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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 10804 | . . . 4 ⊢ (𝑋 ∈ ℂ → (𝑋 + 1) ∈ ℂ) | |
2 | 1 | halfcld 11874 | . . 3 ⊢ (𝑋 ∈ ℂ → ((𝑋 + 1) / 2) ∈ ℂ) |
3 | peano2cnm 10944 | . . 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 10944 | . . 3 ⊢ (𝑋 ∈ ℂ → (𝑋 − 1) ∈ ℂ) | |
6 | 5 | halfcld 11874 | . 2 ⊢ (𝑋 ∈ ℂ → ((𝑋 − 1) / 2) ∈ ℂ) |
7 | 2cnd 11707 | . 2 ⊢ (𝑋 ∈ ℂ → 2 ∈ ℂ) | |
8 | 2ne0 11733 | . . 3 ⊢ 2 ≠ 0 | |
9 | 8 | a1i 11 | . 2 ⊢ (𝑋 ∈ ℂ → 2 ≠ 0) |
10 | 1cnd 10628 | . . . 4 ⊢ (𝑋 ∈ ℂ → 1 ∈ ℂ) | |
11 | 2, 10, 7 | subdird 11089 | . . 3 ⊢ (𝑋 ∈ ℂ → ((((𝑋 + 1) / 2) − 1) · 2) = ((((𝑋 + 1) / 2) · 2) − (1 · 2))) |
12 | 1, 7, 9 | divcan1d 11409 | . . . 4 ⊢ (𝑋 ∈ ℂ → (((𝑋 + 1) / 2) · 2) = (𝑋 + 1)) |
13 | 7 | mulid2d 10651 | . . . 4 ⊢ (𝑋 ∈ ℂ → (1 · 2) = 2) |
14 | 12, 13 | oveq12d 7166 | . . 3 ⊢ (𝑋 ∈ ℂ → ((((𝑋 + 1) / 2) · 2) − (1 · 2)) = ((𝑋 + 1) − 2)) |
15 | 5, 7, 9 | divcan1d 11409 | . . . 4 ⊢ (𝑋 ∈ ℂ → (((𝑋 − 1) / 2) · 2) = (𝑋 − 1)) |
16 | 2m1e1 11755 | . . . . . 6 ⊢ (2 − 1) = 1 | |
17 | 16 | a1i 11 | . . . . 5 ⊢ (𝑋 ∈ ℂ → (2 − 1) = 1) |
18 | 17 | oveq2d 7164 | . . . 4 ⊢ (𝑋 ∈ ℂ → (𝑋 − (2 − 1)) = (𝑋 − 1)) |
19 | id 22 | . . . . 5 ⊢ (𝑋 ∈ ℂ → 𝑋 ∈ ℂ) | |
20 | 19, 7, 10 | subsub3d 11019 | . . . 4 ⊢ (𝑋 ∈ ℂ → (𝑋 − (2 − 1)) = ((𝑋 + 1) − 2)) |
21 | 15, 18, 20 | 3eqtr2rd 2861 | . . 3 ⊢ (𝑋 ∈ ℂ → ((𝑋 + 1) − 2) = (((𝑋 − 1) / 2) · 2)) |
22 | 11, 14, 21 | 3eqtrd 2858 | . 2 ⊢ (𝑋 ∈ ℂ → ((((𝑋 + 1) / 2) − 1) · 2) = (((𝑋 − 1) / 2) · 2)) |
23 | 4, 6, 7, 9, 22 | mulcan2ad 11268 | 1 ⊢ (𝑋 ∈ ℂ → (((𝑋 + 1) / 2) − 1) = ((𝑋 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 (class class class)co 7148 ℂcc 10527 0cc0 10529 1c1 10530 + caddc 10532 · cmul 10534 − cmin 10862 / cdiv 11289 2c2 11684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-2 11692 |
This theorem is referenced by: mod2eq1n2dvds 15688 zob 15700 nno 15725 nn0ob 15727 dignn0flhalflem1 44666 |
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