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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem16 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 31695. (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 6567 | . 2 ⊢ (𝜑 → (𝐵 − 𝐴) = (((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) − ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))) |
6 | 2cnd 10970 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ) | |
7 | knoppndvlem16.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
8 | 7 | nncnd 10913 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
9 | 6, 8 | mulcld 9939 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℂ) |
10 | 2ne0 10990 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
11 | 10 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ≠ 0) |
12 | 7 | nnne0d 10942 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ≠ 0) |
13 | 6, 8, 11, 12 | mulne0d 10558 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ≠ 0) |
14 | knoppndvlem16.j | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℕ0) | |
15 | 14 | nn0zd 11356 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ) |
16 | 15 | znegcld 11360 | . . . . . 6 ⊢ (𝜑 → -𝐽 ∈ ℤ) |
17 | 9, 13, 16 | expclzd 12875 | . . . . 5 ⊢ (𝜑 → ((2 · 𝑁)↑-𝐽) ∈ ℂ) |
18 | 6, 8, 13 | mulne0bad 10561 | . . . . 5 ⊢ (𝜑 → 2 ≠ 0) |
19 | 17, 6, 18 | divcld 10680 | . . . 4 ⊢ (𝜑 → (((2 · 𝑁)↑-𝐽) / 2) ∈ ℂ) |
20 | knoppndvlem16.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
21 | 20 | zcnd 11359 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
22 | 1cnd 9935 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
23 | 21, 22 | addcld 9938 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ ℂ) |
24 | 19, 23, 21 | subdid 10365 | . . 3 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀)) = (((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) − ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))) |
25 | 24 | eqcomd 2616 | . 2 ⊢ (𝜑 → (((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) − ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)) = ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀))) |
26 | 21, 22 | pncan2d 10273 | . . . 4 ⊢ (𝜑 → ((𝑀 + 1) − 𝑀) = 1) |
27 | 26 | oveq2d 6565 | . . 3 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀)) = ((((2 · 𝑁)↑-𝐽) / 2) · 1)) |
28 | 19 | mulid1d 9936 | . . 3 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 1) = (((2 · 𝑁)↑-𝐽) / 2)) |
29 | 27, 28 | eqtrd 2644 | . 2 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · ((𝑀 + 1) − 𝑀)) = (((2 · 𝑁)↑-𝐽) / 2)) |
30 | 5, 25, 29 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (𝐵 − 𝐴) = (((2 · 𝑁)↑-𝐽) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ℤcz 11254 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-exp 12723 |
This theorem is referenced by: knoppndvlem17 31689 knoppndvlem21 31693 |
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