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Mirrors > Home > MPE Home > Th. List > divalglem1 | Structured version Visualization version GIF version |
Description: Lemma for divalg 14964. (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 11260 | . . . 4 ⊢ 𝑁 ∈ ℝ |
3 | 0re 9919 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | 2, 3 | letrii 10041 | . . 3 ⊢ (𝑁 ≤ 0 ∨ 0 ≤ 𝑁) |
5 | divalglem0.2 | . . . . . . . 8 ⊢ 𝐷 ∈ ℤ | |
6 | divalglem1.3 | . . . . . . . 8 ⊢ 𝐷 ≠ 0 | |
7 | nnabscl 13913 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ) | |
8 | 5, 6, 7 | mp2an 704 | . . . . . . 7 ⊢ (abs‘𝐷) ∈ ℕ |
9 | nnge1 10923 | . . . . . . 7 ⊢ ((abs‘𝐷) ∈ ℕ → 1 ≤ (abs‘𝐷)) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ 1 ≤ (abs‘𝐷) |
11 | le0neg1 10415 | . . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁)) | |
12 | 2, 11 | ax-mp 5 | . . . . . . 7 ⊢ (𝑁 ≤ 0 ↔ 0 ≤ -𝑁) |
13 | 2 | renegcli 10221 | . . . . . . . 8 ⊢ -𝑁 ∈ ℝ |
14 | 5 | zrei 11260 | . . . . . . . . . 10 ⊢ 𝐷 ∈ ℝ |
15 | 14 | recni 9931 | . . . . . . . . 9 ⊢ 𝐷 ∈ ℂ |
16 | 15 | abscli 13982 | . . . . . . . 8 ⊢ (abs‘𝐷) ∈ ℝ |
17 | lemulge11 10764 | . . . . . . . 8 ⊢ (((-𝑁 ∈ ℝ ∧ (abs‘𝐷) ∈ ℝ) ∧ (0 ≤ -𝑁 ∧ 1 ≤ (abs‘𝐷))) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) | |
18 | 13, 16, 17 | mpanl12 714 | . . . . . . 7 ⊢ ((0 ≤ -𝑁 ∧ 1 ≤ (abs‘𝐷)) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
19 | 12, 18 | sylanb 488 | . . . . . 6 ⊢ ((𝑁 ≤ 0 ∧ 1 ≤ (abs‘𝐷)) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
20 | 10, 19 | mpan2 703 | . . . . 5 ⊢ (𝑁 ≤ 0 → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
21 | 2 | recni 9931 | . . . . . . 7 ⊢ 𝑁 ∈ ℂ |
22 | 21, 15 | absmuli 13991 | . . . . . 6 ⊢ (abs‘(𝑁 · 𝐷)) = ((abs‘𝑁) · (abs‘𝐷)) |
23 | 2 | absnidi 13966 | . . . . . . 7 ⊢ (𝑁 ≤ 0 → (abs‘𝑁) = -𝑁) |
24 | 23 | oveq1d 6564 | . . . . . 6 ⊢ (𝑁 ≤ 0 → ((abs‘𝑁) · (abs‘𝐷)) = (-𝑁 · (abs‘𝐷))) |
25 | 22, 24 | syl5eq 2656 | . . . . 5 ⊢ (𝑁 ≤ 0 → (abs‘(𝑁 · 𝐷)) = (-𝑁 · (abs‘𝐷))) |
26 | 20, 25 | breqtrrd 4611 | . . . 4 ⊢ (𝑁 ≤ 0 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
27 | le0neg2 10416 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (0 ≤ 𝑁 ↔ -𝑁 ≤ 0)) | |
28 | 2, 27 | ax-mp 5 | . . . . 5 ⊢ (0 ≤ 𝑁 ↔ -𝑁 ≤ 0) |
29 | 2, 14 | remulcli 9933 | . . . . . . . 8 ⊢ (𝑁 · 𝐷) ∈ ℝ |
30 | 29 | recni 9931 | . . . . . . 7 ⊢ (𝑁 · 𝐷) ∈ ℂ |
31 | 30 | absge0i 13983 | . . . . . 6 ⊢ 0 ≤ (abs‘(𝑁 · 𝐷)) |
32 | 30 | abscli 13982 | . . . . . . 7 ⊢ (abs‘(𝑁 · 𝐷)) ∈ ℝ |
33 | 13, 3, 32 | letri 10045 | . . . . . 6 ⊢ ((-𝑁 ≤ 0 ∧ 0 ≤ (abs‘(𝑁 · 𝐷))) → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
34 | 31, 33 | mpan2 703 | . . . . 5 ⊢ (-𝑁 ≤ 0 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
35 | 28, 34 | sylbi 206 | . . . 4 ⊢ (0 ≤ 𝑁 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
36 | 26, 35 | jaoi 393 | . . 3 ⊢ ((𝑁 ≤ 0 ∨ 0 ≤ 𝑁) → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
37 | 4, 36 | ax-mp 5 | . 2 ⊢ -𝑁 ≤ (abs‘(𝑁 · 𝐷)) |
38 | df-neg 10148 | . . . 4 ⊢ -𝑁 = (0 − 𝑁) | |
39 | 38 | breq1i 4590 | . . 3 ⊢ (-𝑁 ≤ (abs‘(𝑁 · 𝐷)) ↔ (0 − 𝑁) ≤ (abs‘(𝑁 · 𝐷))) |
40 | 3, 2, 32 | lesubadd2i 10467 | . . 3 ⊢ ((0 − 𝑁) ≤ (abs‘(𝑁 · 𝐷)) ↔ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))) |
41 | 39, 40 | bitri 263 | . 2 ⊢ (-𝑁 ≤ (abs‘(𝑁 · 𝐷)) ↔ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))) |
42 | 37, 41 | mpbi 219 | 1 ⊢ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ≤ cle 9954 − cmin 10145 -cneg 10146 ℕcn 10897 ℤcz 11254 abscabs 13822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 |
This theorem is referenced by: divalglem2 14956 |
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