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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 9191 |
. 2
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2 | dfsbcq2 2977 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | nfv 1538 |
. . . . 5
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4 | nn0ind-raph.2 |
. . . . 5
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5 | 3, 4 | sbhypf 2798 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | nfv 1538 |
. . . . 5
![]() ![]() ![]() ![]() | |
7 | nn0ind-raph.3 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 6, 7 | sbhypf 2798 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | nfv 1538 |
. . . . 5
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10 | nn0ind-raph.4 |
. . . . 5
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11 | 9, 10 | sbhypf 2798 |
. . . 4
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12 | nfsbc1v 2993 |
. . . . 5
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13 | 1ex 7965 |
. . . . 5
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14 | c0ex 7964 |
. . . . . . 7
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15 | 0nn0 9204 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() | |
16 | eleq1a 2259 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . 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 168 |
. . . . . . . . . . . . . 14
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21 | eqeq2 2197 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 21, 4 | syl6bir 164 |
. . . . . . . . . . . . . . 15
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | pm5.74d 182 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 20, 23 | mpbii 148 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | com12 30 |
. . . . . . . . . . . 12
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26 | 14, 25 | vtocle 2823 |
. . . . . . . . . . 11
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27 | nn0ind-raph.6 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 17, 26, 27 | sylc 62 |
. . . . . . . . . 10
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29 | 28 | adantr 276 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | oveq1 5895 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 0p1e1 9046 |
. . . . . . . . . . . . 13
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32 | 30, 31 | eqtrdi 2236 |
. . . . . . . . . . . 12
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33 | 32 | eqeq2d 2199 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 33, 7 | syl6bir 164 |
. . . . . . . . . 10
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35 | 34 | imp 124 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 29, 35 | mpbird 167 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 36 | ex 115 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 14, 37 | vtocle 2823 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | sbceq1a 2984 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
40 | 38, 39 | mpbid 147 |
. . . . 5
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41 | 12, 13, 40 | vtoclef 2822 |
. . . 4
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42 | nnnn0 9196 |
. . . . 5
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43 | 42, 27 | syl 14 |
. . . 4
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44 | 2, 5, 8, 11, 41, 43 | nnind 8948 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | nfv 1538 |
. . . . 5
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46 | eqeq1 2194 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
47 | 19 | bicomd 141 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 47, 10 | sylan9bb 462 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 18, 48 | mpbii 148 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 49 | ex 115 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51 | 46, 50 | sylbird 170 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 45, 14, 51 | vtoclef 2822 |
. . . 4
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53 | 52 | eqcoms 2190 |
. . 3
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54 | 44, 53 | jaoi 717 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | 1, 54 | 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-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-sep 4133 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-i2m1 7929 ax-0id 7932 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-iota 5190 df-fv 5236 df-ov 5891 df-inn 8933 df-n0 9190 |
This theorem is referenced by: (None) |
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