![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > divalglem1 | Structured version Visualization version GIF version |
Description: Lemma for divalg 15744. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
divalglem0.1 | ⊢ 𝑁 ∈ ℤ |
divalglem0.2 | ⊢ 𝐷 ∈ ℤ |
divalglem1.3 | ⊢ 𝐷 ≠ 0 |
Ref | Expression |
---|---|
divalglem1 | ⊢ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divalglem0.1 | . . . . 5 ⊢ 𝑁 ∈ ℤ | |
2 | 1 | zrei 11975 | . . . 4 ⊢ 𝑁 ∈ ℝ |
3 | 0re 10632 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | 2, 3 | letrii 10754 | . . 3 ⊢ (𝑁 ≤ 0 ∨ 0 ≤ 𝑁) |
5 | divalglem0.2 | . . . . . . . 8 ⊢ 𝐷 ∈ ℤ | |
6 | divalglem1.3 | . . . . . . . 8 ⊢ 𝐷 ≠ 0 | |
7 | nnabscl 14677 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ) | |
8 | 5, 6, 7 | mp2an 691 | . . . . . . 7 ⊢ (abs‘𝐷) ∈ ℕ |
9 | nnge1 11653 | . . . . . . 7 ⊢ ((abs‘𝐷) ∈ ℕ → 1 ≤ (abs‘𝐷)) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ 1 ≤ (abs‘𝐷) |
11 | le0neg1 11137 | . . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁)) | |
12 | 2, 11 | ax-mp 5 | . . . . . . 7 ⊢ (𝑁 ≤ 0 ↔ 0 ≤ -𝑁) |
13 | 2 | renegcli 10936 | . . . . . . . 8 ⊢ -𝑁 ∈ ℝ |
14 | 5 | zrei 11975 | . . . . . . . . . 10 ⊢ 𝐷 ∈ ℝ |
15 | 14 | recni 10644 | . . . . . . . . 9 ⊢ 𝐷 ∈ ℂ |
16 | 15 | abscli 14747 | . . . . . . . 8 ⊢ (abs‘𝐷) ∈ ℝ |
17 | lemulge11 11491 | . . . . . . . 8 ⊢ (((-𝑁 ∈ ℝ ∧ (abs‘𝐷) ∈ ℝ) ∧ (0 ≤ -𝑁 ∧ 1 ≤ (abs‘𝐷))) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) | |
18 | 13, 16, 17 | mpanl12 701 | . . . . . . 7 ⊢ ((0 ≤ -𝑁 ∧ 1 ≤ (abs‘𝐷)) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
19 | 12, 18 | sylanb 584 | . . . . . 6 ⊢ ((𝑁 ≤ 0 ∧ 1 ≤ (abs‘𝐷)) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
20 | 10, 19 | mpan2 690 | . . . . 5 ⊢ (𝑁 ≤ 0 → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
21 | 2 | recni 10644 | . . . . . . 7 ⊢ 𝑁 ∈ ℂ |
22 | 21, 15 | absmuli 14756 | . . . . . 6 ⊢ (abs‘(𝑁 · 𝐷)) = ((abs‘𝑁) · (abs‘𝐷)) |
23 | 2 | absnidi 14730 | . . . . . . 7 ⊢ (𝑁 ≤ 0 → (abs‘𝑁) = -𝑁) |
24 | 23 | oveq1d 7150 | . . . . . 6 ⊢ (𝑁 ≤ 0 → ((abs‘𝑁) · (abs‘𝐷)) = (-𝑁 · (abs‘𝐷))) |
25 | 22, 24 | syl5eq 2845 | . . . . 5 ⊢ (𝑁 ≤ 0 → (abs‘(𝑁 · 𝐷)) = (-𝑁 · (abs‘𝐷))) |
26 | 20, 25 | breqtrrd 5058 | . . . 4 ⊢ (𝑁 ≤ 0 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
27 | le0neg2 11138 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (0 ≤ 𝑁 ↔ -𝑁 ≤ 0)) | |
28 | 2, 27 | ax-mp 5 | . . . . 5 ⊢ (0 ≤ 𝑁 ↔ -𝑁 ≤ 0) |
29 | 2, 14 | remulcli 10646 | . . . . . . . 8 ⊢ (𝑁 · 𝐷) ∈ ℝ |
30 | 29 | recni 10644 | . . . . . . 7 ⊢ (𝑁 · 𝐷) ∈ ℂ |
31 | 30 | absge0i 14748 | . . . . . 6 ⊢ 0 ≤ (abs‘(𝑁 · 𝐷)) |
32 | 30 | abscli 14747 | . . . . . . 7 ⊢ (abs‘(𝑁 · 𝐷)) ∈ ℝ |
33 | 13, 3, 32 | letri 10758 | . . . . . 6 ⊢ ((-𝑁 ≤ 0 ∧ 0 ≤ (abs‘(𝑁 · 𝐷))) → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
34 | 31, 33 | mpan2 690 | . . . . 5 ⊢ (-𝑁 ≤ 0 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
35 | 28, 34 | sylbi 220 | . . . 4 ⊢ (0 ≤ 𝑁 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
36 | 26, 35 | jaoi 854 | . . 3 ⊢ ((𝑁 ≤ 0 ∨ 0 ≤ 𝑁) → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
37 | 4, 36 | ax-mp 5 | . 2 ⊢ -𝑁 ≤ (abs‘(𝑁 · 𝐷)) |
38 | df-neg 10862 | . . . 4 ⊢ -𝑁 = (0 − 𝑁) | |
39 | 38 | breq1i 5037 | . . 3 ⊢ (-𝑁 ≤ (abs‘(𝑁 · 𝐷)) ↔ (0 − 𝑁) ≤ (abs‘(𝑁 · 𝐷))) |
40 | 3, 2, 32 | lesubadd2i 11189 | . . 3 ⊢ ((0 − 𝑁) ≤ (abs‘(𝑁 · 𝐷)) ↔ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))) |
41 | 39, 40 | bitri 278 | . 2 ⊢ (-𝑁 ≤ (abs‘(𝑁 · 𝐷)) ↔ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))) |
42 | 37, 41 | mpbi 233 | 1 ⊢ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∨ wo 844 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ≤ cle 10665 − cmin 10859 -cneg 10860 ℕcn 11625 ℤcz 11969 abscabs 14585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: divalglem2 15736 |
Copyright terms: Public domain | W3C validator |