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Mirrors > Home > ILE Home > Th. List > 2shfti | Unicode version |
Description: Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) |
Ref | Expression |
---|---|
shftfval.1 |
Ref | Expression |
---|---|
2shfti |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shftfval.1 | . . . . . . . . 9 | |
2 | 1 | shftfval 9422 | . . . . . . . 8 |
3 | 2 | breqd 3775 | . . . . . . 7 |
4 | 3 | ad2antrr 457 | . . . . . 6 |
5 | simpr 103 | . . . . . . . 8 | |
6 | simplr 482 | . . . . . . . 8 | |
7 | 5, 6 | subcld 7322 | . . . . . . 7 |
8 | vex 2560 | . . . . . . 7 | |
9 | eleq1 2100 | . . . . . . . . 9 | |
10 | oveq1 5519 | . . . . . . . . . 10 | |
11 | 10 | breq1d 3774 | . . . . . . . . 9 |
12 | 9, 11 | anbi12d 442 | . . . . . . . 8 |
13 | breq2 3768 | . . . . . . . . 9 | |
14 | 13 | anbi2d 437 | . . . . . . . 8 |
15 | eqid 2040 | . . . . . . . 8 | |
16 | 12, 14, 15 | brabg 4006 | . . . . . . 7 |
17 | 7, 8, 16 | sylancl 392 | . . . . . 6 |
18 | 4, 17 | bitrd 177 | . . . . 5 |
19 | subcl 7210 | . . . . . . . 8 | |
20 | 19 | biantrurd 289 | . . . . . . 7 |
21 | 20 | ancoms 255 | . . . . . 6 |
22 | 21 | adantll 445 | . . . . 5 |
23 | sub32 7245 | . . . . . . . . 9 | |
24 | subsub4 7244 | . . . . . . . . 9 | |
25 | 23, 24 | eqtr3d 2074 | . . . . . . . 8 |
26 | 25 | 3expb 1105 | . . . . . . 7 |
27 | 26 | ancoms 255 | . . . . . 6 |
28 | 27 | breq1d 3774 | . . . . 5 |
29 | 18, 22, 28 | 3bitr2d 205 | . . . 4 |
30 | 29 | pm5.32da 425 | . . 3 |
31 | 30 | opabbidv 3823 | . 2 |
32 | ovshftex 9420 | . . . . 5 | |
33 | 1, 32 | mpan 400 | . . . 4 |
34 | shftfvalg 9419 | . . . 4 | |
35 | 33, 34 | sylan2 270 | . . 3 |
36 | 35 | ancoms 255 | . 2 |
37 | addcl 7006 | . . 3 | |
38 | 1 | shftfval 9422 | . . 3 |
39 | 37, 38 | syl 14 | . 2 |
40 | 31, 36, 39 | 3eqtr4d 2082 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 cvv 2557 class class class wbr 3764 copab 3817 (class class class)co 5512 cc 6887 caddc 6892 cmin 7182 cshi 9415 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-shft 9416 |
This theorem is referenced by: shftcan1 9435 |
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