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Mirrors > Home > ILE Home > Th. List > nn0ind | Unicode version |
Description: Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 13-May-2004.) |
Ref | Expression |
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nn0ind.1 |
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nn0ind.2 |
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nn0ind.3 |
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nn0ind.4 |
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nn0ind.5 |
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nn0ind.6 |
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Ref | Expression |
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nn0ind |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0z 8965 |
. 2
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2 | 0z 8963 |
. . 3
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3 | nn0ind.1 |
. . . 4
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4 | nn0ind.2 |
. . . 4
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5 | nn0ind.3 |
. . . 4
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6 | nn0ind.4 |
. . . 4
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7 | nn0ind.5 |
. . . . 5
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8 | 7 | a1i 9 |
. . . 4
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9 | elnn0z 8965 |
. . . . . 6
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10 | nn0ind.6 |
. . . . . 6
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11 | 9, 10 | sylbir 134 |
. . . . 5
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12 | 11 | 3adant1 980 |
. . . 4
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13 | 3, 4, 5, 6, 8, 12 | uzind 9060 |
. . 3
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14 | 2, 13 | mp3an1 1283 |
. 2
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15 | 1, 14 | sylbi 120 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-ltadd 7655 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-iota 5044 df-fun 5081 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-inn 8625 df-n0 8876 df-z 8953 |
This theorem is referenced by: zindd 9067 uzaddcl 9277 frecfzennn 10086 mulexp 10219 expadd 10222 expmul 10225 leexp1a 10235 bernneq 10299 faccl 10368 facdiv 10371 facwordi 10373 faclbnd 10374 faclbnd6 10377 facubnd 10378 bccl 10400 cjexp 10552 absexp 10737 binom 11139 bcxmas 11144 demoivreALT 11324 odd2np1lem 11411 alginv 11568 prmfac1 11670 ennnfonelemhf1o 11765 expcncf 12572 |
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