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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 11093 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 2 ∈ ℂ) | |
3 | 2ne0 11113 | . . . . . 6 ⊢ 2 ≠ 0 | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 2 ≠ 0) |
5 | 1, 2, 4 | divcan1d 10802 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) · 2) = 𝐴) |
6 | 5 | eqcomd 2628 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((𝐴 / 2) · 2)) |
7 | 6 | oveq1d 6665 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (((𝐴 / 2) · 2) − (𝐴 / 2))) |
8 | halfcl 11257 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ) | |
9 | 8, 2 | mulcomd 10061 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) · 2) = (2 · (𝐴 / 2))) |
10 | 9 | oveq1d 6665 | . 2 ⊢ (𝐴 ∈ ℂ → (((𝐴 / 2) · 2) − (𝐴 / 2)) = ((2 · (𝐴 / 2)) − (𝐴 / 2))) |
11 | 2, 8 | mulsubfacd 10492 | . . 3 ⊢ (𝐴 ∈ ℂ → ((2 · (𝐴 / 2)) − (𝐴 / 2)) = ((2 − 1) · (𝐴 / 2))) |
12 | 2m1e1 11135 | . . . . 5 ⊢ (2 − 1) = 1 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → (2 − 1) = 1) |
14 | 13 | oveq1d 6665 | . . 3 ⊢ (𝐴 ∈ ℂ → ((2 − 1) · (𝐴 / 2)) = (1 · (𝐴 / 2))) |
15 | 8 | mulid2d 10058 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · (𝐴 / 2)) = (𝐴 / 2)) |
16 | 11, 14, 15 | 3eqtrd 2660 | . 2 ⊢ (𝐴 ∈ ℂ → ((2 · (𝐴 / 2)) − (𝐴 / 2)) = (𝐴 / 2)) |
17 | 7, 10, 16 | 3eqtrd 2660 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (𝐴 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 · cmul 9941 − cmin 10266 / cdiv 10684 2c2 11070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 |
This theorem is referenced by: fldiv4lem1div2uz2 12637 gausslemma2dlem1a 25090 |
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