Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 2txmxeqx | Structured version Visualization version GIF version |
Description: Two times a complex number minus the number itself results in the number itself. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
Ref | Expression |
---|---|
2txmxeqx | ⊢ (𝑋 ∈ ℂ → ((2 · 𝑋) − 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝑋 ∈ ℂ → 𝑋 ∈ ℂ) | |
2 | 2times 11963 | . 2 ⊢ (𝑋 ∈ ℂ → (2 · 𝑋) = (𝑋 + 𝑋)) | |
3 | 1, 1, 2 | mvrladdd 11242 | 1 ⊢ (𝑋 ∈ ℂ → ((2 · 𝑋) − 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 (class class class)co 7210 ℂcc 10724 · cmul 10731 − cmin 11059 2c2 11882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-po 5465 df-so 5466 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 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-pnf 10866 df-mnf 10867 df-ltxr 10869 df-sub 11061 df-2 11890 |
This theorem is referenced by: subsq2 13776 bpoly4 15618 2lgslem3a 26274 2lgslem3b 26275 2lgslem3c 26276 2lgslem3d 26277 2np3bcnp1 39820 ply1mulgsumlem2 45399 itcovalt2lem2lem2 45691 |
Copyright terms: Public domain | W3C validator |