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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 9709 | . . . . . . . 8 |
3 | 2 | breqd 3796 | . . . . . . 7 |
4 | 3 | ad2antrr 471 | . . . . . 6 |
5 | simpr 108 | . . . . . . . 8 | |
6 | simplr 496 | . . . . . . . 8 | |
7 | 5, 6 | subcld 7419 | . . . . . . 7 |
8 | vex 2604 | . . . . . . 7 | |
9 | eleq1 2141 | . . . . . . . . 9 | |
10 | oveq1 5539 | . . . . . . . . . 10 | |
11 | 10 | breq1d 3795 | . . . . . . . . 9 |
12 | 9, 11 | anbi12d 456 | . . . . . . . 8 |
13 | breq2 3789 | . . . . . . . . 9 | |
14 | 13 | anbi2d 451 | . . . . . . . 8 |
15 | eqid 2081 | . . . . . . . 8 | |
16 | 12, 14, 15 | brabg 4024 | . . . . . . 7 |
17 | 7, 8, 16 | sylancl 404 | . . . . . 6 |
18 | 4, 17 | bitrd 186 | . . . . 5 |
19 | subcl 7307 | . . . . . . . 8 | |
20 | 19 | biantrurd 299 | . . . . . . 7 |
21 | 20 | ancoms 264 | . . . . . 6 |
22 | 21 | adantll 459 | . . . . 5 |
23 | sub32 7342 | . . . . . . . . 9 | |
24 | subsub4 7341 | . . . . . . . . 9 | |
25 | 23, 24 | eqtr3d 2115 | . . . . . . . 8 |
26 | 25 | 3expb 1139 | . . . . . . 7 |
27 | 26 | ancoms 264 | . . . . . 6 |
28 | 27 | breq1d 3795 | . . . . 5 |
29 | 18, 22, 28 | 3bitr2d 214 | . . . 4 |
30 | 29 | pm5.32da 439 | . . 3 |
31 | 30 | opabbidv 3844 | . 2 |
32 | ovshftex 9707 | . . . . 5 | |
33 | 1, 32 | mpan 414 | . . . 4 |
34 | shftfvalg 9706 | . . . 4 | |
35 | 33, 34 | sylan2 280 | . . 3 |
36 | 35 | ancoms 264 | . 2 |
37 | addcl 7098 | . . 3 | |
38 | 1 | shftfval 9709 | . . 3 |
39 | 37, 38 | syl 14 | . 2 |
40 | 31, 36, 39 | 3eqtr4d 2123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 cvv 2601 class class class wbr 3785 copab 3838 (class class class)co 5532 cc 6979 caddc 6984 cmin 7279 cshi 9702 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-shft 9703 |
This theorem is referenced by: shftcan1 9722 |
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