Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fzshftral | Unicode version |
Description: Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.) |
Ref | Expression |
---|---|
fzshftral |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 8362 | . . . 4 | |
2 | fzrevral 9122 | . . . 4 | |
3 | 1, 2 | mp3an3 1257 | . . 3 |
4 | 3 | 3adant3 958 | . 2 |
5 | zsubcl 8392 | . . . . 5 | |
6 | 1, 5 | mpan 414 | . . . 4 |
7 | zsubcl 8392 | . . . . 5 | |
8 | 1, 7 | mpan 414 | . . . 4 |
9 | id 19 | . . . 4 | |
10 | fzrevral 9122 | . . . 4 | |
11 | 6, 8, 9, 10 | syl3an 1211 | . . 3 |
12 | 11 | 3com12 1142 | . 2 |
13 | elfzelz 9045 | . . . . . 6 | |
14 | zsubcl 8392 | . . . . . . 7 | |
15 | oveq2 5540 | . . . . . . . 8 | |
16 | 15 | sbcco3g 2959 | . . . . . . 7 |
17 | 14, 16 | syl 14 | . . . . . 6 |
18 | 13, 17 | sylan2 280 | . . . . 5 |
19 | 18 | ralbidva 2364 | . . . 4 |
20 | 19 | 3ad2ant3 961 | . . 3 |
21 | zcn 8356 | . . . . 5 | |
22 | zcn 8356 | . . . . 5 | |
23 | zcn 8356 | . . . . 5 | |
24 | df-neg 7282 | . . . . . . . . . 10 | |
25 | 24 | oveq2i 5543 | . . . . . . . . 9 |
26 | subneg 7357 | . . . . . . . . . 10 | |
27 | addcom 7245 | . . . . . . . . . 10 | |
28 | 26, 27 | eqtrd 2113 | . . . . . . . . 9 |
29 | 25, 28 | syl5eqr 2127 | . . . . . . . 8 |
30 | 29 | 3adant3 958 | . . . . . . 7 |
31 | df-neg 7282 | . . . . . . . . . 10 | |
32 | 31 | oveq2i 5543 | . . . . . . . . 9 |
33 | subneg 7357 | . . . . . . . . . 10 | |
34 | addcom 7245 | . . . . . . . . . 10 | |
35 | 33, 34 | eqtrd 2113 | . . . . . . . . 9 |
36 | 32, 35 | syl5eqr 2127 | . . . . . . . 8 |
37 | 36 | 3adant2 957 | . . . . . . 7 |
38 | 30, 37 | oveq12d 5550 | . . . . . 6 |
39 | 38 | 3coml 1145 | . . . . 5 |
40 | 21, 22, 23, 39 | syl3an 1211 | . . . 4 |
41 | 40 | raleqdv 2555 | . . 3 |
42 | elfzelz 9045 | . . . . . . . 8 | |
43 | 42 | zcnd 8470 | . . . . . . 7 |
44 | df-neg 7282 | . . . . . . . 8 | |
45 | negsubdi2 7367 | . . . . . . . 8 | |
46 | 44, 45 | syl5eqr 2127 | . . . . . . 7 |
47 | 23, 43, 46 | syl2an 283 | . . . . . 6 |
48 | 47 | sbceq1d 2820 | . . . . 5 |
49 | 48 | ralbidva 2364 | . . . 4 |
50 | 49 | 3ad2ant3 961 | . . 3 |
51 | 20, 41, 50 | 3bitrd 212 | . 2 |
52 | 4, 12, 51 | 3bitrd 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 wral 2348 wsbc 2815 (class class class)co 5532 cc 6979 cc0 6981 caddc 6984 cmin 7279 cneg 7280 cz 8351 cfz 9029 |
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-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 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-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 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-nel 2340 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-int 3637 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-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-fz 9030 |
This theorem is referenced by: fzoshftral 9247 |
Copyright terms: Public domain | W3C validator |