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| 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 480 | . 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 16722 | 1 ⊢ (𝐴 ∈ ℕ → 𝜂) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 1c1 11156 · cmul 11160 − cmin 11492 ℕcn 12266 2c2 12321 ℤ≥cuz 12878 ...cfz 13547 ℙcprime 16708 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-prm 16709 | 
| This theorem is referenced by: exprmfct 16741 lgsquad2lem2 27429 2sqlem6 27467 ostthlem2 27672 fmtnofac2 47556 | 
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