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Theorem aks4d1p8d2 40542
Description: Any prime power dividing a positive integer is less than that integer if that integer has another prime factor. (Contributed by metakunt, 13-Nov-2024.)
Hypotheses
Ref Expression
aks4d1p8d2.1 (𝜑𝑅 ∈ ℕ)
aks4d1p8d2.2 (𝜑𝑁 ∈ ℕ)
aks4d1p8d2.3 (𝜑𝑃 ∈ ℙ)
aks4d1p8d2.4 (𝜑𝑄 ∈ ℙ)
aks4d1p8d2.5 (𝜑𝑃𝑅)
aks4d1p8d2.6 (𝜑𝑄𝑅)
aks4d1p8d2.7 (𝜑 → ¬ 𝑃𝑁)
aks4d1p8d2.8 (𝜑𝑄𝑁)
Assertion
Ref Expression
aks4d1p8d2 (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)

Proof of Theorem aks4d1p8d2
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 aks4d1p8d2.3 . . . . 5 (𝜑𝑃 ∈ ℙ)
2 prmnn 16550 . . . . 5 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
31, 2syl 17 . . . 4 (𝜑𝑃 ∈ ℕ)
43nnred 12168 . . 3 (𝜑𝑃 ∈ ℝ)
5 aks4d1p8d2.1 . . . 4 (𝜑𝑅 ∈ ℕ)
61, 5pccld 16722 . . 3 (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ0)
74, 6reexpcld 14068 . 2 (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ)
8 aks4d1p8d2.4 . . . . 5 (𝜑𝑄 ∈ ℙ)
9 prmnn 16550 . . . . 5 (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
108, 9syl 17 . . . 4 (𝜑𝑄 ∈ ℕ)
1110nnred 12168 . . 3 (𝜑𝑄 ∈ ℝ)
127, 11remulcld 11185 . 2 (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℝ)
135nnred 12168 . 2 (𝜑𝑅 ∈ ℝ)
147recnd 11183 . . . 4 (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℂ)
1514mulid1d 11172 . . 3 (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) = (𝑃↑(𝑃 pCnt 𝑅)))
16 1red 11156 . . . 4 (𝜑 → 1 ∈ ℝ)
173nnrpd 12955 . . . . 5 (𝜑𝑃 ∈ ℝ+)
186nn0zd 12525 . . . . 5 (𝜑 → (𝑃 pCnt 𝑅) ∈ ℤ)
1917, 18rpexpcld 14150 . . . 4 (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ+)
20 prmgt1 16573 . . . . 5 (𝑄 ∈ ℙ → 1 < 𝑄)
218, 20syl 17 . . . 4 (𝜑 → 1 < 𝑄)
2216, 11, 19, 21ltmul2dd 13013 . . 3 (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
2315, 22eqbrtrrd 5129 . 2 (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
243nnzd 12526 . . . . 5 (𝜑𝑃 ∈ ℤ)
2524, 6zexpcld 13993 . . . 4 (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ)
2610nnzd 12526 . . . 4 (𝜑𝑄 ∈ ℤ)
275nnzd 12526 . . . 4 (𝜑𝑅 ∈ ℤ)
2825, 26gcdcomd 16394 . . . . 5 (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))))
29 0lt1 11677 . . . . . . . . . . . . . 14 0 < 1
3029a1i 11 . . . . . . . . . . . . 13 (𝜑 → 0 < 1)
31 0red 11158 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ ℝ)
3231, 16ltnled 11302 . . . . . . . . . . . . 13 (𝜑 → (0 < 1 ↔ ¬ 1 ≤ 0))
3330, 32mpbid 231 . . . . . . . . . . . 12 (𝜑 → ¬ 1 ≤ 0)
3411recnd 11183 . . . . . . . . . . . . . . . . . 18 (𝜑𝑄 ∈ ℂ)
3534exp1d 14046 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑄↑1) = 𝑄)
3635eqcomd 2742 . . . . . . . . . . . . . . . 16 (𝜑𝑄 = (𝑄↑1))
3736oveq2d 7373 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄 pCnt 𝑄) = (𝑄 pCnt (𝑄↑1)))
38 1zzd 12534 . . . . . . . . . . . . . . . 16 (𝜑 → 1 ∈ ℤ)
39 pcid 16745 . . . . . . . . . . . . . . . 16 ((𝑄 ∈ ℙ ∧ 1 ∈ ℤ) → (𝑄 pCnt (𝑄↑1)) = 1)
408, 38, 39syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄 pCnt (𝑄↑1)) = 1)
4137, 40eqtrd 2776 . . . . . . . . . . . . . 14 (𝜑 → (𝑄 pCnt 𝑄) = 1)
42 aks4d1p8d2.8 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑄𝑁)
4342adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑃 = 𝑄) → 𝑄𝑁)
44 breq1 5108 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 = 𝑄 → (𝑃𝑁𝑄𝑁))
4544adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑃 = 𝑄) → (𝑃𝑁𝑄𝑁))
4645bicomd 222 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑃 = 𝑄) → (𝑄𝑁𝑃𝑁))
4746biimpd 228 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑃 = 𝑄) → (𝑄𝑁𝑃𝑁))
4843, 47mpd 15 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑃 = 𝑄) → 𝑃𝑁)
49 aks4d1p8d2.7 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ¬ 𝑃𝑁)
5049adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑃 = 𝑄) → ¬ 𝑃𝑁)
5148, 50pm2.65da 815 . . . . . . . . . . . . . . . . 17 (𝜑 → ¬ 𝑃 = 𝑄)
5251neqcomd 2746 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝑄 = 𝑃)
53 aks4d1p8d2.5 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑃𝑅)
54 pcelnn 16742 . . . . . . . . . . . . . . . . . . . 20 ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℕ) → ((𝑃 pCnt 𝑅) ∈ ℕ ↔ 𝑃𝑅))
551, 5, 54syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑃 pCnt 𝑅) ∈ ℕ ↔ 𝑃𝑅))
5653, 55mpbird 256 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ)
57 prmdvdsexpb 16592 . . . . . . . . . . . . . . . . . 18 ((𝑄 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ (𝑃 pCnt 𝑅) ∈ ℕ) → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ 𝑄 = 𝑃))
588, 1, 56, 57syl3anc 1371 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ 𝑄 = 𝑃))
