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Mirrors > Home > MPE Home > Th. List > 2shfti | Structured version Visualization version 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 13810 | . . . . . . . 8 |
3 | 2 | breqd 4664 | . . . . . . 7 |
4 | ovex 6678 | . . . . . . . 8 | |
5 | vex 3203 | . . . . . . . 8 | |
6 | eleq1 2689 | . . . . . . . . 9 | |
7 | oveq1 6657 | . . . . . . . . . 10 | |
8 | 7 | breq1d 4663 | . . . . . . . . 9 |
9 | 6, 8 | anbi12d 747 | . . . . . . . 8 |
10 | breq2 4657 | . . . . . . . . 9 | |
11 | 10 | anbi2d 740 | . . . . . . . 8 |
12 | eqid 2622 | . . . . . . . 8 | |
13 | 4, 5, 9, 11, 12 | brab 4998 | . . . . . . 7 |
14 | 3, 13 | syl6bb 276 | . . . . . 6 |
15 | 14 | ad2antrr 762 | . . . . 5 |
16 | subcl 10280 | . . . . . . . 8 | |
17 | 16 | biantrurd 529 | . . . . . . 7 |
18 | 17 | ancoms 469 | . . . . . 6 |
19 | 18 | adantll 750 | . . . . 5 |
20 | sub32 10315 | . . . . . . . . 9 | |
21 | subsub4 10314 | . . . . . . . . 9 | |
22 | 20, 21 | eqtr3d 2658 | . . . . . . . 8 |
23 | 22 | 3expb 1266 | . . . . . . 7 |
24 | 23 | ancoms 469 | . . . . . 6 |
25 | 24 | breq1d 4663 | . . . . 5 |
26 | 15, 19, 25 | 3bitr2d 296 | . . . 4 |
27 | 26 | pm5.32da 673 | . . 3 |
28 | 27 | opabbidv 4716 | . 2 |
29 | ovex 6678 | . . . 4 | |
30 | 29 | shftfval 13810 | . . 3 |
31 | 30 | adantl 482 | . 2 |
32 | addcl 10018 | . . 3 | |
33 | 1 | shftfval 13810 | . . 3 |
34 | 32, 33 | syl 17 | . 2 |
35 | 28, 31, 34 | 3eqtr4d 2666 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 cvv 3200 class class class wbr 4653 copab 4712 (class class class)co 6650 cc 9934 caddc 9939 cmin 10266 cshi 13806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-shft 13807 |
This theorem is referenced by: shftcan1 13823 |
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