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Mirrors > Home > MPE Home > Th. List > divalglem1 | Structured version Visualization version GIF version |
Description: Lemma for divalg 15754. (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 11988 | . . . 4 ⊢ 𝑁 ∈ ℝ |
3 | 0re 10643 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | 2, 3 | letrii 10765 | . . 3 ⊢ (𝑁 ≤ 0 ∨ 0 ≤ 𝑁) |
5 | divalglem0.2 | . . . . . . . 8 ⊢ 𝐷 ∈ ℤ | |
6 | divalglem1.3 | . . . . . . . 8 ⊢ 𝐷 ≠ 0 | |
7 | nnabscl 14685 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ) | |
8 | 5, 6, 7 | mp2an 690 | . . . . . . 7 ⊢ (abs‘𝐷) ∈ ℕ |
9 | nnge1 11666 | . . . . . . 7 ⊢ ((abs‘𝐷) ∈ ℕ → 1 ≤ (abs‘𝐷)) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ 1 ≤ (abs‘𝐷) |
11 | le0neg1 11148 | . . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁)) | |
12 | 2, 11 | ax-mp 5 | . . . . . . 7 ⊢ (𝑁 ≤ 0 ↔ 0 ≤ -𝑁) |
13 | 2 | renegcli 10947 | . . . . . . . 8 ⊢ -𝑁 ∈ ℝ |
14 | 5 | zrei 11988 | . . . . . . . . . 10 ⊢ 𝐷 ∈ ℝ |
15 | 14 | recni 10655 | . . . . . . . . 9 ⊢ 𝐷 ∈ ℂ |
16 | 15 | abscli 14755 | . . . . . . . 8 ⊢ (abs‘𝐷) ∈ ℝ |
17 | lemulge11 11502 | . . . . . . . 8 ⊢ (((-𝑁 ∈ ℝ ∧ (abs‘𝐷) ∈ ℝ) ∧ (0 ≤ -𝑁 ∧ 1 ≤ (abs‘𝐷))) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) | |
18 | 13, 16, 17 | mpanl12 700 | . . . . . . 7 ⊢ ((0 ≤ -𝑁 ∧ 1 ≤ (abs‘𝐷)) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
19 | 12, 18 | sylanb 583 | . . . . . 6 ⊢ ((𝑁 ≤ 0 ∧ 1 ≤ (abs‘𝐷)) → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
20 | 10, 19 | mpan2 689 | . . . . 5 ⊢ (𝑁 ≤ 0 → -𝑁 ≤ (-𝑁 · (abs‘𝐷))) |
21 | 2 | recni 10655 | . . . . . . 7 ⊢ 𝑁 ∈ ℂ |
22 | 21, 15 | absmuli 14764 | . . . . . 6 ⊢ (abs‘(𝑁 · 𝐷)) = ((abs‘𝑁) · (abs‘𝐷)) |
23 | 2 | absnidi 14738 | . . . . . . 7 ⊢ (𝑁 ≤ 0 → (abs‘𝑁) = -𝑁) |
24 | 23 | oveq1d 7171 | . . . . . 6 ⊢ (𝑁 ≤ 0 → ((abs‘𝑁) · (abs‘𝐷)) = (-𝑁 · (abs‘𝐷))) |
25 | 22, 24 | syl5eq 2868 | . . . . 5 ⊢ (𝑁 ≤ 0 → (abs‘(𝑁 · 𝐷)) = (-𝑁 · (abs‘𝐷))) |
26 | 20, 25 | breqtrrd 5094 | . . . 4 ⊢ (𝑁 ≤ 0 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
27 | le0neg2 11149 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (0 ≤ 𝑁 ↔ -𝑁 ≤ 0)) | |
28 | 2, 27 | ax-mp 5 | . . . . 5 ⊢ (0 ≤ 𝑁 ↔ -𝑁 ≤ 0) |
29 | 2, 14 | remulcli 10657 | . . . . . . . 8 ⊢ (𝑁 · 𝐷) ∈ ℝ |
30 | 29 | recni 10655 | . . . . . . 7 ⊢ (𝑁 · 𝐷) ∈ ℂ |
31 | 30 | absge0i 14756 | . . . . . 6 ⊢ 0 ≤ (abs‘(𝑁 · 𝐷)) |
32 | 30 | abscli 14755 | . . . . . . 7 ⊢ (abs‘(𝑁 · 𝐷)) ∈ ℝ |
33 | 13, 3, 32 | letri 10769 | . . . . . 6 ⊢ ((-𝑁 ≤ 0 ∧ 0 ≤ (abs‘(𝑁 · 𝐷))) → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
34 | 31, 33 | mpan2 689 | . . . . 5 ⊢ (-𝑁 ≤ 0 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
35 | 28, 34 | sylbi 219 | . . . 4 ⊢ (0 ≤ 𝑁 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
36 | 26, 35 | jaoi 853 | . . 3 ⊢ ((𝑁 ≤ 0 ∨ 0 ≤ 𝑁) → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
37 | 4, 36 | ax-mp 5 | . 2 ⊢ -𝑁 ≤ (abs‘(𝑁 · 𝐷)) |
38 | df-neg 10873 | . . . 4 ⊢ -𝑁 = (0 − 𝑁) | |
39 | 38 | breq1i 5073 | . . 3 ⊢ (-𝑁 ≤ (abs‘(𝑁 · 𝐷)) ↔ (0 − 𝑁) ≤ (abs‘(𝑁 · 𝐷))) |
40 | 3, 2, 32 | lesubadd2i 11200 | . . 3 ⊢ ((0 − 𝑁) ≤ (abs‘(𝑁 · 𝐷)) ↔ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))) |
41 | 39, 40 | bitri 277 | . 2 ⊢ (-𝑁 ≤ (abs‘(𝑁 · 𝐷)) ↔ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))) |
42 | 37, 41 | mpbi 232 | 1 ⊢ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∨ wo 843 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ≤ cle 10676 − cmin 10870 -cneg 10871 ℕcn 11638 ℤcz 11982 abscabs 14593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 |
This theorem is referenced by: divalglem2 15746 |
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