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Mirrors > Home > MPE Home > Th. List > xp1d2m1eqxm1d2 | Structured version Visualization version GIF version |
Description: A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.) |
Ref | Expression |
---|---|
xp1d2m1eqxm1d2 | ⊢ (𝑋 ∈ ℂ → (((𝑋 + 1) / 2) − 1) = ((𝑋 − 1) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2cn 10087 | . . . 4 ⊢ (𝑋 ∈ ℂ → (𝑋 + 1) ∈ ℂ) | |
2 | 1 | halfcld 11154 | . . 3 ⊢ (𝑋 ∈ ℂ → ((𝑋 + 1) / 2) ∈ ℂ) |
3 | peano2cnm 10226 | . . 3 ⊢ (((𝑋 + 1) / 2) ∈ ℂ → (((𝑋 + 1) / 2) − 1) ∈ ℂ) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝑋 ∈ ℂ → (((𝑋 + 1) / 2) − 1) ∈ ℂ) |
5 | peano2cnm 10226 | . . 3 ⊢ (𝑋 ∈ ℂ → (𝑋 − 1) ∈ ℂ) | |
6 | 5 | halfcld 11154 | . 2 ⊢ (𝑋 ∈ ℂ → ((𝑋 − 1) / 2) ∈ ℂ) |
7 | 2cnd 10970 | . 2 ⊢ (𝑋 ∈ ℂ → 2 ∈ ℂ) | |
8 | 2ne0 10990 | . . 3 ⊢ 2 ≠ 0 | |
9 | 8 | a1i 11 | . 2 ⊢ (𝑋 ∈ ℂ → 2 ≠ 0) |
10 | 1cnd 9935 | . . . 4 ⊢ (𝑋 ∈ ℂ → 1 ∈ ℂ) | |
11 | 2, 10, 7 | subdird 10366 | . . 3 ⊢ (𝑋 ∈ ℂ → ((((𝑋 + 1) / 2) − 1) · 2) = ((((𝑋 + 1) / 2) · 2) − (1 · 2))) |
12 | 1, 7, 9 | divcan1d 10681 | . . . 4 ⊢ (𝑋 ∈ ℂ → (((𝑋 + 1) / 2) · 2) = (𝑋 + 1)) |
13 | 7 | mulid2d 9937 | . . . 4 ⊢ (𝑋 ∈ ℂ → (1 · 2) = 2) |
14 | 12, 13 | oveq12d 6567 | . . 3 ⊢ (𝑋 ∈ ℂ → ((((𝑋 + 1) / 2) · 2) − (1 · 2)) = ((𝑋 + 1) − 2)) |
15 | 5, 7, 9 | divcan1d 10681 | . . . 4 ⊢ (𝑋 ∈ ℂ → (((𝑋 − 1) / 2) · 2) = (𝑋 − 1)) |
16 | 2m1e1 11012 | . . . . . 6 ⊢ (2 − 1) = 1 | |
17 | 16 | a1i 11 | . . . . 5 ⊢ (𝑋 ∈ ℂ → (2 − 1) = 1) |
18 | 17 | oveq2d 6565 | . . . 4 ⊢ (𝑋 ∈ ℂ → (𝑋 − (2 − 1)) = (𝑋 − 1)) |
19 | id 22 | . . . . 5 ⊢ (𝑋 ∈ ℂ → 𝑋 ∈ ℂ) | |
20 | 19, 7, 10 | subsub3d 10301 | . . . 4 ⊢ (𝑋 ∈ ℂ → (𝑋 − (2 − 1)) = ((𝑋 + 1) − 2)) |
21 | 15, 18, 20 | 3eqtr2rd 2651 | . . 3 ⊢ (𝑋 ∈ ℂ → ((𝑋 + 1) − 2) = (((𝑋 − 1) / 2) · 2)) |
22 | 11, 14, 21 | 3eqtrd 2648 | . 2 ⊢ (𝑋 ∈ ℂ → ((((𝑋 + 1) / 2) − 1) · 2) = (((𝑋 − 1) / 2) · 2)) |
23 | 4, 6, 7, 9, 22 | mulcan2ad 10542 | 1 ⊢ (𝑋 ∈ ℂ → (((𝑋 + 1) / 2) − 1) = ((𝑋 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 / cdiv 10563 2c2 10947 |
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-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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 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-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-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-2 10956 |
This theorem is referenced by: mod2eq1n2dvds 14909 zob 14921 nno 14936 nn0ob 14938 dignn0flhalflem1 42207 |
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