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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 483 | . 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 16029 | 1 ⊢ (𝐴 ∈ ℕ → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ‘cfv 6355 (class class class)co 7156 1c1 10538 · cmul 10542 − cmin 10870 ℕcn 11638 2c2 11693 ℤ≥cuz 12244 ...cfz 12893 ℙcprime 16015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-prm 16016 |
This theorem is referenced by: exprmfct 16048 lgsquad2lem2 25961 2sqlem6 25999 ostthlem2 26204 fmtnofac2 43751 |
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