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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem12 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 32525. (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 10055 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
2 | 2re 11090 | . . . . . 6 ⊢ 2 ∈ ℝ | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℝ) |
4 | knoppndvlem12.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
5 | nnre 11027 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
7 | 3, 6 | remulcld 10070 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ) |
8 | knoppndvlem12.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
9 | 8 | knoppndvlem3 32505 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
10 | 9 | simpld 475 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
11 | 10 | recnd 10068 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
12 | 11 | abscld 14175 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ) |
13 | 7, 12 | remulcld 10070 | . . . . 5 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ) |
14 | 1lt2 11194 | . . . . . 6 ⊢ 1 < 2 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 < 2) |
16 | 2t1e2 11176 | . . . . . . . . 9 ⊢ (2 · 1) = 2 | |
17 | 16 | eqcomi 2631 | . . . . . . . 8 ⊢ 2 = (2 · 1) |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 = (2 · 1)) |
19 | 6, 12 | remulcld 10070 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 · (abs‘𝐶)) ∈ ℝ) |
20 | 2rp 11837 | . . . . . . . . 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 11928 | . . . . . . 7 ⊢ (𝜑 → (2 · 1) < (2 · (𝑁 · (abs‘𝐶)))) |
24 | 18, 23 | eqbrtrd 4675 | . . . . . 6 ⊢ (𝜑 → 2 < (2 · (𝑁 · (abs‘𝐶)))) |
25 | 3 | recnd 10068 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ) |
26 | 6 | recnd 10068 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
27 | 12 | recnd 10068 | . . . . . . . 8 ⊢ (𝜑 → (abs‘𝐶) ∈ ℂ) |
28 | 25, 26, 27 | mulassd 10063 | . . . . . . 7 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) = (2 · (𝑁 · (abs‘𝐶)))) |
29 | 28 | eqcomd 2628 | . . . . . 6 ⊢ (𝜑 → (2 · (𝑁 · (abs‘𝐶))) = ((2 · 𝑁) · (abs‘𝐶))) |
30 | 24, 29 | breqtrd 4679 | . . . . 5 ⊢ (𝜑 → 2 < ((2 · 𝑁) · (abs‘𝐶))) |
31 | 1, 3, 13, 15, 30 | lttrd 10198 | . . . 4 ⊢ (𝜑 → 1 < ((2 · 𝑁) · (abs‘𝐶))) |
32 | 1, 31 | jca 554 | . . 3 ⊢ (𝜑 → (1 ∈ ℝ ∧ 1 < ((2 · 𝑁) · (abs‘𝐶)))) |
33 | ltne 10134 | . . 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 11134 | . . . . 5 ⊢ (1 + 1) = 2 | |
36 | 35 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 + 1) = 2) |
37 | 36, 30 | eqbrtrd 4675 | . . 3 ⊢ (𝜑 → (1 + 1) < ((2 · 𝑁) · (abs‘𝐶))) |
38 | 1, 1, 13 | ltaddsubd 10627 | . . 3 ⊢ (𝜑 → ((1 + 1) < ((2 · 𝑁) · (abs‘𝐶)) ↔ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
39 | 37, 38 | mpbid 222 | . 2 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) |
40 | 34, 39 | jca 554 | 1 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 − cmin 10266 -cneg 10267 ℕcn 11020 2c2 11070 ℝ+crp 11832 (,)cioo 12175 abscabs 13974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ioo 12179 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 |
This theorem is referenced by: knoppndvlem14 32516 knoppndvlem15 32517 knoppndvlem17 32519 knoppndvlem20 32522 |
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