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Mirrors > Home > MPE Home > Th. List > divalglem1 | Structured version Visualization version GIF version |
Description: Lemma for divalg 16112. (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 12325 | . . . 4 ⊢ 𝑁 ∈ ℝ |
3 | 0re 10977 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | 2, 3 | letrii 11100 | . . 3 ⊢ (𝑁 ≤ 0 ∨ 0 ≤ 𝑁) |
5 | divalglem0.2 | . . . . . . . 8 ⊢ 𝐷 ∈ ℤ | |
6 | divalglem1.3 | . . . . . . . 8 ⊢ 𝐷 ≠ 0 | |
7 | nnabscl 15037 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ) | |
8 | 5, 6, 7 | mp2an 689 | . . . . . . 7 ⊢ (abs‘𝐷) ∈ ℕ |
9 | nnge1 12001 | . . . . . . 7 ⊢ ((abs‘𝐷) ∈ ℕ → 1 ≤ (abs‘𝐷)) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ 1 ≤ (abs‘𝐷) |
11 | le0neg1 11483 | . . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁)) | |
12 | 2, 11 | ax-mp 5 | . . . . . . 7 ⊢ (𝑁 ≤ 0 ↔ 0 ≤ -𝑁) |
13 | 2 | renegcli 11282 | . . . . . . . 8 ⊢ -𝑁 ∈ ℝ |
14 | 5 | zrei 12325 | . . . . . . . . . 10 ⊢ 𝐷 ∈ ℝ |
15 | 14 | recni 10989 | . . . . . . . . 9 ⊢ 𝐷 ∈ ℂ |
16 | 15 | abscli 15107 | . . . . . . . 8 ⊢ (abs‘𝐷) ∈ ℝ |
17 | lemulge11 11837 | . . . . . . . 8 ⊢ (((-𝑁 ∈ ℝ ∧ (abs‘𝐷) ∈ ℝ) ∧ (0 ≤ -𝑁 ∧ 1 ≤ (abs‘𝐷))) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) | |
18 | 13, 16, 17 | mpanl12 699 | . . . . . . 7 ⊢ ((0 ≤ -𝑁 ∧ 1 ≤ (abs‘𝐷)) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
19 | 12, 18 | sylanb 581 | . . . . . 6 ⊢ ((𝑁 ≤ 0 ∧ 1 ≤ (abs‘𝐷)) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
20 | 10, 19 | mpan2 688 | . . . . 5 ⊢ (𝑁 ≤ 0 → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
21 | 2 | recni 10989 | . . . . . . 7 ⊢ 𝑁 ∈ ℂ |
22 | 21, 15 | absmuli 15116 | . . . . . 6 ⊢ (abs‘(𝑁 · 𝐷)) = ((abs‘𝑁) · (abs‘𝐷)) |
23 | 2 | absnidi 15090 | . . . . . . 7 ⊢ (𝑁 ≤ 0 → (abs‘𝑁) = -𝑁) |
24 | 23 | oveq1d 7290 | . . . . . 6 ⊢ (𝑁 ≤ 0 → ((abs‘𝑁) · (abs‘𝐷)) = (-𝑁 · (abs‘𝐷))) |
25 | 22, 24 | eqtrid 2790 | . . . . 5 ⊢ (𝑁 ≤ 0 → (abs‘(𝑁 · 𝐷)) = (-𝑁 · (abs‘𝐷))) |
26 | 20, 25 | breqtrrd 5102 | . . . 4 ⊢ (𝑁 ≤ 0 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
27 | le0neg2 11484 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (0 ≤ 𝑁 ↔ -𝑁 ≤ 0)) | |
28 | 2, 27 | ax-mp 5 | . . . . 5 ⊢ (0 ≤ 𝑁 ↔ -𝑁 ≤ 0) |
29 | 2, 14 | remulcli 10991 | . . . . . . . 8 ⊢ (𝑁 · 𝐷) ∈ ℝ |
30 | 29 | recni 10989 | . . . . . . 7 ⊢ (𝑁 · 𝐷) ∈ ℂ |
31 | 30 | absge0i 15108 | . . . . . 6 ⊢ 0 ≤ (abs‘(𝑁 · 𝐷)) |
32 | 30 | abscli 15107 | . . . . . . 7 ⊢ (abs‘(𝑁 · 𝐷)) ∈ ℝ |
33 | 13, 3, 32 | letri 11104 | . . . . . 6 ⊢ ((-𝑁 ≤ 0 ∧ 0 ≤ (abs‘(𝑁 · 𝐷))) → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
34 | 31, 33 | mpan2 688 | . . . . 5 ⊢ (-𝑁 ≤ 0 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
35 | 28, 34 | sylbi 216 | . . . 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 11208 | . . . 4 ⊢ -𝑁 = (0 − 𝑁) | |
39 | 38 | breq1i 5081 | . . 3 ⊢ (-𝑁 ≤ (abs‘(𝑁 · 𝐷)) ↔ (0 − 𝑁) ≤ (abs‘(𝑁 · 𝐷))) |
40 | 3, 2, 32 | lesubadd2i 11535 | . . 3 ⊢ ((0 − 𝑁) ≤ (abs‘(𝑁 · 𝐷)) ↔ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))) |
41 | 39, 40 | bitri 274 | . 2 ⊢ (-𝑁 ≤ (abs‘(𝑁 · 𝐷)) ↔ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))) |
42 | 37, 41 | mpbi 229 | 1 ⊢ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∨ wo 844 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ≤ cle 11010 − cmin 11205 -cneg 11206 ℕcn 11973 ℤcz 12319 abscabs 14945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 |
This theorem is referenced by: divalglem2 16104 |
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