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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem16 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 32650. (Contributed by Asger C. Ipsen, 19-Jul-2021.) |
Ref | Expression |
---|---|
knoppndvlem16.a | ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) |
knoppndvlem16.b | ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) |
knoppndvlem16.j | ⊢ (𝜑 → 𝐽 ∈ ℕ0) |
knoppndvlem16.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
knoppndvlem16.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
knoppndvlem16 | ⊢ (𝜑 → (𝐵 − 𝐴) = (((2 · 𝑁)↑-𝐽) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | knoppndvlem16.b | . . . 4 ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))) |
3 | knoppndvlem16.a | . . . 4 ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)) |
5 | 2, 4 | oveq12d 6708 | . 2 ⊢ (𝜑 → (𝐵 − 𝐴) = (((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) − ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))) |
6 | 2cnd 11131 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ) | |
7 | knoppndvlem16.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
8 | 7 | nncnd 11074 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
9 | 6, 8 | mulcld 10098 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℂ) |
10 | 2ne0 11151 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
11 | 10 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ≠ 0) |
12 | 7 | nnne0d 11103 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ≠ 0) |
13 | 6, 8, 11, 12 | mulne0d 10717 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ≠ 0) |
14 | knoppndvlem16.j | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℕ0) | |
15 | 14 | nn0zd 11518 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ) |
16 | 15 | znegcld 11522 | . . . . . 6 ⊢ (𝜑 → -𝐽 ∈ ℤ) |
17 | 9, 13, 16 | expclzd 13053 | . . . . 5 ⊢ (𝜑 → ((2 · 𝑁)↑-𝐽) ∈ ℂ) |
18 | 6, 8, 13 | mulne0bad 10720 | . . . . 5 ⊢ (𝜑 → 2 ≠ 0) |
19 | 17, 6, 18 | divcld 10839 | . . . 4 ⊢ (𝜑 → (((2 · 𝑁)↑-𝐽) / 2) ∈ ℂ) |
20 | knoppndvlem16.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
21 | 20 | zcnd 11521 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
22 | 1cnd 10094 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
23 | 21, 22 | addcld 10097 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ ℂ) |
24 | 19, 23, 21 | subdid 10524 | . . 3 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀)) = (((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) − ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))) |
25 | 24 | eqcomd 2657 | . 2 ⊢ (𝜑 → (((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) − ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)) = ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀))) |
26 | 21, 22 | pncan2d 10432 | . . . 4 ⊢ (𝜑 → ((𝑀 + 1) − 𝑀) = 1) |
27 | 26 | oveq2d 6706 | . . 3 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀)) = ((((2 · 𝑁)↑-𝐽) / 2) · 1)) |
28 | 19 | mulid1d 10095 | . . 3 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 1) = (((2 · 𝑁)↑-𝐽) / 2)) |
29 | 27, 28 | eqtrd 2685 | . 2 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀)) = (((2 · 𝑁)↑-𝐽) / 2)) |
30 | 5, 25, 29 | 3eqtrd 2689 | 1 ⊢ (𝜑 → (𝐵 − 𝐴) = (((2 · 𝑁)↑-𝐽) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 (class class class)co 6690 0cc0 9974 1c1 9975 + caddc 9977 · cmul 9979 − cmin 10304 -cneg 10305 / cdiv 10722 ℕcn 11058 2c2 11108 ℕ0cn0 11330 ℤcz 11415 ↑cexp 12900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-n0 11331 df-z 11416 df-uz 11726 df-seq 12842 df-exp 12901 |
This theorem is referenced by: knoppndvlem17 32644 knoppndvlem21 32648 |
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