ∈ (1...𝑀) → ((𝐵‘𝑗) < 𝐸 → (𝐴‘𝑀) < 𝐸))) |
37 | 2, 8, 36 | rexlimd 3026 | . 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 384 ∧ w3a 1037 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ∃wrex 2913 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 ≤ cle 10075 ℕcn 11020 ℤ≥cuz 11687 ℝ+crp 11832 ...cfz 12326 seqcseq 12801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 |
This theorem is referenced by: stoweidlem48 40265 |
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