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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 16806 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁) | |
4 | 1, 2, 3 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁) |
5 | prmnn 16618 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
6 | 1, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
7 | 6 | nnzd 12589 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ) |
8 | 1, 2 | pccld 16792 | . . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℕ0) |
9 | 7, 8 | zexpcld 14058 | . . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℤ) |
10 | 7 | zcnd 12671 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
11 | 0red 11221 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ) | |
12 | 1red 11219 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ) | |
13 | 7 | zred 12670 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
14 | 0lt1 11740 | . . . . . . . . . 10 ⊢ 0 < 1 | |
15 | 14 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 0 < 1) |
16 | prmgt1 16641 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
17 | 1, 16 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 1 < 𝑃) |
18 | 11, 12, 13, 15, 17 | lttrd 11379 | . . . . . . . 8 ⊢ (𝜑 → 0 < 𝑃) |
19 | 11, 18 | ltned 11354 | . . . . . . 7 ⊢ (𝜑 → 0 ≠ 𝑃) |
20 | 19 | necomd 2990 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 0) |
21 | 8 | nn0zd 12588 | . . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℤ) |
22 | 10, 20, 21 | expne0d 14122 | . . . . 5 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑁)) ≠ 0) |
23 | 2 | nnzd 12589 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
24 | dvdsval2 16207 | . . . . 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 1368 | . . . 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 16462 | . 2 ⊢ (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = ((𝑃↑(𝑃 pCnt 𝑁)) gcd (𝑁 / (𝑃↑(𝑃 pCnt 𝑁))))) |
28 | pcndvds2 16810 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ¬ 𝑃 ∥ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) | |
29 | 1, 2, 28 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝑁 / (𝑃↑(𝑃 pCnt 𝑁)))) |
30 | coprm 16655 | . . . . 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 583 | . . . 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 16812 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((𝑃 pCnt 𝑁) ∈ ℕ ↔ 𝑃 ∥ 𝑁)) | |
35 | 1, 2, 34 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → ((𝑃 pCnt 𝑁) ∈ ℕ ↔ 𝑃 ∥ 𝑁)) |
36 | 33, 35 | mpbird 257 | . . . 4 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℕ) |
37 | rpexp 16667 | . . . 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 1368 | . . 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 2766 | 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 1533 ∈ wcel 2098 ≠ wne 2934 class class class wbr 5141 (class class class)co 7405 0cc0 11112 1c1 11113 < clt 11252 / cdiv 11875 ℕcn 12216 ℤcz 12562 ↑cexp 14032 ∥ cdvds 16204 gcd cgcd 16442 ℙcprime 16615 pCnt cpc 16778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-fz 13491 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16205 df-gcd 16443 df-prm 16616 df-pc 16779 |
This theorem is referenced by: aks4d1p8 41469 |
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