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Mirrors > Home > MPE Home > Th. List > divalglem1 | Structured version Visualization version GIF version |
Description: Lemma for divalg 15860. (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 12080 | . . . 4 ⊢ 𝑁 ∈ ℝ |
3 | 0re 10733 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | 2, 3 | letrii 10855 | . . 3 ⊢ (𝑁 ≤ 0 ∨ 0 ≤ 𝑁) |
5 | divalglem0.2 | . . . . . . . 8 ⊢ 𝐷 ∈ ℤ | |
6 | divalglem1.3 | . . . . . . . 8 ⊢ 𝐷 ≠ 0 | |
7 | nnabscl 14787 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ) | |
8 | 5, 6, 7 | mp2an 692 | . . . . . . 7 ⊢ (abs‘𝐷) ∈ ℕ |
9 | nnge1 11756 | . . . . . . 7 ⊢ ((abs‘𝐷) ∈ ℕ → 1 ≤ (abs‘𝐷)) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ 1 ≤ (abs‘𝐷) |
11 | le0neg1 11238 | . . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁)) | |
12 | 2, 11 | ax-mp 5 | . . . . . . 7 ⊢ (𝑁 ≤ 0 ↔ 0 ≤ -𝑁) |
13 | 2 | renegcli 11037 | . . . . . . . 8 ⊢ -𝑁 ∈ ℝ |
14 | 5 | zrei 12080 | . . . . . . . . . 10 ⊢ 𝐷 ∈ ℝ |
15 | 14 | recni 10745 | . . . . . . . . 9 ⊢ 𝐷 ∈ ℂ |
16 | 15 | abscli 14857 | . . . . . . . 8 ⊢ (abs‘𝐷) ∈ ℝ |
17 | lemulge11 11592 | . . . . . . . 8 ⊢ (((-𝑁 ∈ ℝ ∧ (abs‘𝐷) ∈ ℝ) ∧ (0 ≤ -𝑁 ∧ 1 ≤ (abs‘𝐷))) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) | |
18 | 13, 16, 17 | mpanl12 702 | . . . . . . 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 691 | . . . . 5 ⊢ (𝑁 ≤ 0 → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
21 | 2 | recni 10745 | . . . . . . 7 ⊢ 𝑁 ∈ ℂ |
22 | 21, 15 | absmuli 14866 | . . . . . 6 ⊢ (abs‘(𝑁 · 𝐷)) = ((abs‘𝑁) · (abs‘𝐷)) |
23 | 2 | absnidi 14840 | . . . . . . 7 ⊢ (𝑁 ≤ 0 → (abs‘𝑁) = -𝑁) |
24 | 23 | oveq1d 7197 | . . . . . 6 ⊢ (𝑁 ≤ 0 → ((abs‘𝑁) · (abs‘𝐷)) = (-𝑁 · (abs‘𝐷))) |
25 | 22, 24 | syl5eq 2786 | . . . . 5 ⊢ (𝑁 ≤ 0 → (abs‘(𝑁 · 𝐷)) = (-𝑁 · (abs‘𝐷))) |
26 | 20, 25 | breqtrrd 5068 | . . . 4 ⊢ (𝑁 ≤ 0 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
27 | le0neg2 11239 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (0 ≤ 𝑁 ↔ -𝑁 ≤ 0)) | |
28 | 2, 27 | ax-mp 5 | . . . . 5 ⊢ (0 ≤ 𝑁 ↔ -𝑁 ≤ 0) |
29 | 2, 14 | remulcli 10747 | . . . . . . . 8 ⊢ (𝑁 · 𝐷) ∈ ℝ |
30 | 29 | recni 10745 | . . . . . . 7 ⊢ (𝑁 · 𝐷) ∈ ℂ |
31 | 30 | absge0i 14858 | . . . . . 6 ⊢ 0 ≤ (abs‘(𝑁 · 𝐷)) |
32 | 30 | abscli 14857 | . . . . . . 7 ⊢ (abs‘(𝑁 · 𝐷)) ∈ ℝ |
33 | 13, 3, 32 | letri 10859 | . . . . . 6 ⊢ ((-𝑁 ≤ 0 ∧ 0 ≤ (abs‘(𝑁 · 𝐷))) → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
34 | 31, 33 | mpan2 691 | . . . . 5 ⊢ (-𝑁 ≤ 0 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
35 | 28, 34 | sylbi 220 | . . . 4 ⊢ (0 ≤ 𝑁 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
36 | 26, 35 | jaoi 856 | . . 3 ⊢ ((𝑁 ≤ 0 ∨ 0 ≤ 𝑁) → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
37 | 4, 36 | ax-mp 5 | . 2 ⊢ -𝑁 ≤ (abs‘(𝑁 · 𝐷)) |
38 | df-neg 10963 | . . . 4 ⊢ -𝑁 = (0 − 𝑁) | |
39 | 38 | breq1i 5047 | . . 3 ⊢ (-𝑁 ≤ (abs‘(𝑁 · 𝐷)) ↔ (0 − 𝑁) ≤ (abs‘(𝑁 · 𝐷))) |
40 | 3, 2, 32 | lesubadd2i 11290 | . . 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 846 ∈ wcel 2114 ≠ wne 2935 class class class wbr 5040 ‘cfv 6349 (class class class)co 7182 ℝcr 10626 0cc0 10627 1c1 10628 + caddc 10630 · cmul 10632 ≤ cle 10766 − cmin 10960 -cneg 10961 ℕcn 11728 ℤcz 12074 abscabs 14695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-sup 8991 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-n0 11989 df-z 12075 df-uz 12337 df-rp 12485 df-seq 13473 df-exp 13534 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 |
This theorem is referenced by: divalglem2 15852 |
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