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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 9297 |
. 2
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2 | 0z 9295 |
. . 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 9297 |
. . . . . 6
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10 | nn0ind.6 |
. . . . . 6
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11 | 9, 10 | sylbir 135 |
. . . . 5
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12 | 11 | 3adant1 1017 |
. . . 4
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13 | 3, 4, 5, 6, 8, 12 | uzind 9395 |
. . 3
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14 | 2, 13 | mp3an1 1335 |
. 2
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15 | 1, 14 | sylbi 121 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-inn 8951 df-n0 9208 df-z 9285 |
This theorem is referenced by: zindd 9402 uzaddcl 9618 frecfzennn 10459 mulexp 10593 expadd 10596 expmul 10599 leexp1a 10609 bernneq 10675 modqexp 10681 nn0ltexp2 10724 faccl 10750 facdiv 10753 facwordi 10755 faclbnd 10756 faclbnd6 10759 facubnd 10760 bccl 10782 cjexp 10937 absexp 11123 binom 11527 bcxmas 11532 fprodfac 11658 demoivreALT 11816 odd2np1lem 11912 alginv 12082 prmfac1 12187 pcfac 12385 ennnfonelemhf1o 12467 mhmmulg 13120 srgmulgass 13360 srgpcomp 13361 lmodvsmmulgdi 13656 cnfldexp 13897 expcncf 14569 |
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