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Mirrors > Home > ILE Home > Th. List > prmind | GIF version |
Description: Perform induction over the multiplicative structure of ℕ. If a property 𝜑(𝑥) holds for the primes and 1 and is preserved under multiplication, then it holds for every positive integer. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Ref | Expression |
---|---|
prmind.1 | ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) |
prmind.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
prmind.3 | ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃)) |
prmind.4 | ⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏)) |
prmind.5 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) |
prmind.6 | ⊢ 𝜓 |
prmind.7 | ⊢ (𝑥 ∈ ℙ → 𝜑) |
prmind.8 | ⊢ ((𝑦 ∈ (ℤ≥‘2) ∧ 𝑧 ∈ (ℤ≥‘2)) → ((𝜒 ∧ 𝜃) → 𝜏)) |
Ref | Expression |
---|---|
prmind | ⊢ (𝐴 ∈ ℕ → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmind.1 | . 2 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
2 | prmind.2 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
3 | prmind.3 | . 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃)) | |
4 | prmind.4 | . 2 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏)) | |
5 | prmind.5 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) | |
6 | prmind.6 | . 2 ⊢ 𝜓 | |
7 | prmind.7 | . . 3 ⊢ (𝑥 ∈ ℙ → 𝜑) | |
8 | 7 | adantr 274 | . 2 ⊢ ((𝑥 ∈ ℙ ∧ ∀𝑦 ∈ (1...(𝑥 − 1))𝜒) → 𝜑) |
9 | prmind.8 | . 2 ⊢ ((𝑦 ∈ (ℤ≥‘2) ∧ 𝑧 ∈ (ℤ≥‘2)) → ((𝜒 ∧ 𝜃) → 𝜏)) | |
10 | 1, 2, 3, 4, 5, 6, 8, 9 | prmind2 11801 | 1 ⊢ (𝐴 ∈ ℕ → 𝜂) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ‘cfv 5123 (class class class)co 5774 1c1 7621 · cmul 7625 − cmin 7933 ℕcn 8720 2c2 8771 ℤ≥cuz 9326 ...cfz 9790 ℙcprime 11788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-1o 6313 df-2o 6314 df-er 6429 df-en 6635 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-fzo 9920 df-fl 10043 df-mod 10096 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-dvds 11494 df-prm 11789 |
This theorem is referenced by: exprmfct 11818 |
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