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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem1 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 36009. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
Ref | Expression |
---|---|
knoppndvlem1.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
knoppndvlem1.j | ⊢ (𝜑 → 𝐽 ∈ ℤ) |
knoppndvlem1.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
Ref | Expression |
---|---|
knoppndvlem1 | ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 12317 | . . . . . 6 ⊢ 2 ∈ ℝ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℝ) |
3 | knoppndvlem1.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | nnz 12610 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
6 | 5 | zred 12697 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
7 | 2, 6 | remulcld 11275 | . . . 4 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ) |
8 | 2 | recnd 11273 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℂ) |
9 | 6 | recnd 11273 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
10 | 2ne0 12347 | . . . . . 6 ⊢ 2 ≠ 0 | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ≠ 0) |
12 | 0red 11248 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ) | |
13 | 1red 11246 | . . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ) | |
14 | 0lt1 11767 | . . . . . . . . 9 ⊢ 0 < 1 | |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 0 < 1) |
16 | nnge1 12271 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
17 | 3, 16 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝑁) |
18 | 12, 13, 6, 15, 17 | ltletrd 11405 | . . . . . . 7 ⊢ (𝜑 → 0 < 𝑁) |
19 | 12, 18 | ltned 11381 | . . . . . 6 ⊢ (𝜑 → 0 ≠ 𝑁) |
20 | 19 | necomd 2993 | . . . . 5 ⊢ (𝜑 → 𝑁 ≠ 0) |
21 | 8, 9, 11, 20 | mulne0d 11897 | . . . 4 ⊢ (𝜑 → (2 · 𝑁) ≠ 0) |
22 | knoppndvlem1.j | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℤ) | |
23 | 22 | znegcld 12699 | . . . 4 ⊢ (𝜑 → -𝐽 ∈ ℤ) |
24 | 7, 21, 23 | reexpclzd 14244 | . . 3 ⊢ (𝜑 → ((2 · 𝑁)↑-𝐽) ∈ ℝ) |
25 | 24, 2, 11 | redivcld 12073 | . 2 ⊢ (𝜑 → (((2 · 𝑁)↑-𝐽) / 2) ∈ ℝ) |
26 | knoppndvlem1.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
27 | 26 | zred 12697 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
28 | 25, 27 | remulcld 11275 | 1 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ≠ wne 2937 class class class wbr 5148 (class class class)co 7420 ℝcr 11138 0cc0 11139 1c1 11140 · cmul 11144 < clt 11279 ≤ cle 11280 -cneg 11476 / cdiv 11902 ℕcn 12243 2c2 12298 ℤcz 12589 ↑cexp 14059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 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 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 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 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 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 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 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-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-n0 12504 df-z 12590 df-uz 12854 df-seq 14000 df-exp 14060 |
This theorem is referenced by: knoppndvlem6 35992 knoppndvlem7 35993 knoppndvlem10 35996 knoppndvlem14 36000 knoppndvlem15 36001 knoppndvlem17 36003 |
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