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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 3927 | . . . . . . . . . . 11 | |
2 | 1 | anbi2d 459 | . . . . . . . . . 10 |
3 | fzind.1 | . . . . . . . . . 10 | |
4 | 2, 3 | imbi12d 233 | . . . . . . . . 9 |
5 | breq1 3927 | . . . . . . . . . . 11 | |
6 | 5 | anbi2d 459 | . . . . . . . . . 10 |
7 | fzind.2 | . . . . . . . . . 10 | |
8 | 6, 7 | imbi12d 233 | . . . . . . . . 9 |
9 | breq1 3927 | . . . . . . . . . . 11 | |
10 | 9 | anbi2d 459 | . . . . . . . . . 10 |
11 | fzind.3 | . . . . . . . . . 10 | |
12 | 10, 11 | imbi12d 233 | . . . . . . . . 9 |
13 | breq1 3927 | . . . . . . . . . . 11 | |
14 | 13 | anbi2d 459 | . . . . . . . . . 10 |
15 | fzind.4 | . . . . . . . . . 10 | |
16 | 14, 15 | imbi12d 233 | . . . . . . . . 9 |
17 | fzind.5 | . . . . . . . . . 10 | |
18 | 17 | 3expib 1184 | . . . . . . . . 9 |
19 | zre 9051 | . . . . . . . . . . . . . 14 | |
20 | zre 9051 | . . . . . . . . . . . . . 14 | |
21 | p1le 8600 | . . . . . . . . . . . . . . 15 | |
22 | 21 | 3expia 1183 | . . . . . . . . . . . . . 14 |
23 | 19, 20, 22 | syl2an 287 | . . . . . . . . . . . . 13 |
24 | 23 | imdistanda 444 | . . . . . . . . . . . 12 |
25 | 24 | imim1d 75 | . . . . . . . . . . 11 |
26 | 25 | 3ad2ant2 1003 | . . . . . . . . . 10 |
27 | zltp1le 9101 | . . . . . . . . . . . . . . . . . . . . . 22 | |
28 | 27 | adantlr 468 | . . . . . . . . . . . . . . . . . . . . 21 |
29 | 28 | expcom 115 | . . . . . . . . . . . . . . . . . . . 20 |
30 | 29 | pm5.32d 445 | . . . . . . . . . . . . . . . . . . 19 |
31 | 30 | adantl 275 | . . . . . . . . . . . . . . . . . 18 |
32 | fzind.6 | . . . . . . . . . . . . . . . . . . . . 21 | |
33 | 32 | expcom 115 | . . . . . . . . . . . . . . . . . . . 20 |
34 | 33 | 3expa 1181 | . . . . . . . . . . . . . . . . . . 19 |
35 | 34 | com12 30 | . . . . . . . . . . . . . . . . . 18 |
36 | 31, 35 | sylbird 169 | . . . . . . . . . . . . . . . . 17 |
37 | 36 | ex 114 | . . . . . . . . . . . . . . . 16 |
38 | 37 | com23 78 | . . . . . . . . . . . . . . 15 |
39 | 38 | expd 256 | . . . . . . . . . . . . . 14 |
40 | 39 | 3impib 1179 | . . . . . . . . . . . . 13 |
41 | 40 | com23 78 | . . . . . . . . . . . 12 |
42 | 41 | impd 252 | . . . . . . . . . . 11 |
43 | 42 | a2d 26 | . . . . . . . . . 10 |
44 | 26, 43 | syld 45 | . . . . . . . . 9 |
45 | 4, 8, 12, 16, 18, 44 | uzind 9155 | . . . . . . . 8 |
46 | 45 | expcomd 1417 | . . . . . . 7 |
47 | 46 | 3expb 1182 | . . . . . 6 |
48 | 47 | expcom 115 | . . . . 5 |
49 | 48 | com23 78 | . . . 4 |
50 | 49 | 3impia 1178 | . . 3 |
51 | 50 | impd 252 | . 2 |
52 | 51 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 class class class wbr 3924 (class class class)co 5767 cr 7612 c1 7614 caddc 7616 clt 7793 cle 7794 cz 9047 |
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-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 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-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 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-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 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-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 |
This theorem is referenced by: fnn0ind 9160 fzind2 10009 |
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