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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem12 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 33484. (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 10495 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
2 | 2re 11565 | . . . . . 6 ⊢ 2 ∈ ℝ | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℝ) |
4 | knoppndvlem12.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
5 | nnre 11499 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
7 | 3, 6 | remulcld 10524 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ) |
8 | knoppndvlem12.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
9 | 8 | knoppndvlem3 33464 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
10 | 9 | simpld 495 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
11 | 10 | recnd 10522 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
12 | 11 | abscld 14634 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ) |
13 | 7, 12 | remulcld 10524 | . . . . 5 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ) |
14 | 1lt2 11662 | . . . . . 6 ⊢ 1 < 2 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 < 2) |
16 | 2t1e2 11654 | . . . . . . . . 9 ⊢ (2 · 1) = 2 | |
17 | 16 | eqcomi 2806 | . . . . . . . 8 ⊢ 2 = (2 · 1) |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 = (2 · 1)) |
19 | 6, 12 | remulcld 10524 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 · (abs‘𝐶)) ∈ ℝ) |
20 | 2rp 12248 | . . . . . . . . 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 12341 | . . . . . . 7 ⊢ (𝜑 → (2 · 1) < (2 · (𝑁 · (abs‘𝐶)))) |
24 | 18, 23 | eqbrtrd 4990 | . . . . . 6 ⊢ (𝜑 → 2 < (2 · (𝑁 · (abs‘𝐶)))) |
25 | 3 | recnd 10522 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ) |
26 | 6 | recnd 10522 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
27 | 12 | recnd 10522 | . . . . . . . 8 ⊢ (𝜑 → (abs‘𝐶) ∈ ℂ) |
28 | 25, 26, 27 | mulassd 10517 | . . . . . . 7 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) = (2 · (𝑁 · (abs‘𝐶)))) |
29 | 28 | eqcomd 2803 | . . . . . 6 ⊢ (𝜑 → (2 · (𝑁 · (abs‘𝐶))) = ((2 · 𝑁) · (abs‘𝐶))) |
30 | 24, 29 | breqtrd 4994 | . . . . 5 ⊢ (𝜑 → 2 < ((2 · 𝑁) · (abs‘𝐶))) |
31 | 1, 3, 13, 15, 30 | lttrd 10654 | . . . 4 ⊢ (𝜑 → 1 < ((2 · 𝑁) · (abs‘𝐶))) |
32 | 1, 31 | jca 512 | . . 3 ⊢ (𝜑 → (1 ∈ ℝ ∧ 1 < ((2 · 𝑁) · (abs‘𝐶)))) |
33 | ltne 10590 | . . 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 11616 | . . . . 5 ⊢ (1 + 1) = 2 | |
36 | 35 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 + 1) = 2) |
37 | 36, 30 | eqbrtrd 4990 | . . 3 ⊢ (𝜑 → (1 + 1) < ((2 · 𝑁) · (abs‘𝐶))) |
38 | 1, 1, 13 | ltaddsubd 11094 | . . 3 ⊢ (𝜑 → ((1 + 1) < ((2 · 𝑁) · (abs‘𝐶)) ↔ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
39 | 37, 38 | mpbid 233 | . 2 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) |
40 | 34, 39 | jca 512 | 1 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 ℝcr 10389 1c1 10391 + caddc 10393 · cmul 10395 < clt 10528 − cmin 10723 -cneg 10724 ℕcn 11492 2c2 11546 ℝ+crp 12243 (,)cioo 12592 abscabs 14431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-sup 8759 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-n0 11752 df-z 11836 df-uz 12098 df-rp 12244 df-ioo 12596 df-seq 13224 df-exp 13284 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 |
This theorem is referenced by: knoppndvlem14 33475 knoppndvlem15 33476 knoppndvlem17 33478 knoppndvlem20 33481 |
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