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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 270 | . 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 10502 | 1 ⊢ (𝐴 ∈ ℕ → 𝜂) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∀wral 2348 ‘cfv 4922 (class class class)co 5532 1c1 6982 · cmul 6986 − cmin 7279 ℕcn 8039 2c2 8089 ℤ≥cuz 8619 ...cfz 9029 ℙcprime 10489 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 ax-caucvg 7096 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-1o 6024 df-2o 6025 df-er 6129 df-en 6245 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-uz 8620 df-q 8705 df-rp 8735 df-fz 9030 df-fzo 9153 df-fl 9274 df-mod 9325 df-iseq 9432 df-iexp 9476 df-cj 9729 df-re 9730 df-im 9731 df-rsqrt 9884 df-abs 9885 df-dvds 10196 df-prm 10490 |
This theorem is referenced by: exprmfct 10519 |
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