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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem12 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 33875. (Contributed by Asger C. Ipsen, 29-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
Ref | Expression |
---|---|
knoppndvlem12.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
knoppndvlem12.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
knoppndvlem12.1 | ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) |
Ref | Expression |
---|---|
knoppndvlem12 | ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 10644 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
2 | 2re 11714 | . . . . . 6 ⊢ 2 ∈ ℝ | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℝ) |
4 | knoppndvlem12.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
5 | nnre 11647 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
7 | 3, 6 | remulcld 10673 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ) |
8 | knoppndvlem12.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
9 | 8 | knoppndvlem3 33855 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
10 | 9 | simpld 497 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
11 | 10 | recnd 10671 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
12 | 11 | abscld 14798 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ) |
13 | 7, 12 | remulcld 10673 | . . . . 5 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ) |
14 | 1lt2 11811 | . . . . . 6 ⊢ 1 < 2 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 < 2) |
16 | 2t1e2 11803 | . . . . . . . . 9 ⊢ (2 · 1) = 2 | |
17 | 16 | eqcomi 2832 | . . . . . . . 8 ⊢ 2 = (2 · 1) |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 = (2 · 1)) |
19 | 6, 12 | remulcld 10673 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 · (abs‘𝐶)) ∈ ℝ) |
20 | 2rp 12397 | . . . . . . . . 9 ⊢ 2 ∈ ℝ+ | |
21 | 20 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ+) |
22 | knoppndvlem12.1 | . . . . . . . 8 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) | |
23 | 1, 19, 21, 22 | ltmul2dd 12490 | . . . . . . 7 ⊢ (𝜑 → (2 · 1) < (2 · (𝑁 · (abs‘𝐶)))) |
24 | 18, 23 | eqbrtrd 5090 | . . . . . 6 ⊢ (𝜑 → 2 < (2 · (𝑁 · (abs‘𝐶)))) |
25 | 3 | recnd 10671 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ) |
26 | 6 | recnd 10671 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
27 | 12 | recnd 10671 | . . . . . . . 8 ⊢ (𝜑 → (abs‘𝐶) ∈ ℂ) |
28 | 25, 26, 27 | mulassd 10666 | . . . . . . 7 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) = (2 · (𝑁 · (abs‘𝐶)))) |
29 | 28 | eqcomd 2829 | . . . . . 6 ⊢ (𝜑 → (2 · (𝑁 · (abs‘𝐶))) = ((2 · 𝑁) · (abs‘𝐶))) |
30 | 24, 29 | breqtrd 5094 | . . . . 5 ⊢ (𝜑 → 2 < ((2 · 𝑁) · (abs‘𝐶))) |
31 | 1, 3, 13, 15, 30 | lttrd 10803 | . . . 4 ⊢ (𝜑 → 1 < ((2 · 𝑁) · (abs‘𝐶))) |
32 | 1, 31 | jca 514 | . . 3 ⊢ (𝜑 → (1 ∈ ℝ ∧ 1 < ((2 · 𝑁) · (abs‘𝐶)))) |
33 | ltne 10739 | . . 3 ⊢ ((1 ∈ ℝ ∧ 1 < ((2 · 𝑁) · (abs‘𝐶))) → ((2 · 𝑁) · (abs‘𝐶)) ≠ 1) | |
34 | 32, 33 | syl 17 | . 2 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ≠ 1) |
35 | 1p1e2 11765 | . . . . 5 ⊢ (1 + 1) = 2 | |
36 | 35 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 + 1) = 2) |
37 | 36, 30 | eqbrtrd 5090 | . . 3 ⊢ (𝜑 → (1 + 1) < ((2 · 𝑁) · (abs‘𝐶))) |
38 | 1, 1, 13 | ltaddsubd 11242 | . . 3 ⊢ (𝜑 → ((1 + 1) < ((2 · 𝑁) · (abs‘𝐶)) ↔ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
39 | 37, 38 | mpbid 234 | . 2 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) |
40 | 34, 39 | jca 514 | 1 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 − cmin 10872 -cneg 10873 ℕcn 11640 2c2 11695 ℝ+crp 12392 (,)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-1st 7691 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-ioo 12745 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 knoppndvlem15 33867 knoppndvlem17 33869 knoppndvlem20 33872 |
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