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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ppncan 7997 | . . . . . 6 | |
2 | 1 | 3anidm13 1274 | . . . . 5 |
3 | 2times 8841 | . . . . . 6 | |
4 | 3 | adantr 274 | . . . . 5 |
5 | 2, 4 | eqtr4d 2173 | . . . 4 |
6 | 5 | oveq1d 5782 | . . 3 |
7 | addcl 7738 | . . . 4 | |
8 | subcl 7954 | . . . 4 | |
9 | 2cn 8784 | . . . . . 6 | |
10 | 2ap0 8806 | . . . . . 6 # | |
11 | 9, 10 | pm3.2i 270 | . . . . 5 # |
12 | divdirap 8450 | . . . . 5 # | |
13 | 11, 12 | mp3an3 1304 | . . . 4 |
14 | 7, 8, 13 | syl2anc 408 | . . 3 |
15 | divcanap3 8451 | . . . . 5 # | |
16 | 9, 10, 15 | mp3an23 1307 | . . . 4 |
17 | 16 | adantr 274 | . . 3 |
18 | 6, 14, 17 | 3eqtr3d 2178 | . 2 |
19 | pnncan 7996 | . . . . . 6 | |
20 | 19 | 3anidm23 1275 | . . . . 5 |
21 | 2times 8841 | . . . . . 6 | |
22 | 21 | adantl 275 | . . . . 5 |
23 | 20, 22 | eqtr4d 2173 | . . . 4 |
24 | 23 | oveq1d 5782 | . . 3 |
25 | divsubdirap 8461 | . . . . 5 # | |
26 | 11, 25 | mp3an3 1304 | . . . 4 |
27 | 7, 8, 26 | syl2anc 408 | . . 3 |
28 | divcanap3 8451 | . . . . 5 # | |
29 | 9, 10, 28 | mp3an23 1307 | . . . 4 |
30 | 29 | adantl 275 | . . 3 |
31 | 24, 27, 30 | 3eqtr3d 2178 | . 2 |
32 | 18, 31 | jca 304 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 class class class wbr 3924 (class class class)co 5767 cc 7611 cc0 7613 caddc 7616 cmul 7618 cmin 7926 # cap 8336 cdiv 8425 c2 8764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-2 8772 |
This theorem is referenced by: addsin 11438 subsin 11439 addcos 11442 subcos 11443 ioo2bl 12701 |
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