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Mirrors > Home > MPE Home > Th. List > nnindALT | Structured version Visualization version GIF version |
Description: Principle of Mathematical
Induction (inference schema). The last four
hypotheses give us the substitution instances we need; the first two are
the induction step and the basis.
This ALT version of nnind 11650 has a different hypothesis order. It may be easier to use with the Metamath program Proof Assistant, because "MM-PA> ASSIGN LAST" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> MINIMIZE_WITH nnind / MAYGROW". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
nnindALT.6 | ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃)) |
nnindALT.5 | ⊢ 𝜓 |
nnindALT.1 | ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) |
nnindALT.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
nnindALT.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) |
nnindALT.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
Ref | Expression |
---|---|
nnindALT | ⊢ (𝐴 ∈ ℕ → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnindALT.1 | . 2 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
2 | nnindALT.2 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
3 | nnindALT.3 | . 2 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) | |
4 | nnindALT.4 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
5 | nnindALT.5 | . 2 ⊢ 𝜓 | |
6 | nnindALT.6 | . 2 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃)) | |
7 | 1, 2, 3, 4, 5, 6 | nnind 11650 | 1 ⊢ (𝐴 ∈ ℕ → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 (class class class)co 7150 1c1 10532 + caddc 10534 ℕcn 11632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-1cn 10589 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-nn 11633 |
This theorem is referenced by: (None) |
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