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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 16741 | . . . . 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 16555 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
6 | 1, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
7 | 6 | nnzd 12531 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ) |
8 | 1, 2 | pccld 16727 | . . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℕ0) |
9 | 7, 8 | zexpcld 13999 | . . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℤ) |
10 | 7 | zcnd 12613 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
11 | 0red 11163 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ) | |
12 | 1red 11161 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ) | |
13 | 7 | zred 12612 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
14 | 0lt1 11682 | . . . . . . . . . 10 ⊢ 0 < 1 | |
15 | 14 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 0 < 1) |
16 | prmgt1 16578 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
17 | 1, 16 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 1 < 𝑃) |
18 | 11, 12, 13, 15, 17 | lttrd 11321 | . . . . . . . 8 ⊢ (𝜑 → 0 < 𝑃) |
19 | 11, 18 | ltned 11296 | . . . . . . 7 ⊢ (𝜑 → 0 ≠ 𝑃) |
20 | 19 | necomd 2996 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 0) |
21 | 8 | nn0zd 12530 | . . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℤ) |
22 | 10, 20, 21 | expne0d 14063 | . . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ≠ 0) |
23 | 2 | nnzd 12531 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
24 | dvdsval2 16144 | . . . . 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 1372 | . . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁 ↔ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) ∈ ℤ)) |
26 | 4, 25 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) ∈ ℤ) |
27 | 26, 9 | gcdcomd 16399 | . 2 ⊢ (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = ((𝑃↑(𝑃 pCnt 𝑁)) gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))))) |
28 | pcndvds2 16745 | . . . . 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 16592 | . . . . 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 231 | . . 3 ⊢ (𝜑 → (𝑃 gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) = 1) |
33 | aks4d1p8d3.3 | . . . . 5 ⊢ (𝜑 → 𝑃 ∥ 𝑁) | |
34 | pcelnn 16747 | . . . . . 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 16603 | . . . 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 1372 | . . 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 2773 | 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 205 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 class class class wbr 5106 (class class class)co 7358 0cc0 11056 1c1 11057 < clt 11194 / cdiv 11817 ℕcn 12158 ℤcz 12504 ↑cexp 13973 ∥ cdvds 16141 gcd cgcd 16379 ℙcprime 16552 pCnt cpc 16713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-rp 12921 df-fz 13431 df-fl 13703 df-mod 13781 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-dvds 16142 df-gcd 16380 df-prm 16553 df-pc 16714 |
This theorem is referenced by: aks4d1p8 40590 |
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