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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem13 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 33875. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
Ref | Expression |
---|---|
knoppndvlem13.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
knoppndvlem13.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
knoppndvlem13.1 | ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) |
Ref | Expression |
---|---|
knoppndvlem13 | ⊢ (𝜑 → 𝐶 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | knoppndvlem13.1 | . . . 4 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 0) → 1 < (𝑁 · (abs‘𝐶))) |
3 | 0lt1 11164 | . . . . . 6 ⊢ 0 < 1 | |
4 | 0re 10645 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
5 | 1re 10643 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | ltnsymi 10761 | . . . . . 6 ⊢ (0 < 1 → ¬ 1 < 0) |
7 | 3, 6 | ax-mp 5 | . . . . 5 ⊢ ¬ 1 < 0 |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 0) → ¬ 1 < 0) |
9 | id 22 | . . . . . . . . . 10 ⊢ (𝐶 = 0 → 𝐶 = 0) | |
10 | 9 | abs00bd 14653 | . . . . . . . . 9 ⊢ (𝐶 = 0 → (abs‘𝐶) = 0) |
11 | 10 | oveq2d 7174 | . . . . . . . 8 ⊢ (𝐶 = 0 → (𝑁 · (abs‘𝐶)) = (𝑁 · 0)) |
12 | 11 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝑁 · (abs‘𝐶)) = (𝑁 · 0)) |
13 | knoppndvlem13.n | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
14 | nncn 11648 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
15 | 13, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
16 | 15 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 0) → 𝑁 ∈ ℂ) |
17 | 16 | mul01d 10841 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝑁 · 0) = 0) |
18 | 12, 17 | eqtrd 2858 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝑁 · (abs‘𝐶)) = 0) |
19 | 18 | eqcomd 2829 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 0) → 0 = (𝑁 · (abs‘𝐶))) |
20 | 19 | breq2d 5080 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 0) → (1 < 0 ↔ 1 < (𝑁 · (abs‘𝐶)))) |
21 | 8, 20 | mtbid 326 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 0) → ¬ 1 < (𝑁 · (abs‘𝐶))) |
22 | 2, 21 | pm2.65da 815 | . 2 ⊢ (𝜑 → ¬ 𝐶 = 0) |
23 | 22 | neqned 3025 | 1 ⊢ (𝜑 → 𝐶 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 · cmul 10544 < clt 10677 -cneg 10873 ℕcn 11640 (,)cioo 12741 abscabs 14595 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 |
This theorem is referenced by: knoppndvlem14 33866 knoppndvlem17 33869 |
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