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| Mirrors > Home > MPE Home > Th. List > divalglem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for divalg 16440. (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 12619 | . . . 4 ⊢ 𝑁 ∈ ℝ |
| 3 | 0re 11263 | . . . 4 ⊢ 0 ∈ ℝ | |
| 4 | 2, 3 | letrii 11386 | . . 3 ⊢ (𝑁 ≤ 0 ∨ 0 ≤ 𝑁) |
| 5 | divalglem0.2 | . . . . . . . 8 ⊢ 𝐷 ∈ ℤ | |
| 6 | divalglem1.3 | . . . . . . . 8 ⊢ 𝐷 ≠ 0 | |
| 7 | nnabscl 15364 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ) | |
| 8 | 5, 6, 7 | mp2an 692 | . . . . . . 7 ⊢ (abs‘𝐷) ∈ ℕ |
| 9 | nnge1 12294 | . . . . . . 7 ⊢ ((abs‘𝐷) ∈ ℕ → 1 ≤ (abs‘𝐷)) | |
| 10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ 1 ≤ (abs‘𝐷) |
| 11 | le0neg1 11771 | . . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁)) | |
| 12 | 2, 11 | ax-mp 5 | . . . . . . 7 ⊢ (𝑁 ≤ 0 ↔ 0 ≤ -𝑁) |
| 13 | 2 | renegcli 11570 | . . . . . . . 8 ⊢ -𝑁 ∈ ℝ |
| 14 | 5 | zrei 12619 | . . . . . . . . . 10 ⊢ 𝐷 ∈ ℝ |
| 15 | 14 | recni 11275 | . . . . . . . . 9 ⊢ 𝐷 ∈ ℂ |
| 16 | 15 | abscli 15434 | . . . . . . . 8 ⊢ (abs‘𝐷) ∈ ℝ |
| 17 | lemulge11 12130 | . . . . . . . 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 581 | . . . . . 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 11275 | . . . . . . 7 ⊢ 𝑁 ∈ ℂ |
| 22 | 21, 15 | absmuli 15443 | . . . . . 6 ⊢ (abs‘(𝑁 · 𝐷)) = ((abs‘𝑁) · (abs‘𝐷)) |
| 23 | 2 | absnidi 15417 | . . . . . . 7 ⊢ (𝑁 ≤ 0 → (abs‘𝑁) = -𝑁) |
| 24 | 23 | oveq1d 7446 | . . . . . 6 ⊢ (𝑁 ≤ 0 → ((abs‘𝑁) · (abs‘𝐷)) = (-𝑁 · (abs‘𝐷))) |
| 25 | 22, 24 | eqtrid 2789 | . . . . 5 ⊢ (𝑁 ≤ 0 → (abs‘(𝑁 · 𝐷)) = (-𝑁 · (abs‘𝐷))) |
| 26 | 20, 25 | breqtrrd 5171 | . . . 4 ⊢ (𝑁 ≤ 0 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
| 27 | le0neg2 11772 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (0 ≤ 𝑁 ↔ -𝑁 ≤ 0)) | |
| 28 | 2, 27 | ax-mp 5 | . . . . 5 ⊢ (0 ≤ 𝑁 ↔ -𝑁 ≤ 0) |
| 29 | 2, 14 | remulcli 11277 | . . . . . . . 8 ⊢ (𝑁 · 𝐷) ∈ ℝ |
| 30 | 29 | recni 11275 | . . . . . . 7 ⊢ (𝑁 · 𝐷) ∈ ℂ |
| 31 | 30 | absge0i 15435 | . . . . . 6 ⊢ 0 ≤ (abs‘(𝑁 · 𝐷)) |
| 32 | 30 | abscli 15434 | . . . . . . 7 ⊢ (abs‘(𝑁 · 𝐷)) ∈ ℝ |
| 33 | 13, 3, 32 | letri 11390 | . . . . . 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 217 | . . . 4 ⊢ (0 ≤ 𝑁 → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
| 36 | 26, 35 | jaoi 858 | . . 3 ⊢ ((𝑁 ≤ 0 ∨ 0 ≤ 𝑁) → -𝑁 ≤ (abs‘(𝑁 · 𝐷))) |
| 37 | 4, 36 | ax-mp 5 | . 2 ⊢ -𝑁 ≤ (abs‘(𝑁 · 𝐷)) |
| 38 | df-neg 11495 | . . . 4 ⊢ -𝑁 = (0 − 𝑁) | |
| 39 | 38 | breq1i 5150 | . . 3 ⊢ (-𝑁 ≤ (abs‘(𝑁 · 𝐷)) ↔ (0 − 𝑁) ≤ (abs‘(𝑁 · 𝐷))) |
| 40 | 3, 2, 32 | lesubadd2i 11823 | . . 3 ⊢ ((0 − 𝑁) ≤ (abs‘(𝑁 · 𝐷)) ↔ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))) |
| 41 | 39, 40 | bitri 275 | . 2 ⊢ (-𝑁 ≤ (abs‘(𝑁 · 𝐷)) ↔ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))) |
| 42 | 37, 41 | mpbi 230 | 1 ⊢ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∨ wo 848 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 -cneg 11493 ℕcn 12266 ℤcz 12613 abscabs 15273 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 |
| This theorem is referenced by: divalglem2 16432 |
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