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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 12120 | . . 3 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) | |
2 | 1 | oveq1d 7287 | . 2 ⊢ (𝐴 ∈ ℂ → ((2 · 𝐴) / 2) = ((𝐴 + 𝐴) / 2)) |
3 | 2cn 12059 | . . 3 ⊢ 2 ∈ ℂ | |
4 | 2ne0 12088 | . . 3 ⊢ 2 ≠ 0 | |
5 | divcan3 11670 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝐴) / 2) = 𝐴) | |
6 | 3, 4, 5 | mp3an23 1452 | . 2 ⊢ (𝐴 ∈ ℂ → ((2 · 𝐴) / 2) = 𝐴) |
7 | 2cnne0 12194 | . . . 4 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
8 | divdir 11669 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝐴 + 𝐴) / 2) = ((𝐴 / 2) + (𝐴 / 2))) | |
9 | 7, 8 | mp3an3 1449 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐴 + 𝐴) / 2) = ((𝐴 / 2) + (𝐴 / 2))) |
10 | 9 | anidms 567 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 𝐴) / 2) = ((𝐴 / 2) + (𝐴 / 2))) |
11 | 2, 6, 10 | 3eqtr3rd 2789 | 1 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 (class class class)co 7272 ℂcc 10880 0cc0 10882 + caddc 10885 · cmul 10887 / cdiv 11643 2c2 12039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-2 12047 |
This theorem is referenced by: halfpos 12214 lt2halves 12219 2halvesd 12230 pcoass 24198 pidiv2halves 25635 sincos4thpi 25681 efeq1 25695 cxpsqrt 25869 dvsqrt 25906 dvcnsqrt 25908 subfacval3 33160 dnibndlem5 34671 dnibndlem10 34676 infleinflem1 42891 smflimlem4 44288 |
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