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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 11891 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 2 ∈ ℂ) | |
3 | 2ne0 11917 | . . . . . 6 ⊢ 2 ≠ 0 | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 2 ≠ 0) |
5 | 1, 2, 4 | divcan1d 11592 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) · 2) = 𝐴) |
6 | 5 | eqcomd 2740 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((𝐴 / 2) · 2)) |
7 | 6 | oveq1d 7217 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (((𝐴 / 2) · 2) − (𝐴 / 2))) |
8 | halfcl 12038 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ) | |
9 | 8, 2 | mulcomd 10837 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) · 2) = (2 · (𝐴 / 2))) |
10 | 9 | oveq1d 7217 | . 2 ⊢ (𝐴 ∈ ℂ → (((𝐴 / 2) · 2) − (𝐴 / 2)) = ((2 · (𝐴 / 2)) − (𝐴 / 2))) |
11 | 2, 8 | mulsubfacd 11276 | . . 3 ⊢ (𝐴 ∈ ℂ → ((2 · (𝐴 / 2)) − (𝐴 / 2)) = ((2 − 1) · (𝐴 / 2))) |
12 | 2m1e1 11939 | . . . . 5 ⊢ (2 − 1) = 1 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → (2 − 1) = 1) |
14 | 13 | oveq1d 7217 | . . 3 ⊢ (𝐴 ∈ ℂ → ((2 − 1) · (𝐴 / 2)) = (1 · (𝐴 / 2))) |
15 | 8 | mulid2d 10834 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · (𝐴 / 2)) = (𝐴 / 2)) |
16 | 11, 14, 15 | 3eqtrd 2778 | . 2 ⊢ (𝐴 ∈ ℂ → ((2 · (𝐴 / 2)) − (𝐴 / 2)) = (𝐴 / 2)) |
17 | 7, 10, 16 | 3eqtrd 2778 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (𝐴 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 (class class class)co 7202 ℂcc 10710 0cc0 10712 1c1 10713 · cmul 10717 − cmin 11045 / cdiv 11472 2c2 11868 |
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 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
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 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-po 5457 df-so 5458 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-2 11876 |
This theorem is referenced by: fldiv4lem1div2uz2 13394 gausslemma2dlem1a 26218 |
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