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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 10586 | . . . . . . . 8 |
3 | 2 | breqd 3935 | . . . . . . 7 |
4 | 3 | ad2antrr 479 | . . . . . 6 |
5 | simpr 109 | . . . . . . . 8 | |
6 | simplr 519 | . . . . . . . 8 | |
7 | 5, 6 | subcld 8066 | . . . . . . 7 |
8 | vex 2684 | . . . . . . 7 | |
9 | eleq1 2200 | . . . . . . . . 9 | |
10 | oveq1 5774 | . . . . . . . . . 10 | |
11 | 10 | breq1d 3934 | . . . . . . . . 9 |
12 | 9, 11 | anbi12d 464 | . . . . . . . 8 |
13 | breq2 3928 | . . . . . . . . 9 | |
14 | 13 | anbi2d 459 | . . . . . . . 8 |
15 | eqid 2137 | . . . . . . . 8 | |
16 | 12, 14, 15 | brabg 4186 | . . . . . . 7 |
17 | 7, 8, 16 | sylancl 409 | . . . . . 6 |
18 | 4, 17 | bitrd 187 | . . . . 5 |
19 | subcl 7954 | . . . . . . . 8 | |
20 | 19 | biantrurd 303 | . . . . . . 7 |
21 | 20 | ancoms 266 | . . . . . 6 |
22 | 21 | adantll 467 | . . . . 5 |
23 | sub32 7989 | . . . . . . . . 9 | |
24 | subsub4 7988 | . . . . . . . . 9 | |
25 | 23, 24 | eqtr3d 2172 | . . . . . . . 8 |
26 | 25 | 3expb 1182 | . . . . . . 7 |
27 | 26 | ancoms 266 | . . . . . 6 |
28 | 27 | breq1d 3934 | . . . . 5 |
29 | 18, 22, 28 | 3bitr2d 215 | . . . 4 |
30 | 29 | pm5.32da 447 | . . 3 |
31 | 30 | opabbidv 3989 | . 2 |
32 | ovshftex 10584 | . . . . 5 | |
33 | 1, 32 | mpan 420 | . . . 4 |
34 | shftfvalg 10583 | . . . 4 | |
35 | 33, 34 | sylan2 284 | . . 3 |
36 | 35 | ancoms 266 | . 2 |
37 | addcl 7738 | . . 3 | |
38 | 1 | shftfval 10586 | . . 3 |
39 | 37, 38 | syl 14 | . 2 |
40 | 31, 36, 39 | 3eqtr4d 2180 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 cvv 2681 class class class wbr 3924 copab 3983 (class class class)co 5767 cc 7611 caddc 7616 cmin 7926 cshi 10579 |
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-coll 4038 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-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
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-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 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-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-shft 10580 |
This theorem is referenced by: shftcan1 10599 |
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