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Mirrors > Home > MPE Home > Th. List > nn0ind | Structured version Visualization version GIF 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 |
---|---|
nn0ind.1 | ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) |
nn0ind.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
nn0ind.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) |
nn0ind.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
nn0ind.5 | ⊢ 𝜓 |
nn0ind.6 | ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
nn0ind | ⊢ (𝐴 ∈ ℕ0 → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0z 11267 | . 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴)) | |
2 | 0z 11265 | . . 3 ⊢ 0 ∈ ℤ | |
3 | nn0ind.1 | . . . 4 ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) | |
4 | nn0ind.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
5 | nn0ind.3 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) | |
6 | nn0ind.4 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
7 | nn0ind.5 | . . . . 5 ⊢ 𝜓 | |
8 | 7 | a1i 11 | . . . 4 ⊢ (0 ∈ ℤ → 𝜓) |
9 | elnn0z 11267 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)) | |
10 | nn0ind.6 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃)) | |
11 | 9, 10 | sylbir 224 | . . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) → (𝜒 → 𝜃)) |
12 | 11 | 3adant1 1072 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) → (𝜒 → 𝜃)) |
13 | 3, 4, 5, 6, 8, 12 | uzind 11345 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → 𝜏) |
14 | 2, 13 | mp3an1 1403 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → 𝜏) |
15 | 1, 14 | sylbi 206 | 1 ⊢ (𝐴 ∈ ℕ0 → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 ℕ0cn0 11169 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 |
This theorem is referenced by: nn0indALT 11349 nn0indd 11350 zindd 11354 fzennn 12629 mulexp 12761 expadd 12764 expmul 12767 leexp1a 12781 bernneq 12852 modexp 12861 faccl 12932 facdiv 12936 facwordi 12938 faclbnd 12939 faclbnd6 12948 facubnd 12949 bccl 12971 brfi1indALT 13137 brfi1indALTOLD 13143 wrdind 13328 wrd2ind 13329 cshweqrep 13418 rtrclreclem4 13649 relexpindlem 13651 cjexp 13738 absexp 13892 iseraltlem2 14261 binom 14401 bcxmas 14406 climcndslem1 14420 binomfallfac 14611 demoivreALT 14770 ruclem8 14805 odd2np1lem 14902 bitsinv1 15002 sadcadd 15018 sadadd2 15020 saddisjlem 15024 smu01lem 15045 smumullem 15052 alginv 15126 prmfac1 15269 pcfac 15441 ramcl 15571 mhmmulg 17406 psgnunilem3 17739 sylow1lem1 17836 efgsrel 17970 efgsfo 17975 efgred 17984 srgmulgass 18354 srgpcomp 18355 srgbinom 18368 lmodvsmmulgdi 18721 assamulgscm 19171 mplcoe3 19287 cnfldexp 19598 tmdmulg 21706 expcn 22483 dvnadd 23498 dvnres 23500 dvnfre 23521 ply1divex 23700 fta1g 23731 plyco 23801 dgrco 23835 dvnply2 23846 plydivex 23856 fta1 23867 cxpmul2 24235 facgam 24592 dchrisumlem1 24978 qabvle 25114 qabvexp 25115 ostth2lem2 25123 rusgranumwlk 26484 eupath2 26507 ex-ind-dvds 26710 subfacval2 30423 cvmliftlem7 30527 bccolsum 30878 faclim 30885 faclim2 30887 heiborlem4 32783 mzpexpmpt 36326 pell14qrexpclnn0 36448 rmxypos 36532 jm2.17a 36545 jm2.17b 36546 rmygeid 36549 jm2.19lem3 36576 hbtlem5 36717 cnsrexpcl 36754 relexpiidm 37015 fperiodmullem 38458 stoweidlem17 38910 stoweidlem19 38912 wallispilem3 38960 fmtnorec2 39993 rusgrnumwwlk 41178 eupth2 41407 lmodvsmdi 41957 |
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