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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem16 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 32525. (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 6668 | . 2 ⊢ (𝜑 → (𝐵 − 𝐴) = (((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) − ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))) |
6 | 2cnd 11093 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ) | |
7 | knoppndvlem16.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
8 | 7 | nncnd 11036 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
9 | 6, 8 | mulcld 10060 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℂ) |
10 | 2ne0 11113 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
11 | 10 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ≠ 0) |
12 | 7 | nnne0d 11065 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ≠ 0) |
13 | 6, 8, 11, 12 | mulne0d 10679 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ≠ 0) |
14 | knoppndvlem16.j | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℕ0) | |
15 | 14 | nn0zd 11480 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ) |
16 | 15 | znegcld 11484 | . . . . . 6 ⊢ (𝜑 → -𝐽 ∈ ℤ) |
17 | 9, 13, 16 | expclzd 13013 | . . . . 5 ⊢ (𝜑 → ((2 · 𝑁)↑-𝐽) ∈ ℂ) |
18 | 6, 8, 13 | mulne0bad 10682 | . . . . 5 ⊢ (𝜑 → 2 ≠ 0) |
19 | 17, 6, 18 | divcld 10801 | . . . 4 ⊢ (𝜑 → (((2 · 𝑁)↑-𝐽) / 2) ∈ ℂ) |
20 | knoppndvlem16.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
21 | 20 | zcnd 11483 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
22 | 1cnd 10056 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
23 | 21, 22 | addcld 10059 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ ℂ) |
24 | 19, 23, 21 | subdid 10486 | . . 3 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀)) = (((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) − ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))) |
25 | 24 | eqcomd 2628 | . 2 ⊢ (𝜑 → (((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) − ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)) = ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀))) |
26 | 21, 22 | pncan2d 10394 | . . . 4 ⊢ (𝜑 → ((𝑀 + 1) − 𝑀) = 1) |
27 | 26 | oveq2d 6666 | . . 3 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀)) = ((((2 · 𝑁)↑-𝐽) / 2) · 1)) |
28 | 19 | mulid1d 10057 | . . 3 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 1) = (((2 · 𝑁)↑-𝐽) / 2)) |
29 | 27, 28 | eqtrd 2656 | . 2 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀)) = (((2 · 𝑁)↑-𝐽) / 2)) |
30 | 5, 25, 29 | 3eqtrd 2660 | 1 ⊢ (𝜑 → (𝐵 − 𝐴) = (((2 · 𝑁)↑-𝐽) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ0cn0 11292 ℤcz 11377 ↑cexp 12860 |
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 |
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-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 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-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861 |
This theorem is referenced by: knoppndvlem17 32519 knoppndvlem21 32523 |
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