| Mathbox for metakunt |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > aks4d1p8d3 | Structured version Visualization version GIF version | ||
| Description: The remainder of a division with its maximal prime power is coprime with that prime power. (Contributed by metakunt, 13-Nov-2024.) |
| Ref | Expression |
|---|---|
| aks4d1p8d3.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
| aks4d1p8d3.2 | ⊢ (𝜑 → 𝑃 ∈ ℙ) |
| aks4d1p8d3.3 | ⊢ (𝜑 → 𝑃 ∥ 𝑁) |
| Ref | Expression |
|---|---|
| aks4d1p8d3 | ⊢ (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = 1) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | aks4d1p8d3.2 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ) | |
| 2 | aks4d1p8d3.1 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
| 3 | pcdvds 16832 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁) | |
| 4 | 1, 2, 3 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁) |
| 5 | prmnn 16640 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
| 6 | 1, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
| 7 | 6 | nnzd 12547 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ) |
| 8 | 1, 2 | pccld 16818 | . . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℕ0) |
| 9 | 7, 8 | zexpcld 14046 | . . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℤ) |
| 10 | 7 | zcnd 12631 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
| 11 | 0red 11144 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 12 | 1red 11142 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 13 | 7 | zred 12630 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
| 14 | 0lt1 11669 | . . . . . . . . . 10 ⊢ 0 < 1 | |
| 15 | 14 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 0 < 1) |
| 16 | prmgt1 16664 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
| 17 | 1, 16 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 1 < 𝑃) |
| 18 | 11, 12, 13, 15, 17 | lttrd 11304 | . . . . . . . 8 ⊢ (𝜑 → 0 < 𝑃) |
| 19 | 11, 18 | ltned 11279 | . . . . . . 7 ⊢ (𝜑 → 0 ≠ 𝑃) |
| 20 | 19 | necomd 2988 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 0) |
| 21 | 8 | nn0zd 12546 | . . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℤ) |
| 22 | 10, 20, 21 | expne0d 14111 | . . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ≠ 0) |
| 23 | 2 | nnzd 12547 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 24 | dvdsval2 16221 | . . . . 5 ⊢ (((𝑃↑(𝑃 pCnt 𝑁)) ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑁)) ≠ 0 ∧ 𝑁 ∈ ℤ) → ((𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁 ↔ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) ∈ ℤ)) | |
| 25 | 9, 22, 23, 24 | syl3anc 1374 | . . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁 ↔ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) ∈ ℤ)) |
| 26 | 4, 25 | mpbid 232 | . . 3 ⊢ (𝜑 → (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) ∈ ℤ) |
| 27 | 26, 9 | gcdcomd 16480 | . 2 ⊢ (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = ((𝑃↑(𝑃 pCnt 𝑁)) gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))))) |
| 28 | pcndvds2 16836 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ¬ 𝑃 ∥ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) | |
| 29 | 1, 2, 28 | syl2anc 585 | . . . 4 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) |
| 30 | coprm 16678 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) ∈ ℤ) → (¬ 𝑃 ∥ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) ↔ (𝑃 gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) = 1)) | |
| 31 | 1, 26, 30 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (¬ 𝑃 ∥ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) ↔ (𝑃 gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) = 1)) |
| 32 | 29, 31 | mpbid 232 | . . 3 ⊢ (𝜑 → (𝑃 gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) = 1) |
| 33 | aks4d1p8d3.3 | . . . . 5 ⊢ (𝜑 → 𝑃 ∥ 𝑁) | |
| 34 | pcelnn 16838 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((𝑃 pCnt 𝑁) ∈ ℕ ↔ 𝑃 ∥ 𝑁)) | |
| 35 | 1, 2, 34 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → ((𝑃 pCnt 𝑁) ∈ ℕ ↔ 𝑃 ∥ 𝑁)) |
| 36 | 33, 35 | mpbird 257 | . . . 4 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℕ) |
| 37 | rpexp 16689 | . . . 4 ⊢ ((𝑃 ∈ ℤ ∧ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) ∈ ℤ ∧ (𝑃 pCnt 𝑁) ∈ ℕ) → (((𝑃↑(𝑃 pCnt 𝑁)) gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) = 1 ↔ (𝑃 gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) = 1)) | |
| 38 | 7, 26, 36, 37 | syl3anc 1374 | . . 3 ⊢ (𝜑 → (((𝑃↑(𝑃 pCnt 𝑁)) gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) = 1 ↔ (𝑃 gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) = 1)) |
| 39 | 32, 38 | mpbird 257 | . 2 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑁)) gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) = 1) |
| 40 | 27, 39 | eqtrd 2772 | 1 ⊢ (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = 1) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5086 (class class class)co 7364 0cc0 11035 1c1 11036 < clt 11176 / cdiv 11804 ℕcn 12171 ℤcz 12521 ↑cexp 14020 ∥ cdvds 16218 gcd cgcd 16460 ℙcprime 16637 pCnt cpc 16804 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112 ax-pre-sup 11113 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-om 7815 df-1st 7939 df-2nd 7940 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9352 df-inf 9353 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-div 11805 df-nn 12172 df-2 12241 df-3 12242 df-n0 12435 df-z 12522 df-uz 12786 df-q 12896 df-rp 12940 df-fz 13459 df-fl 13748 df-mod 13826 df-seq 13961 df-exp 14021 df-cj 15058 df-re 15059 df-im 15060 df-sqrt 15194 df-abs 15195 df-dvds 16219 df-gcd 16461 df-prm 16638 df-pc 16805 |
| This theorem is referenced by: aks4d1p8 42548 |
| Copyright terms: Public domain | W3C validator |