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Mirrors > Home > ILE Home > Th. List > nn0ind-raph | 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. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.) |
Ref | Expression |
---|---|
nn0ind-raph.1 |
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nn0ind-raph.2 |
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nn0ind-raph.3 |
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nn0ind-raph.4 |
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nn0ind-raph.5 |
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nn0ind-raph.6 |
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Ref | Expression |
---|---|
nn0ind-raph |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 8880 |
. 2
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2 | dfsbcq2 2881 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | nfv 1491 |
. . . . 5
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4 | nn0ind-raph.2 |
. . . . 5
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5 | 3, 4 | sbhypf 2706 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | nfv 1491 |
. . . . 5
![]() ![]() ![]() ![]() | |
7 | nn0ind-raph.3 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 6, 7 | sbhypf 2706 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | nfv 1491 |
. . . . 5
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10 | nn0ind-raph.4 |
. . . . 5
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11 | 9, 10 | sbhypf 2706 |
. . . 4
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12 | nfsbc1v 2896 |
. . . . 5
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13 | 1ex 7682 |
. . . . 5
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14 | c0ex 7681 |
. . . . . . 7
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15 | 0nn0 8893 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() | |
16 | eleq1a 2186 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 15, 16 | ax-mp 7 |
. . . . . . . . . . 11
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18 | nn0ind-raph.5 |
. . . . . . . . . . . . . . 15
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19 | nn0ind-raph.1 |
. . . . . . . . . . . . . . 15
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20 | 18, 19 | mpbiri 167 |
. . . . . . . . . . . . . 14
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21 | eqeq2 2124 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 21, 4 | syl6bir 163 |
. . . . . . . . . . . . . . 15
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | pm5.74d 181 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 20, 23 | mpbii 147 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | com12 30 |
. . . . . . . . . . . 12
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26 | 14, 25 | vtocle 2731 |
. . . . . . . . . . 11
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27 | nn0ind-raph.6 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 17, 26, 27 | sylc 62 |
. . . . . . . . . 10
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29 | 28 | adantr 272 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | oveq1 5735 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 0p1e1 8741 |
. . . . . . . . . . . . 13
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32 | 30, 31 | syl6eq 2163 |
. . . . . . . . . . . 12
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33 | 32 | eqeq2d 2126 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 33, 7 | syl6bir 163 |
. . . . . . . . . 10
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35 | 34 | imp 123 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 29, 35 | mpbird 166 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 36 | ex 114 |
. . . . . . 7
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38 | 14, 37 | vtocle 2731 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | sbceq1a 2887 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
40 | 38, 39 | mpbid 146 |
. . . . 5
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41 | 12, 13, 40 | vtoclef 2730 |
. . . 4
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42 | nnnn0 8885 |
. . . . 5
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43 | 42, 27 | syl 14 |
. . . 4
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44 | 2, 5, 8, 11, 41, 43 | nnind 8643 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | nfv 1491 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
46 | eqeq1 2121 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
47 | 19 | bicomd 140 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 47, 10 | sylan9bb 455 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 18, 48 | mpbii 147 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 49 | ex 114 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51 | 46, 50 | sylbird 169 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 45, 14, 51 | vtoclef 2730 |
. . . 4
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53 | 52 | eqcoms 2118 |
. . 3
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54 | 44, 53 | jaoi 688 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | 1, 54 | 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-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-cnex 7633 ax-resscn 7634 ax-1cn 7635 ax-1re 7636 ax-icn 7637 ax-addcl 7638 ax-addrcl 7639 ax-mulcl 7640 ax-addcom 7642 ax-i2m1 7647 ax-0id 7650 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-sbc 2879 df-un 3041 df-in 3043 df-ss 3050 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-iota 5046 df-fv 5089 df-ov 5731 df-inn 8628 df-n0 8879 |
This theorem is referenced by: (None) |
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