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Mirrors > Home > ILE Home > Th. List > xp1d2m1eqxm1d2 | Unicode version |
Description: A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.) |
Ref | Expression |
---|---|
xp1d2m1eqxm1d2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2cn 8105 |
. . . 4
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2 | 1 | halfcld 9176 |
. . 3
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3 | peano2cnm 8236 |
. . 3
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4 | 2, 3 | syl 14 |
. 2
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5 | peano2cnm 8236 |
. . 3
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6 | 5 | halfcld 9176 |
. 2
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7 | 2cnd 9005 |
. 2
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8 | 2ap0 9025 |
. . 3
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9 | 8 | a1i 9 |
. 2
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10 | 1cnd 7986 |
. . . 4
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11 | 2, 10, 7 | subdird 8385 |
. . 3
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12 | 1, 7, 9 | divcanap1d 8761 |
. . . 4
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13 | 7 | mulid2d 7989 |
. . . 4
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14 | 12, 13 | oveq12d 5906 |
. . 3
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15 | 5, 7, 9 | divcanap1d 8761 |
. . . 4
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16 | 2m1e1 9050 |
. . . . . 6
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17 | 16 | a1i 9 |
. . . . 5
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18 | 17 | oveq2d 5904 |
. . . 4
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19 | id 19 |
. . . . 5
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20 | 19, 7, 10 | subsub3d 8311 |
. . . 4
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21 | 15, 18, 20 | 3eqtr2rd 2227 |
. . 3
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22 | 11, 14, 21 | 3eqtrd 2224 |
. 2
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23 | 4, 6, 7, 9, 22 | mulcanap2ad 8634 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-po 4308 df-iso 4309 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-2 8991 |
This theorem is referenced by: zob 11909 nno 11924 nn0ob 11926 |
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