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Mirrors > Home > MPE Home > Th. List > prmind | Structured version Visualization version 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 474 | . 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 15803 | 1 ⊢ (𝐴 ∈ ℕ → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ‘cfv 6135 (class class class)co 6922 1c1 10273 · cmul 10277 − cmin 10606 ℕcn 11374 2c2 11430 ℤ≥cuz 11992 ...cfz 12643 ℙcprime 15790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-fz 12644 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-dvds 15388 df-prm 15791 |
This theorem is referenced by: exprmfct 15820 lgsquad2lem2 25562 2sqlem6 25600 ostthlem2 25769 fmtnofac2 42502 |
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