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Mirrors > Home > ILE Home > Th. List > fzind | Unicode version |
Description: Induction on the integers from to inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.) |
Ref | Expression |
---|---|
fzind.1 | |
fzind.2 | |
fzind.3 | |
fzind.4 | |
fzind.5 | |
fzind.6 |
Ref | Expression |
---|---|
fzind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4001 | . . . . . . . . . . 11 | |
2 | 1 | anbi2d 464 | . . . . . . . . . 10 |
3 | fzind.1 | . . . . . . . . . 10 | |
4 | 2, 3 | imbi12d 234 | . . . . . . . . 9 |
5 | breq1 4001 | . . . . . . . . . . 11 | |
6 | 5 | anbi2d 464 | . . . . . . . . . 10 |
7 | fzind.2 | . . . . . . . . . 10 | |
8 | 6, 7 | imbi12d 234 | . . . . . . . . 9 |
9 | breq1 4001 | . . . . . . . . . . 11 | |
10 | 9 | anbi2d 464 | . . . . . . . . . 10 |
11 | fzind.3 | . . . . . . . . . 10 | |
12 | 10, 11 | imbi12d 234 | . . . . . . . . 9 |
13 | breq1 4001 | . . . . . . . . . . 11 | |
14 | 13 | anbi2d 464 | . . . . . . . . . 10 |
15 | fzind.4 | . . . . . . . . . 10 | |
16 | 14, 15 | imbi12d 234 | . . . . . . . . 9 |
17 | fzind.5 | . . . . . . . . . 10 | |
18 | 17 | 3expib 1206 | . . . . . . . . 9 |
19 | zre 9228 | . . . . . . . . . . . . . 14 | |
20 | zre 9228 | . . . . . . . . . . . . . 14 | |
21 | p1le 8777 | . . . . . . . . . . . . . . 15 | |
22 | 21 | 3expia 1205 | . . . . . . . . . . . . . 14 |
23 | 19, 20, 22 | syl2an 289 | . . . . . . . . . . . . 13 |
24 | 23 | imdistanda 448 | . . . . . . . . . . . 12 |
25 | 24 | imim1d 75 | . . . . . . . . . . 11 |
26 | 25 | 3ad2ant2 1019 | . . . . . . . . . 10 |
27 | zltp1le 9278 | . . . . . . . . . . . . . . . . . . . . . 22 | |
28 | 27 | adantlr 477 | . . . . . . . . . . . . . . . . . . . . 21 |
29 | 28 | expcom 116 | . . . . . . . . . . . . . . . . . . . 20 |
30 | 29 | pm5.32d 450 | . . . . . . . . . . . . . . . . . . 19 |
31 | 30 | adantl 277 | . . . . . . . . . . . . . . . . . 18 |
32 | fzind.6 | . . . . . . . . . . . . . . . . . . . . 21 | |
33 | 32 | expcom 116 | . . . . . . . . . . . . . . . . . . . 20 |
34 | 33 | 3expa 1203 | . . . . . . . . . . . . . . . . . . 19 |
35 | 34 | com12 30 | . . . . . . . . . . . . . . . . . 18 |
36 | 31, 35 | sylbird 170 | . . . . . . . . . . . . . . . . 17 |
37 | 36 | ex 115 | . . . . . . . . . . . . . . . 16 |
38 | 37 | com23 78 | . . . . . . . . . . . . . . 15 |
39 | 38 | expd 258 | . . . . . . . . . . . . . 14 |
40 | 39 | 3impib 1201 | . . . . . . . . . . . . 13 |
41 | 40 | com23 78 | . . . . . . . . . . . 12 |
42 | 41 | impd 254 | . . . . . . . . . . 11 |
43 | 42 | a2d 26 | . . . . . . . . . 10 |
44 | 26, 43 | syld 45 | . . . . . . . . 9 |
45 | 4, 8, 12, 16, 18, 44 | uzind 9335 | . . . . . . . 8 |
46 | 45 | expcomd 1439 | . . . . . . 7 |
47 | 46 | 3expb 1204 | . . . . . 6 |
48 | 47 | expcom 116 | . . . . 5 |
49 | 48 | com23 78 | . . . 4 |
50 | 49 | 3impia 1200 | . . 3 |
51 | 50 | impd 254 | . 2 |
52 | 51 | impcom 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 w3a 978 wceq 1353 wcel 2146 class class class wbr 3998 (class class class)co 5865 cr 7785 c1 7787 caddc 7789 clt 7966 cle 7967 cz 9224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-addcom 7886 ax-addass 7888 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-0id 7894 ax-rnegex 7895 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-ltadd 7902 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-br 3999 df-opab 4060 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-iota 5170 df-fun 5210 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-inn 8891 df-n0 9148 df-z 9225 |
This theorem is referenced by: fnn0ind 9340 fzind2 10207 |
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