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Mirrors > Home > ILE Home > Th. List > shftfvalg | Unicode version |
Description: The value of the sequence shifter operation is a function on . is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
shftfvalg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 103 | . 2 | |
2 | simpl 102 | . 2 | |
3 | simplr 482 | . . . . . . . . . . 11 | |
4 | simpll 481 | . . . . . . . . . . 11 | |
5 | 3, 4 | subcld 7322 | . . . . . . . . . 10 |
6 | vex 2560 | . . . . . . . . . . 11 | |
7 | breldmg 4541 | . . . . . . . . . . 11 | |
8 | 6, 7 | mp3an2 1220 | . . . . . . . . . 10 |
9 | 5, 8 | sylancom 397 | . . . . . . . . 9 |
10 | npcan 7220 | . . . . . . . . . . . 12 | |
11 | 10 | eqcomd 2045 | . . . . . . . . . . 11 |
12 | 11 | ancoms 255 | . . . . . . . . . 10 |
13 | 12 | adantr 261 | . . . . . . . . 9 |
14 | oveq1 5519 | . . . . . . . . . . 11 | |
15 | 14 | eqeq2d 2051 | . . . . . . . . . 10 |
16 | 15 | rspcev 2656 | . . . . . . . . 9 |
17 | 9, 13, 16 | syl2anc 391 | . . . . . . . 8 |
18 | vex 2560 | . . . . . . . . 9 | |
19 | eqeq1 2046 | . . . . . . . . . 10 | |
20 | 19 | rexbidv 2327 | . . . . . . . . 9 |
21 | 18, 20 | elab 2687 | . . . . . . . 8 |
22 | 17, 21 | sylibr 137 | . . . . . . 7 |
23 | brelrng 4565 | . . . . . . . . 9 | |
24 | 6, 23 | mp3an2 1220 | . . . . . . . 8 |
25 | 5, 24 | sylancom 397 | . . . . . . 7 |
26 | 22, 25 | jca 290 | . . . . . 6 |
27 | 26 | expl 360 | . . . . 5 |
28 | 27 | ssopab2dv 4015 | . . . 4 |
29 | df-xp 4351 | . . . 4 | |
30 | 28, 29 | syl6sseqr 2992 | . . 3 |
31 | dmexg 4596 | . . . . 5 | |
32 | abrexexg 5745 | . . . . 5 | |
33 | 31, 32 | syl 14 | . . . 4 |
34 | rnexg 4597 | . . . 4 | |
35 | xpexg 4452 | . . . 4 | |
36 | 33, 34, 35 | syl2anc 391 | . . 3 |
37 | ssexg 3896 | . . 3 | |
38 | 30, 36, 37 | syl2an 273 | . 2 |
39 | elex 2566 | . . 3 | |
40 | breq 3766 | . . . . . 6 | |
41 | 40 | anbi2d 437 | . . . . 5 |
42 | 41 | opabbidv 3823 | . . . 4 |
43 | oveq2 5520 | . . . . . . 7 | |
44 | 43 | breq1d 3774 | . . . . . 6 |
45 | 44 | anbi2d 437 | . . . . 5 |
46 | 45 | opabbidv 3823 | . . . 4 |
47 | df-shft 9416 | . . . 4 | |
48 | 42, 46, 47 | ovmpt2g 5635 | . . 3 |
49 | 39, 48 | syl3an1 1168 | . 2 |
50 | 1, 2, 38, 49 | syl3anc 1135 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 cab 2026 wrex 2307 cvv 2557 wss 2917 class class class wbr 3764 copab 3817 cxp 4343 cdm 4345 crn 4346 (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-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: ovshftex 9420 shftfibg 9421 2shfti 9432 |
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