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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 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 1890 | . . 3 ⊢ Ⅎ𝑗𝜑 | |
3 | fmul01lt1.3 | . . . . 5 ⊢ Ⅎ𝑗𝐴 | |
4 | nfcv 2947 | . . . . 5 ⊢ Ⅎ𝑗𝑀 | |
5 | 3, 4 | nffv 6540 | . . . 4 ⊢ Ⅎ𝑗(𝐴‘𝑀) |
6 | nfcv 2947 | . . . 4 ⊢ Ⅎ𝑗 < | |
7 | nfcv 2947 | . . . 4 ⊢ Ⅎ𝑗𝐸 | |
8 | 5, 6, 7 | nfbr 5003 | . . 3 ⊢ Ⅎ𝑗(𝐴‘𝑀) < 𝐸 |
9 | fmul01lt1.1 | . . . . 5 ⊢ Ⅎ𝑖𝐵 | |
10 | fmul01lt1.2 | . . . . . 6 ⊢ Ⅎ𝑖𝜑 | |
11 | nfv 1890 | . . . . . 6 ⊢ Ⅎ𝑖 𝑗 ∈ (1...𝑀) | |
12 | nfcv 2947 | . . . . . . . 8 ⊢ Ⅎ𝑖𝑗 | |
13 | 9, 12 | nffv 6540 | . . . . . . 7 ⊢ Ⅎ𝑖(𝐵‘𝑗) |
14 | nfcv 2947 | . . . . . . 7 ⊢ Ⅎ𝑖 < | |
15 | nfcv 2947 | . . . . . . 7 ⊢ Ⅎ𝑖𝐸 | |
16 | 13, 14, 15 | nfbr 5003 | . . . . . 6 ⊢ Ⅎ𝑖(𝐵‘𝑗) < 𝐸 |
17 | 10, 11, 16 | nf3an 1881 | . . . . 5 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) |
18 | fmul01lt1.4 | . . . . 5 ⊢ 𝐴 = seq1( · , 𝐵) | |
19 | 1zzd 11851 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 1 ∈ ℤ) | |
20 | fmul01lt1.5 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
21 | elnnuz 12120 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ≥‘1)) | |
22 | 20, 21 | sylib 219 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1)) |
23 | 22 | 3ad2ant1 1124 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝑀 ∈ (ℤ≥‘1)) |
24 | fmul01lt1.6 | . . . . . . 7 ⊢ (𝜑 → 𝐵:(1...𝑀)⟶ℝ) | |
25 | 24 | ffvelrnda 6707 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
26 | 25 | 3ad2antl1 1176 | . . . . 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 1176 | . . . . 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 1176 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ≤ 1) |
31 | fmul01lt1.9 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
32 | 31 | 3ad2ant1 1124 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝐸 ∈ ℝ+) |
33 | simp2 1128 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → 𝑗 ∈ (1...𝑀)) | |
34 | simp3 1129 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → (𝐵‘𝑗) < 𝐸) | |
35 | 9, 17, 18, 19, 23, 26, 28, 30, 32, 33, 34 | fmul01lt1lem2 41362 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ (𝐵‘𝑗) < 𝐸) → (𝐴‘𝑀) < 𝐸) |
36 | 35 | 3exp 1110 | . . 3 ⊢ (𝜑 → (𝑗 ∈ (1...𝑀) → ((𝐵‘𝑗) < 𝐸 → (𝐴‘𝑀) < 𝐸))) |
37 | 2, 8, 36 | rexlimd 3275 | . 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 396 ∧ w3a 1078 = wceq 1520 Ⅎwnf 1763 ∈ wcel 2079 Ⅎwnfc 2931 ∃wrex 3104 class class class wbr 4956 ⟶wf 6213 ‘cfv 6217 (class class class)co 7007 ℝcr 10371 0cc0 10372 1c1 10373 · cmul 10377 < clt 10510 ≤ cle 10511 ℕcn 11475 ℤ≥cuz 12082 ℝ+crp 12228 ...cfz 12731 seqcseq 13207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-n0 11735 df-z 11819 df-uz 12083 df-rp 12229 df-fz 12732 df-fzo 12873 df-seq 13208 |
This theorem is referenced by: stoweidlem48 41829 |
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