![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nn0ind-raph | Structured version Visualization version 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 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
nn0ind-raph.2 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
nn0ind-raph.3 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
nn0ind-raph.4 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
nn0ind-raph.5 |
![]() ![]() |
nn0ind-raph.6 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
nn0ind-raph |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 10899 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | dfsbcq2 3281 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | nfv 1771 |
. . . . 5
![]() ![]() ![]() ![]() | |
4 | nn0ind-raph.2 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 3, 4 | sbhypf 3106 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | nfv 1771 |
. . . . 5
![]() ![]() ![]() ![]() | |
7 | nn0ind-raph.3 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 6, 7 | sbhypf 3106 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | nfv 1771 |
. . . . 5
![]() ![]() ![]() ![]() | |
10 | nn0ind-raph.4 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 9, 10 | sbhypf 3106 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | nfsbc1v 3298 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 1ex 9663 |
. . . . 5
![]() ![]() ![]() ![]() | |
14 | c0ex 9662 |
. . . . . . 7
![]() ![]() ![]() ![]() | |
15 | 0nn0 10912 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() | |
16 | eleq1a 2534 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | nn0ind-raph.5 |
. . . . . . . . . . . . . . 15
![]() ![]() | |
19 | nn0ind-raph.1 |
. . . . . . . . . . . . . . 15
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 18, 19 | mpbiri 241 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | eqeq2 2472 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 21, 4 | syl6bir 237 |
. . . . . . . . . . . . . . 15
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | pm5.74d 255 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 20, 23 | mpbii 216 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | com12 32 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 14, 25 | vtocle 3134 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | nn0ind-raph.6 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 17, 26, 27 | sylc 62 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 28 | adantr 471 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | oveq1 6321 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 0p1e1 10748 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
32 | 30, 31 | syl6eq 2511 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 32 | eqeq2d 2471 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 33, 7 | syl6bir 237 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 34 | imp 435 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 29, 35 | mpbird 240 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 36 | ex 440 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 14, 37 | vtocle 3134 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | sbceq1a 3289 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
40 | 38, 39 | mpbid 215 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 12, 13, 40 | vtoclef 3133 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | nnnn0 10904 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
43 | 42, 27 | syl 17 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | 2, 5, 8, 11, 41, 43 | nnind 10654 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | nfv 1771 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
46 | eqeq1 2465 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
47 | 19 | bicomd 206 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 47, 10 | sylan9bb 711 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 18, 48 | mpbii 216 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 49 | ex 440 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51 | 46, 50 | sylbird 243 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 45, 14, 51 | vtoclef 3133 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 52 | eqcoms 2469 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | 44, 53 | jaoi 385 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | 1, 54 | sylbi 200 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 ax-resscn 9621 ax-1cn 9622 ax-icn 9623 ax-addcl 9624 ax-addrcl 9625 ax-mulcl 9626 ax-mulrcl 9627 ax-mulcom 9628 ax-addass 9629 ax-mulass 9630 ax-distr 9631 ax-i2m1 9632 ax-1ne0 9633 ax-1rid 9634 ax-rnegex 9635 ax-rrecex 9636 ax-cnre 9637 ax-pre-lttri 9638 ax-pre-lttrn 9639 ax-pre-ltadd 9640 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-nel 2635 df-ral 2753 df-rex 2754 df-reu 2755 df-rab 2757 df-v 3058 df-sbc 3279 df-csb 3375 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-iun 4293 df-br 4416 df-opab 4475 df-mpt 4476 df-tr 4511 df-eprel 4763 df-id 4767 df-po 4773 df-so 4774 df-fr 4811 df-we 4813 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-pred 5398 df-ord 5444 df-on 5445 df-lim 5446 df-suc 5447 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-f1 5605 df-fo 5606 df-f1o 5607 df-fv 5608 df-ov 6317 df-om 6719 df-wrecs 7053 df-recs 7115 df-rdg 7153 df-er 7388 df-en 7595 df-dom 7596 df-sdom 7597 df-pnf 9702 df-mnf 9703 df-ltxr 9705 df-nn 10637 df-n0 10898 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |