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Mirrors > Home > MPE Home > Th. List > pcprmpw | Structured version Visualization version GIF version |
Description: Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Ref | Expression |
---|---|
pcprmpw | ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛) ↔ 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmz 15227 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
2 | 1 | adantr 480 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ℤ) |
3 | zexpcl 12737 | . . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝑛 ∈ ℕ0) → (𝑃↑𝑛) ∈ ℤ) | |
4 | 2, 3 | sylan 487 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ 𝑛 ∈ ℕ0) → (𝑃↑𝑛) ∈ ℤ) |
5 | iddvds 14833 | . . . . . 6 ⊢ ((𝑃↑𝑛) ∈ ℤ → (𝑃↑𝑛) ∥ (𝑃↑𝑛)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ 𝑛 ∈ ℕ0) → (𝑃↑𝑛) ∥ (𝑃↑𝑛)) |
7 | breq1 4586 | . . . . 5 ⊢ (𝐴 = (𝑃↑𝑛) → (𝐴 ∥ (𝑃↑𝑛) ↔ (𝑃↑𝑛) ∥ (𝑃↑𝑛))) | |
8 | 6, 7 | syl5ibrcom 236 | . . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ 𝑛 ∈ ℕ0) → (𝐴 = (𝑃↑𝑛) → 𝐴 ∥ (𝑃↑𝑛))) |
9 | 8 | reximdva 3000 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛) → ∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃↑𝑛))) |
10 | pcprmpw2 15424 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃↑𝑛) ↔ 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))) | |
11 | 9, 10 | sylibd 228 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))) |
12 | pccl 15392 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃 pCnt 𝐴) ∈ ℕ0) | |
13 | oveq2 6557 | . . . . . 6 ⊢ (𝑛 = (𝑃 pCnt 𝐴) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt 𝐴))) | |
14 | 13 | eqeq2d 2620 | . . . . 5 ⊢ (𝑛 = (𝑃 pCnt 𝐴) → (𝐴 = (𝑃↑𝑛) ↔ 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))) |
15 | 14 | rspcev 3282 | . . . 4 ⊢ (((𝑃 pCnt 𝐴) ∈ ℕ0 ∧ 𝐴 = (𝑃↑(𝑃 pCnt 𝐴))) → ∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛)) |
16 | 15 | ex 449 | . . 3 ⊢ ((𝑃 pCnt 𝐴) ∈ ℕ0 → (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → ∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛))) |
17 | 12, 16 | syl 17 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → ∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛))) |
18 | 11, 17 | impbid 201 | 1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛) ↔ 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 (class class class)co 6549 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 ↑cexp 12722 ∥ cdvds 14821 ℙcprime 15223 pCnt cpc 15379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380 |
This theorem is referenced by: pgpfi1 17833 pgpfi 17843 pgpfi2 17844 fislw 17863 |
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