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| 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 16830 | . . . . 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 16638 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
| 6 | 1, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
| 7 | 6 | nnzd 12545 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ) |
| 8 | 1, 2 | pccld 16816 | . . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℕ0) |
| 9 | 7, 8 | zexpcld 14044 | . . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℤ) |
| 10 | 7 | zcnd 12629 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
| 11 | 0red 11142 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 12 | 1red 11140 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 13 | 7 | zred 12628 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
| 14 | 0lt1 11667 | . . . . . . . . . 10 ⊢ 0 < 1 | |
| 15 | 14 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 0 < 1) |
| 16 | prmgt1 16662 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
| 17 | 1, 16 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 1 < 𝑃) |
| 18 | 11, 12, 13, 15, 17 | lttrd 11302 | . . . . . . . 8 ⊢ (𝜑 → 0 < 𝑃) |
| 19 | 11, 18 | ltned 11277 | . . . . . . 7 ⊢ (𝜑 → 0 ≠ 𝑃) |
| 20 | 19 | necomd 2988 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 0) |
| 21 | 8 | nn0zd 12544 | . . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℤ) |
| 22 | 10, 20, 21 | expne0d 14109 | . . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ≠ 0) |
| 23 | 2 | nnzd 12545 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 24 | dvdsval2 16219 | . . . . 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 16478 | . 2 ⊢ (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = ((𝑃↑(𝑃 pCnt 𝑁)) gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))))) |
| 28 | pcndvds2 16834 | . . . . 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 16676 | . . . . 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 16836 | . . . . . 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 16687 | . . . 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 7362 0cc0 11033 1c1 11034 < clt 11174 / cdiv 11802 ℕcn 12169 ℤcz 12519 ↑cexp 14018 ∥ cdvds 16216 gcd cgcd 16458 ℙcprime 16635 pCnt cpc 16802 |
| 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 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 |
| 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 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-er 8638 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-sup 9350 df-inf 9351 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-z 12520 df-uz 12784 df-q 12894 df-rp 12938 df-fz 13457 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-dvds 16217 df-gcd 16459 df-prm 16636 df-pc 16803 |
| This theorem is referenced by: aks4d1p8 42546 |
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