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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 11853 | . 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 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 + caddc 10529 / cdiv 11286 2c2 11680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-2 11688 |
This theorem is referenced by: reccn2 14945 mertenslem1 15232 sin01bnd 15530 prmreclem5 16246 4sqlem6 16269 4sqlem10 16273 4sqlem15 16285 4sqlem16 16286 blhalf 23012 methaus 23127 nrginvrcnlem 23297 opnreen 23436 iscau3 23882 ovollb2lem 24092 ovolunlem1a 24100 itg2cnlem2 24366 ulmcn 24994 ulmdvlem1 24995 cxpcn3lem 25336 chordthmlem4 25421 lgamgulmlem3 25616 ftalem2 25659 chtub 25796 lgsqrlem2 25931 lgseisenlem2 25960 lgsquadlem1 25964 2sqlem8 26010 mulog2sumlem1 26118 vmalogdivsum 26123 pntibndlem2 26175 lt2addrd 30501 le2halvesd 30505 dnizphlfeqhlf 33928 poimirlem29 35086 heicant 35092 mblfinlem4 35097 itg2addnclem 35108 ftc1anclem6 35135 ftc1anclem8 35137 heibor1lem 35247 suplesup 41971 lptre2pt 42282 0ellimcdiv 42291 ioodvbdlimc1lem2 42574 ioodvbdlimc2lem 42576 dirkertrigeqlem2 42741 dirkercncflem1 42745 sge0xaddlem1 43072 hoiqssbllem2 43262 |
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