5958notbid 317 . . . . . . . . . . . . . . . 16 (𝜑 → (¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ ¬ 𝑄 = 𝑃))
6052, 59mpbird 256 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)))
613, 6nnexpcld 14148 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℕ)
62 pceq0 16743 . . . . . . . . . . . . . . . 16 ((𝑄 ∈ ℙ ∧ (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℕ) → ((𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = 0 ↔ ¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅))))
638, 61, 62syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = 0 ↔ ¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅))))
6460, 63mpbird 256 . . . . . . . . . . . . . 14 (𝜑 → (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = 0)
6541, 64breq12d 5118 . . . . . . . . . . . . 13 (𝜑 → ((𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ 1 ≤ 0))
6665notbid 317 . . . . . . . . . . . 12 (𝜑 → (¬ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ ¬ 1 ≤ 0))
6733, 66mpbird 256 . . . . . . . . . . 11 (𝜑 → ¬ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
6867adantr 481 . . . . . . . . . 10 ((𝜑𝑝 = 𝑄) → ¬ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
69 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑝 = 𝑄) → 𝑝 = 𝑄)
7069oveq1d 7372 . . . . . . . . . . . 12 ((𝜑𝑝 = 𝑄) → (𝑝 pCnt 𝑄) = (𝑄 pCnt 𝑄))
7169oveq1d 7372 . . . . . . . . . . . 12 ((𝜑𝑝 = 𝑄) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
7270, 71breq12d 5118 . . . . . . . . . . 11 ((𝜑𝑝 = 𝑄) → ((𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
7372notbid 317 . . . . . . . . . 10 ((𝜑𝑝 = 𝑄) → (¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ ¬ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
7468, 73mpbird 256 . . . . . . . . 9 ((𝜑𝑝 = 𝑄) → ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
7574, 8rspcime 3584 . . . . . . . 8 (𝜑 → ∃𝑝 ∈ ℙ ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
76 rexnal 3103 . . . . . . . . 9 (∃𝑝 ∈ ℙ ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ ¬ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
7776a1i 11 . . . . . . . 8 (𝜑 → (∃𝑝 ∈ ℙ ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ ¬ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
7875, 77mpbid 231 . . . . . . 7 (𝜑 → ¬ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
79 pc2dvds 16751 . . . . . . . . 9 ((𝑄 ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ) → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
8026, 25, 79syl2anc 584 . . . . . . . 8 (𝜑 → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
8180notbid 317 . . . . . . 7 (𝜑 → (¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ ¬ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
8278, 81mpbird 256 . . . . . 6 (𝜑 → ¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)))
83 coprm 16587 . . . . . . 7 ((𝑄 ∈ ℙ ∧ (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ) → (¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))) = 1))
848, 25, 83syl2anc 584 . . . . . 6 (𝜑 → (¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))) = 1))
8582, 84mpbid 231 . . . . 5 (𝜑 → (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))) = 1)
8628, 85eqtrd 2776 . . . 4 (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = 1)
87 pcdvds 16736 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
881, 5, 87syl2anc 584 . . . 4 (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
89 aks4d1p8d2.6 . . . 4 (𝜑𝑄𝑅)
9025, 26, 27, 86, 88, 89coprmdvds2d 40459 . . 3 (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∥ 𝑅)
9125, 26zmulcld 12613 . . . 4 (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℤ)
92 dvdsle 16192 . . . 4 ((((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℤ ∧ 𝑅 ∈ ℕ) → (((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∥ 𝑅 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ≤ 𝑅))
9391, 5, 92syl2anc 584 . . 3 (𝜑 → (((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∥ 𝑅 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ≤ 𝑅))
9490, 93mpd 15 . 2 (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ≤ 𝑅)
957, 12, 13, 23, 94ltletrd 11315 1 (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073   class class class wbr 5105  (class class class)co 7357  0cc0 11051  1c1 11052   · cmul 11056   < clt 11189  cle 11190  cn 12153  cz 12499  cexp 13967  cdvds 16136   gcd cgcd 16374  cprime 16547   pCnt cpc 16708
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-fz 13425  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-dvds 16137  df-gcd 16375  df-prm 16548  df-pc 16709
This theorem is referenced by:  aks4d1p8  40544
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