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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 16898 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁) | |
4 | 1, 2, 3 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁) |
5 | prmnn 16708 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
6 | 1, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
7 | 6 | nnzd 12638 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ) |
8 | 1, 2 | pccld 16884 | . . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℕ0) |
9 | 7, 8 | zexpcld 14125 | . . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℤ) |
10 | 7 | zcnd 12721 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
11 | 0red 11262 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ) | |
12 | 1red 11260 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ) | |
13 | 7 | zred 12720 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
14 | 0lt1 11783 | . . . . . . . . . 10 ⊢ 0 < 1 | |
15 | 14 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 0 < 1) |
16 | prmgt1 16731 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
17 | 1, 16 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 1 < 𝑃) |
18 | 11, 12, 13, 15, 17 | lttrd 11420 | . . . . . . . 8 ⊢ (𝜑 → 0 < 𝑃) |
19 | 11, 18 | ltned 11395 | . . . . . . 7 ⊢ (𝜑 → 0 ≠ 𝑃) |
20 | 19 | necomd 2994 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 0) |
21 | 8 | nn0zd 12637 | . . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℤ) |
22 | 10, 20, 21 | expne0d 14189 | . . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ≠ 0) |
23 | 2 | nnzd 12638 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
24 | dvdsval2 16290 | . . . . 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 1370 | . . . 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 16548 | . 2 ⊢ (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = ((𝑃↑(𝑃 pCnt 𝑁)) gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))))) |
28 | pcndvds2 16902 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ¬ 𝑃 ∥ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) | |
29 | 1, 2, 28 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) |
30 | coprm 16745 | . . . . 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 584 | . . . 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 16904 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((𝑃 pCnt 𝑁) ∈ ℕ ↔ 𝑃 ∥ 𝑁)) | |
35 | 1, 2, 34 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((𝑃 pCnt 𝑁) ∈ ℕ ↔ 𝑃 ∥ 𝑁)) |
36 | 33, 35 | mpbird 257 | . . . 4 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℕ) |
37 | rpexp 16756 | . . . 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 1370 | . . 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 2775 | 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 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 (class class class)co 7431 0cc0 11153 1c1 11154 < clt 11293 / cdiv 11918 ℕcn 12264 ℤcz 12611 ↑cexp 14099 ∥ cdvds 16287 gcd cgcd 16528 ℙcprime 16705 pCnt cpc 16870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-fz 13545 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-prm 16706 df-pc 16871 |
This theorem is referenced by: aks4d1p8 42069 |
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