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Theorem prarloclemarch 6544
 Description: A version of the Archimedean property. This variation is "stronger" than archnqq 6543 in the sense that we provide an integer which is larger than a given rational 𝐴 even after being multiplied by a second rational 𝐵. (Contributed by Jim Kingdon, 30-Nov-2019.)
Assertion
Ref Expression
prarloclemarch ((𝐴Q𝐵Q) → ∃𝑥N 𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem prarloclemarch
StepHypRef Expression
1 recclnq 6518 . . . 4 (𝐵Q → (*Q𝐵) ∈ Q)
2 mulclnq 6502 . . . 4 ((𝐴Q ∧ (*Q𝐵) ∈ Q) → (𝐴 ·Q (*Q𝐵)) ∈ Q)
31, 2sylan2 274 . . 3 ((𝐴Q𝐵Q) → (𝐴 ·Q (*Q𝐵)) ∈ Q)
4 archnqq 6543 . . 3 ((𝐴 ·Q (*Q𝐵)) ∈ Q → ∃𝑥N (𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1𝑜⟩] ~Q )
53, 4syl 14 . 2 ((𝐴Q𝐵Q) → ∃𝑥N (𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1𝑜⟩] ~Q )
6 simpll 489 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → 𝐴Q)
7 1pi 6441 . . . . . . . . . . 11 1𝑜N
8 opelxpi 4401 . . . . . . . . . . 11 ((𝑥N ∧ 1𝑜N) → ⟨𝑥, 1𝑜⟩ ∈ (N × N))
97, 8mpan2 409 . . . . . . . . . 10 (𝑥N → ⟨𝑥, 1𝑜⟩ ∈ (N × N))
10 enqex 6486 . . . . . . . . . . 11 ~Q ∈ V
1110ecelqsi 6188 . . . . . . . . . 10 (⟨𝑥, 1𝑜⟩ ∈ (N × N) → [⟨𝑥, 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
129, 11syl 14 . . . . . . . . 9 (𝑥N → [⟨𝑥, 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
13 df-nqqs 6474 . . . . . . . . 9 Q = ((N × N) / ~Q )
1412, 13syl6eleqr 2145 . . . . . . . 8 (𝑥N → [⟨𝑥, 1𝑜⟩] ~QQ)
1514adantl 266 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → [⟨𝑥, 1𝑜⟩] ~QQ)
16 simplr 490 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → 𝐵Q)
17 mulclnq 6502 . . . . . . 7 (([⟨𝑥, 1𝑜⟩] ~QQ𝐵Q) → ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ∈ Q)
1815, 16, 17syl2anc 397 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ∈ Q)
1916, 1syl 14 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → (*Q𝐵) ∈ Q)
20 ltmnqg 6527 . . . . . 6 ((𝐴Q ∧ ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ∈ Q ∧ (*Q𝐵) ∈ Q) → (𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ↔ ((*Q𝐵) ·Q 𝐴) <Q ((*Q𝐵) ·Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵))))
216, 18, 19, 20syl3anc 1144 . . . . 5 (((𝐴Q𝐵Q) ∧ 𝑥N) → (𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ↔ ((*Q𝐵) ·Q 𝐴) <Q ((*Q𝐵) ·Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵))))
22 mulcomnqg 6509 . . . . . . 7 (((*Q𝐵) ∈ Q𝐴Q) → ((*Q𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q𝐵)))
2319, 6, 22syl2anc 397 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((*Q𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q𝐵)))
24 mulcomnqg 6509 . . . . . . . 8 (((*Q𝐵) ∈ Q ∧ ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ∈ Q) → ((*Q𝐵) ·Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵)) = (([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)))
2519, 18, 24syl2anc 397 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((*Q𝐵) ·Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵)) = (([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)))
26 mulassnqg 6510 . . . . . . . . 9 (([⟨𝑥, 1𝑜⟩] ~QQ𝐵Q ∧ (*Q𝐵) ∈ Q) → (([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)) = ([⟨𝑥, 1𝑜⟩] ~Q ·Q (𝐵 ·Q (*Q𝐵))))
2715, 16, 19, 26syl3anc 1144 . . . . . . . 8 (((𝐴Q𝐵Q) ∧ 𝑥N) → (([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)) = ([⟨𝑥, 1𝑜⟩] ~Q ·Q (𝐵 ·Q (*Q𝐵))))
28 recidnq 6519 . . . . . . . . . 10 (𝐵Q → (𝐵 ·Q (*Q𝐵)) = 1Q)
2928oveq2d 5553 . . . . . . . . 9 (𝐵Q → ([⟨𝑥, 1𝑜⟩] ~Q ·Q (𝐵 ·Q (*Q𝐵))) = ([⟨𝑥, 1𝑜⟩] ~Q ·Q 1Q))
3016, 29syl 14 . . . . . . . 8 (((𝐴Q𝐵Q) ∧ 𝑥N) → ([⟨𝑥, 1𝑜⟩] ~Q ·Q (𝐵 ·Q (*Q𝐵))) = ([⟨𝑥, 1𝑜⟩] ~Q ·Q 1Q))
31 mulidnq 6515 . . . . . . . . 9 ([⟨𝑥, 1𝑜⟩] ~QQ → ([⟨𝑥, 1𝑜⟩] ~Q ·Q 1Q) = [⟨𝑥, 1𝑜⟩] ~Q )
3215, 31syl 14 . . . . . . . 8 (((𝐴Q𝐵Q) ∧ 𝑥N) → ([⟨𝑥, 1𝑜⟩] ~Q ·Q 1Q) = [⟨𝑥, 1𝑜⟩] ~Q )
3327, 30, 323eqtrd 2090 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → (([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)) = [⟨𝑥, 1𝑜⟩] ~Q )
3425, 33eqtrd 2086 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((*Q𝐵) ·Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵)) = [⟨𝑥, 1𝑜⟩] ~Q )
3523, 34breq12d 3802 . . . . 5 (((𝐴Q𝐵Q) ∧ 𝑥N) → (((*Q𝐵) ·Q 𝐴) <Q ((*Q𝐵) ·Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵)) ↔ (𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1𝑜⟩] ~Q ))
3621, 35bitrd 181 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑥N) → (𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ↔ (𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1𝑜⟩] ~Q ))
3736biimprd 151 . . 3 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1𝑜⟩] ~Q𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵)))
3837reximdva 2436 . 2 ((𝐴Q𝐵Q) → (∃𝑥N (𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1𝑜⟩] ~Q → ∃𝑥N 𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵)))
395, 38mpd 13 1 ((𝐴Q𝐵Q) → ∃𝑥N 𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1257   ∈ wcel 1407  ∃wrex 2322  ⟨cop 3403   class class class wbr 3789   × cxp 4368  ‘cfv 4927  (class class class)co 5537  1𝑜c1o 6022  [cec 6132   / cqs 6133  Ncnpi 6398   ~Q ceq 6405  Qcnq 6406  1Qc1q 6407   ·Q cmq 6409  *Qcrq 6410
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