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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem13 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 36121. (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 479 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 0) → 1 < (𝑁 · (abs‘𝐶))) |
3 | 0lt1 11767 | . . . . . 6 ⊢ 0 < 1 | |
4 | 0re 11247 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
5 | 1re 11245 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | ltnsymi 11364 | . . . . . 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 15272 | . . . . . . . . 9 ⊢ (𝐶 = 0 → (abs‘𝐶) = 0) |
11 | 10 | oveq2d 7434 | . . . . . . . 8 ⊢ (𝐶 = 0 → (𝑁 · (abs‘𝐶)) = (𝑁 · 0)) |
12 | 11 | adantl 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝑁 · (abs‘𝐶)) = (𝑁 · 0)) |
13 | knoppndvlem13.n | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
14 | nncn 12251 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
15 | 13, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
16 | 15 | adantr 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 0) → 𝑁 ∈ ℂ) |
17 | 16 | mul01d 11444 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝑁 · 0) = 0) |
18 | 12, 17 | eqtrd 2765 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝑁 · (abs‘𝐶)) = 0) |
19 | 18 | eqcomd 2731 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 0) → 0 = (𝑁 · (abs‘𝐶))) |
20 | 19 | breq2d 5161 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 0) → (1 < 0 ↔ 1 < (𝑁 · (abs‘𝐶)))) |
21 | 8, 20 | mtbid 323 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 0) → ¬ 1 < (𝑁 · (abs‘𝐶))) |
22 | 2, 21 | pm2.65da 815 | . 2 ⊢ (𝜑 → ¬ 𝐶 = 0) |
23 | 22 | neqned 2936 | 1 ⊢ (𝜑 → 𝐶 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 class class class wbr 5149 ‘cfv 6548 (class class class)co 7418 ℂcc 11137 0cc0 11139 1c1 11140 · cmul 11144 < clt 11279 -cneg 11476 ℕcn 12243 (,)cioo 13357 abscabs 15215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6306 df-ord 6373 df-on 6374 df-lim 6375 df-suc 6376 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7374 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-seq 14001 df-exp 14061 df-cj 15080 df-re 15081 df-im 15082 df-sqrt 15216 df-abs 15217 |
This theorem is referenced by: knoppndvlem14 36112 knoppndvlem17 36115 |
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