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| Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for knoppndv 36535. (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 12340 | . . . . . 6 ⊢ 2 ∈ ℝ | |
| 2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℝ) |
| 3 | knoppndvlem1.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
| 4 | nnz 12634 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
| 5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 6 | 5 | zred 12722 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 7 | 2, 6 | remulcld 11291 | . . . 4 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ) |
| 8 | 2 | recnd 11289 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℂ) |
| 9 | 6 | recnd 11289 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 10 | 2ne0 12370 | . . . . . 6 ⊢ 2 ≠ 0 | |
| 11 | 10 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ≠ 0) |
| 12 | 0red 11264 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 13 | 1red 11262 | . . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 14 | 0lt1 11785 | . . . . . . . . 9 ⊢ 0 < 1 | |
| 15 | 14 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 0 < 1) |
| 16 | nnge1 12294 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
| 17 | 3, 16 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝑁) |
| 18 | 12, 13, 6, 15, 17 | ltletrd 11421 | . . . . . . 7 ⊢ (𝜑 → 0 < 𝑁) |
| 19 | 12, 18 | ltned 11397 | . . . . . 6 ⊢ (𝜑 → 0 ≠ 𝑁) |
| 20 | 19 | necomd 2996 | . . . . 5 ⊢ (𝜑 → 𝑁 ≠ 0) |
| 21 | 8, 9, 11, 20 | mulne0d 11915 | . . . 4 ⊢ (𝜑 → (2 · 𝑁) ≠ 0) |
| 22 | knoppndvlem1.j | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℤ) | |
| 23 | 22 | znegcld 12724 | . . . 4 ⊢ (𝜑 → -𝐽 ∈ ℤ) |
| 24 | 7, 21, 23 | reexpclzd 14288 | . . 3 ⊢ (𝜑 → ((2 · 𝑁)↑-𝐽) ∈ ℝ) |
| 25 | 24, 2, 11 | redivcld 12095 | . 2 ⊢ (𝜑 → (((2 · 𝑁)↑-𝐽) / 2) ∈ ℝ) |
| 26 | knoppndvlem1.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 27 | 26 | zred 12722 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
| 28 | 25, 27 | remulcld 11291 | 1 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 ≤ cle 11296 -cneg 11493 / cdiv 11920 ℕcn 12266 2c2 12321 ℤcz 12613 ↑cexp 14102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: knoppndvlem6 36518 knoppndvlem7 36519 knoppndvlem10 36522 knoppndvlem14 36526 knoppndvlem15 36527 knoppndvlem17 36529 |
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