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Mirrors > Home > MPE Home > Th. List > 2halves | Structured version Visualization version GIF version |
Description: Two halves make a whole. (Contributed by NM, 11-Apr-2005.) |
Ref | Expression |
---|---|
2halves | ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2times 11352 | . . 3 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) | |
2 | 1 | oveq1d 6811 | . 2 ⊢ (𝐴 ∈ ℂ → ((2 · 𝐴) / 2) = ((𝐴 + 𝐴) / 2)) |
3 | 2cn 11297 | . . 3 ⊢ 2 ∈ ℂ | |
4 | 2ne0 11319 | . . 3 ⊢ 2 ≠ 0 | |
5 | divcan3 10917 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝐴) / 2) = 𝐴) | |
6 | 3, 4, 5 | mp3an23 1564 | . 2 ⊢ (𝐴 ∈ ℂ → ((2 · 𝐴) / 2) = 𝐴) |
7 | 2cnne0 11449 | . . . 4 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
8 | divdir 10916 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝐴 + 𝐴) / 2) = ((𝐴 / 2) + (𝐴 / 2))) | |
9 | 7, 8 | mp3an3 1561 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐴 + 𝐴) / 2) = ((𝐴 / 2) + (𝐴 / 2))) |
10 | 9 | anidms 556 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 𝐴) / 2) = ((𝐴 / 2) + (𝐴 / 2))) |
11 | 2, 6, 10 | 3eqtr3rd 2814 | 1 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 (class class class)co 6796 ℂcc 10140 0cc0 10142 + caddc 10145 · cmul 10147 / cdiv 10890 2c2 11276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-po 5171 df-so 5172 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-2 11285 |
This theorem is referenced by: halfpos 11469 lt2halves 11474 2halvesd 11485 pcoass 23043 pidiv2halves 24440 sincos4thpi 24486 efeq1 24496 cxpsqrt 24670 dvsqrt 24704 dvcnsqrt 24706 subfacval3 31509 dnibndlem5 32809 dnibndlem10 32814 infleinflem1 40097 smflimlem4 41497 |
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