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Mirrors > Home > MPE Home > Th. List > nn0indd | Structured version Visualization version GIF version |
Description: Principle of Mathematical Induction (inference schema) on nonnegative integers, a deduction version. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
Ref | Expression |
---|---|
nn0indd.1 | ⊢ (𝑥 = 0 → (𝜓 ↔ 𝜒)) |
nn0indd.2 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) |
nn0indd.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) |
nn0indd.4 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) |
nn0indd.5 | ⊢ (𝜑 → 𝜒) |
nn0indd.6 | ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
nn0indd | ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0indd.1 | . . . 4 ⊢ (𝑥 = 0 → (𝜓 ↔ 𝜒)) | |
2 | 1 | imbi2d 340 | . . 3 ⊢ (𝑥 = 0 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
3 | nn0indd.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) | |
4 | 3 | imbi2d 340 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜃))) |
5 | nn0indd.3 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) | |
6 | 5 | imbi2d 340 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜏))) |
7 | nn0indd.4 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) | |
8 | 7 | imbi2d 340 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜂))) |
9 | nn0indd.5 | . . 3 ⊢ (𝜑 → 𝜒) | |
10 | nn0indd.6 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ 𝜃) → 𝜏) | |
11 | 10 | ex 412 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → (𝜃 → 𝜏)) |
12 | 11 | expcom 413 | . . . 4 ⊢ (𝑦 ∈ ℕ0 → (𝜑 → (𝜃 → 𝜏))) |
13 | 12 | a2d 29 | . . 3 ⊢ (𝑦 ∈ ℕ0 → ((𝜑 → 𝜃) → (𝜑 → 𝜏))) |
14 | 2, 4, 6, 8, 9, 13 | nn0ind 12443 | . 2 ⊢ (𝐴 ∈ ℕ0 → (𝜑 → 𝜂)) |
15 | 14 | impcom 407 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1537 ∈ wcel 2101 (class class class)co 7295 0cc0 10899 1c1 10900 + caddc 10902 ℕ0cn0 12261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-n0 12262 df-z 12348 |
This theorem is referenced by: faclbnd6 14041 cjexp 14889 absexp 15044 bcxmas 15575 mhppwdeg 21368 tmdmulg 23271 omndmul2 31366 omndmul 31368 breprexp 32641 sticksstones22 40150 factwoffsmonot 40189 dvnxpaek 43518 itcovalendof 46055 itcovalpc 46058 |
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