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Mirrors > Home > MPE Home > Th. List > 2halvesd | Structured version Visualization version GIF version |
Description: Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
2timesd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
2halvesd | ⊢ (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2timesd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | 2halves 12492 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℂcc 11151 + caddc 11156 / cdiv 11918 2c2 12319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 |
This theorem is referenced by: reccn2 15630 mertenslem1 15917 sin01bnd 16218 prmreclem5 16954 4sqlem6 16977 4sqlem10 16981 4sqlem15 16993 4sqlem16 16994 blhalf 24431 methaus 24549 nrginvrcnlem 24728 opnreen 24867 iscau3 25326 ovollb2lem 25537 ovolunlem1a 25545 itg2cnlem2 25812 ulmcn 26457 ulmdvlem1 26458 cxpcn3lem 26805 chordthmlem4 26893 lgamgulmlem3 27089 ftalem2 27132 chtub 27271 lgsqrlem2 27406 lgseisenlem2 27435 lgsquadlem1 27439 2sqlem8 27485 mulog2sumlem1 27593 vmalogdivsum 27598 pntibndlem2 27650 lt2addrd 32762 le2halvesd 32766 dnizphlfeqhlf 36459 poimirlem29 37636 heicant 37642 mblfinlem4 37647 itg2addnclem 37658 ftc1anclem6 37685 ftc1anclem8 37687 heibor1lem 37796 aks4d1p1p4 42053 suplesup 45289 lptre2pt 45596 0ellimcdiv 45605 ioodvbdlimc1lem2 45888 ioodvbdlimc2lem 45890 dirkertrigeqlem2 46055 dirkercncflem1 46059 sge0xaddlem1 46389 hoiqssbllem2 46579 |
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