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Mirrors > Home > MPE Home > Th. List > subhalfhalf | Structured version Visualization version GIF version |
Description: Subtracting the half of a number from the number yields the half of the number. (Contributed by AV, 28-Jun-2021.) |
Ref | Expression |
---|---|
subhalfhalf | ⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (𝐴 / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
2 | 2cnd 11453 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 2 ∈ ℂ) | |
3 | 2ne0 11486 | . . . . . 6 ⊢ 2 ≠ 0 | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 2 ≠ 0) |
5 | 1, 2, 4 | divcan1d 11152 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) · 2) = 𝐴) |
6 | 5 | eqcomd 2783 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((𝐴 / 2) · 2)) |
7 | 6 | oveq1d 6937 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (((𝐴 / 2) · 2) − (𝐴 / 2))) |
8 | halfcl 11607 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ) | |
9 | 8, 2 | mulcomd 10398 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) · 2) = (2 · (𝐴 / 2))) |
10 | 9 | oveq1d 6937 | . 2 ⊢ (𝐴 ∈ ℂ → (((𝐴 / 2) · 2) − (𝐴 / 2)) = ((2 · (𝐴 / 2)) − (𝐴 / 2))) |
11 | 2, 8 | mulsubfacd 10836 | . . 3 ⊢ (𝐴 ∈ ℂ → ((2 · (𝐴 / 2)) − (𝐴 / 2)) = ((2 − 1) · (𝐴 / 2))) |
12 | 2m1e1 11508 | . . . . 5 ⊢ (2 − 1) = 1 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → (2 − 1) = 1) |
14 | 13 | oveq1d 6937 | . . 3 ⊢ (𝐴 ∈ ℂ → ((2 − 1) · (𝐴 / 2)) = (1 · (𝐴 / 2))) |
15 | 8 | mulid2d 10395 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · (𝐴 / 2)) = (𝐴 / 2)) |
16 | 11, 14, 15 | 3eqtrd 2817 | . 2 ⊢ (𝐴 ∈ ℂ → ((2 · (𝐴 / 2)) − (𝐴 / 2)) = (𝐴 / 2)) |
17 | 7, 10, 16 | 3eqtrd 2817 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (𝐴 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 (class class class)co 6922 ℂcc 10270 0cc0 10272 1c1 10273 · cmul 10277 − cmin 10606 / cdiv 11032 2c2 11430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-2 11438 |
This theorem is referenced by: fldiv4lem1div2uz2 12956 gausslemma2dlem1a 25542 |
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