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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 12342 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 2 ∈ ℂ) | |
3 | 2ne0 12368 | . . . . . 6 ⊢ 2 ≠ 0 | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 2 ≠ 0) |
5 | 1, 2, 4 | divcan1d 12042 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) · 2) = 𝐴) |
6 | 5 | eqcomd 2732 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((𝐴 / 2) · 2)) |
7 | 6 | oveq1d 7439 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (((𝐴 / 2) · 2) − (𝐴 / 2))) |
8 | halfcl 12489 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ) | |
9 | 8, 2 | mulcomd 11285 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) · 2) = (2 · (𝐴 / 2))) |
10 | 9 | oveq1d 7439 | . 2 ⊢ (𝐴 ∈ ℂ → (((𝐴 / 2) · 2) − (𝐴 / 2)) = ((2 · (𝐴 / 2)) − (𝐴 / 2))) |
11 | 2, 8 | mulsubfacd 11725 | . . 3 ⊢ (𝐴 ∈ ℂ → ((2 · (𝐴 / 2)) − (𝐴 / 2)) = ((2 − 1) · (𝐴 / 2))) |
12 | 2m1e1 12390 | . . . . 5 ⊢ (2 − 1) = 1 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → (2 − 1) = 1) |
14 | 13 | oveq1d 7439 | . . 3 ⊢ (𝐴 ∈ ℂ → ((2 − 1) · (𝐴 / 2)) = (1 · (𝐴 / 2))) |
15 | 8 | mullidd 11282 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · (𝐴 / 2)) = (𝐴 / 2)) |
16 | 11, 14, 15 | 3eqtrd 2770 | . 2 ⊢ (𝐴 ∈ ℂ → ((2 · (𝐴 / 2)) − (𝐴 / 2)) = (𝐴 / 2)) |
17 | 7, 10, 16 | 3eqtrd 2770 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (𝐴 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 (class class class)co 7424 ℂcc 11156 0cc0 11158 1c1 11159 · cmul 11163 − cmin 11494 / cdiv 11921 2c2 12319 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-2 12327 |
This theorem is referenced by: fldiv4lem1div2uz2 13856 gausslemma2dlem1a 27394 |
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