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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem20 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 36142. (Contributed by Asger C. Ipsen, 18-Aug-2021.) |
Ref | Expression |
---|---|
knoppndvlem20.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
knoppndvlem20.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
knoppndvlem20.1 | ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) |
Ref | Expression |
---|---|
knoppndvlem20 | ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | knoppndvlem20.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
2 | knoppndvlem20.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
3 | knoppndvlem20.1 | . . . . 5 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) | |
4 | 1, 2, 3 | knoppndvlem12 36131 | . . . 4 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
5 | 4 | simprd 494 | . . 3 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) |
6 | 2re 12324 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
7 | 6 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ) |
8 | 2 | nnred 12265 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
9 | 7, 8 | remulcld 11281 | . . . . . . 7 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ) |
10 | 1 | knoppndvlem3 36122 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
11 | 10 | simpld 493 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
12 | 11 | recnd 11279 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
13 | 12 | abscld 15424 | . . . . . . 7 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ) |
14 | 9, 13 | remulcld 11281 | . . . . . 6 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ) |
15 | 1red 11252 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ) | |
16 | 14, 15 | resubcld 11679 | . . . . 5 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ) |
17 | 0red 11254 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
18 | 0lt1 11773 | . . . . . . 7 ⊢ 0 < 1 | |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 1) |
20 | 17, 15, 16, 19, 5 | lttrd 11412 | . . . . 5 ⊢ (𝜑 → 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) |
21 | 16, 20 | elrpd 13053 | . . . 4 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ+) |
22 | 21 | recgt1d 13070 | . . 3 ⊢ (𝜑 → (1 < (((2 · 𝑁) · (abs‘𝐶)) − 1) ↔ (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) < 1)) |
23 | 5, 22 | mpbid 231 | . 2 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) < 1) |
24 | 21 | rprecred 13067 | . . . 4 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ) |
25 | 24, 15 | jca 510 | . . 3 ⊢ (𝜑 → ((1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ ∧ 1 ∈ ℝ)) |
26 | difrp 13052 | . . 3 ⊢ (((1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) < 1 ↔ (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+)) | |
27 | 25, 26 | syl 17 | . 2 ⊢ (𝜑 → ((1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) < 1 ↔ (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+)) |
28 | 23, 27 | mpbid 231 | 1 ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ≠ wne 2929 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 ℝcr 11144 0cc0 11145 1c1 11146 · cmul 11150 < clt 11285 − cmin 11481 -cneg 11482 / cdiv 11908 ℕcn 12250 2c2 12305 ℝ+crp 13014 (,)cioo 13364 abscabs 15222 |
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 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 |
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 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9472 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-ioo 13368 df-seq 14008 df-exp 14068 df-cj 15087 df-re 15088 df-im 15089 df-sqrt 15223 df-abs 15224 |
This theorem is referenced by: knoppndvlem21 36140 knoppndvlem22 36141 |
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