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Mirrors > Home > ILE Home > Th. List > halfaddsub | Unicode version |
Description: Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.) |
Ref | Expression |
---|---|
halfaddsub |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ppncan 8229 |
. . . . . 6
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2 | 1 | 3anidm13 1307 |
. . . . 5
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3 | 2times 9077 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 3 | adantr 276 |
. . . . 5
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5 | 2, 4 | eqtr4d 2225 |
. . . 4
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6 | 5 | oveq1d 5911 |
. . 3
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7 | addcl 7966 |
. . . 4
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8 | subcl 8186 |
. . . 4
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9 | 2cn 9020 |
. . . . . 6
![]() ![]() ![]() ![]() | |
10 | 2ap0 9042 |
. . . . . 6
![]() ![]() ![]() | |
11 | 9, 10 | pm3.2i 272 |
. . . . 5
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12 | divdirap 8684 |
. . . . 5
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13 | 11, 12 | mp3an3 1337 |
. . . 4
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14 | 7, 8, 13 | syl2anc 411 |
. . 3
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15 | divcanap3 8685 |
. . . . 5
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16 | 9, 10, 15 | mp3an23 1340 |
. . . 4
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17 | 16 | adantr 276 |
. . 3
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18 | 6, 14, 17 | 3eqtr3d 2230 |
. 2
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19 | pnncan 8228 |
. . . . . 6
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20 | 19 | 3anidm23 1308 |
. . . . 5
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21 | 2times 9077 |
. . . . . 6
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22 | 21 | adantl 277 |
. . . . 5
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23 | 20, 22 | eqtr4d 2225 |
. . . 4
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24 | 23 | oveq1d 5911 |
. . 3
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25 | divsubdirap 8695 |
. . . . 5
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26 | 11, 25 | mp3an3 1337 |
. . . 4
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27 | 7, 8, 26 | syl2anc 411 |
. . 3
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28 | divcanap3 8685 |
. . . . 5
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29 | 9, 10, 28 | mp3an23 1340 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 29 | adantl 277 |
. . 3
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31 | 24, 27, 30 | 3eqtr3d 2230 |
. 2
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32 | 18, 31 | jca 306 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 df-2 9008 |
This theorem is referenced by: addsin 11782 subsin 11783 addcos 11786 subcos 11787 ioo2bl 14500 |
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