Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10806 | . . . 4 ⊢ (𝑋 ∈ ℂ → (𝑋 + 1) ∈ ℂ) | |
2 | 1 | halfcld 11876 | . . 3 ⊢ (𝑋 ∈ ℂ → ((𝑋 + 1) / 2) ∈ ℂ) |
3 | peano2cnm 10946 | . . 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 10946 | . . 3 ⊢ (𝑋 ∈ ℂ → (𝑋 − 1) ∈ ℂ) | |
6 | 5 | halfcld 11876 | . 2 ⊢ (𝑋 ∈ ℂ → ((𝑋 − 1) / 2) ∈ ℂ) |
7 | 2cnd 11709 | . 2 ⊢ (𝑋 ∈ ℂ → 2 ∈ ℂ) | |
8 | 2ne0 11735 | . . 3 ⊢ 2 ≠ 0 | |
9 | 8 | a1i 11 | . 2 ⊢ (𝑋 ∈ ℂ → 2 ≠ 0) |
10 | 1cnd 10630 | . . . 4 ⊢ (𝑋 ∈ ℂ → 1 ∈ ℂ) | |
11 | 2, 10, 7 | subdird 11091 | . . 3 ⊢ (𝑋 ∈ ℂ → ((((𝑋 + 1) / 2) − 1) · 2) = ((((𝑋 + 1) / 2) · 2) − (1 · 2))) |
12 | 1, 7, 9 | divcan1d 11411 | . . . 4 ⊢ (𝑋 ∈ ℂ → (((𝑋 + 1) / 2) · 2) = (𝑋 + 1)) |
13 | 7 | mulid2d 10653 | . . . 4 ⊢ (𝑋 ∈ ℂ → (1 · 2) = 2) |
14 | 12, 13 | oveq12d 7168 | . . 3 ⊢ (𝑋 ∈ ℂ → ((((𝑋 + 1) / 2) · 2) − (1 · 2)) = ((𝑋 + 1) − 2)) |
15 | 5, 7, 9 | divcan1d 11411 | . . . 4 ⊢ (𝑋 ∈ ℂ → (((𝑋 − 1) / 2) · 2) = (𝑋 − 1)) |
16 | 2m1e1 11757 | . . . . . 6 ⊢ (2 − 1) = 1 | |
17 | 16 | a1i 11 | . . . . 5 ⊢ (𝑋 ∈ ℂ → (2 − 1) = 1) |
18 | 17 | oveq2d 7166 | . . . 4 ⊢ (𝑋 ∈ ℂ → (𝑋 − (2 − 1)) = (𝑋 − 1)) |
19 | id 22 | . . . . 5 ⊢ (𝑋 ∈ ℂ → 𝑋 ∈ ℂ) | |
20 | 19, 7, 10 | subsub3d 11021 | . . . 4 ⊢ (𝑋 ∈ ℂ → (𝑋 − (2 − 1)) = ((𝑋 + 1) − 2)) |
21 | 15, 18, 20 | 3eqtr2rd 2863 | . . 3 ⊢ (𝑋 ∈ ℂ → ((𝑋 + 1) − 2) = (((𝑋 − 1) / 2) · 2)) |
22 | 11, 14, 21 | 3eqtrd 2860 | . 2 ⊢ (𝑋 ∈ ℂ → ((((𝑋 + 1) / 2) − 1) · 2) = (((𝑋 − 1) / 2) · 2)) |
23 | 4, 6, 7, 9, 22 | mulcan2ad 11270 | 1 ⊢ (𝑋 ∈ ℂ → (((𝑋 + 1) / 2) − 1) = ((𝑋 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 − cmin 10864 / cdiv 11291 2c2 11686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-2 11694 |
This theorem is referenced by: mod2eq1n2dvds 15690 zob 15702 nno 15727 nn0ob 15729 dignn0flhalflem1 44668 |
Copyright terms: Public domain | W3C validator |