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Mirrors > Home > MPE Home > Th. List > 1mhlfehlf | Structured version Visualization version GIF version |
Description: Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.) |
Ref | Expression |
---|---|
1mhlfehlf | ⊢ (1 − (1 / 2)) = (1 / 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 12339 | . . 3 ⊢ 2 ∈ ℂ | |
2 | ax-1cn 11211 | . . 3 ⊢ 1 ∈ ℂ | |
3 | 2cnne0 12474 | . . 3 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
4 | divsubdir 11959 | . . 3 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 − 1) / 2) = ((2 / 2) − (1 / 2))) | |
5 | 1, 2, 3, 4 | mp3an 1460 | . 2 ⊢ ((2 − 1) / 2) = ((2 / 2) − (1 / 2)) |
6 | 2m1e1 12390 | . . 3 ⊢ (2 − 1) = 1 | |
7 | 6 | oveq1i 7441 | . 2 ⊢ ((2 − 1) / 2) = (1 / 2) |
8 | 2div2e1 12405 | . . 3 ⊢ (2 / 2) = 1 | |
9 | 8 | oveq1i 7441 | . 2 ⊢ ((2 / 2) − (1 / 2)) = (1 − (1 / 2)) |
10 | 5, 7, 9 | 3eqtr3ri 2772 | 1 ⊢ (1 − (1 / 2)) = (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 − cmin 11490 / 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: geo2sum 15906 geoihalfsum 15915 pcoass 25071 aaliou3lem3 26401 ang180lem3 26869 coinflippvt 34466 dnibndlem3 36463 oddfl 45228 dirkertrigeqlem3 46056 |
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