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Theorem prarloclemarch 7233
Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7232 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 7207 . . . 4 (𝐵Q → (*Q𝐵) ∈ Q)
2 mulclnq 7191 . . . 4 ((𝐴Q ∧ (*Q𝐵) ∈ Q) → (𝐴 ·Q (*Q𝐵)) ∈ Q)
31, 2sylan2 284 . . 3 ((𝐴Q𝐵Q) → (𝐴 ·Q (*Q𝐵)) ∈ Q)
4 archnqq 7232 . . 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 518 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → 𝐴Q)
7 1pi 7130 . . . . . . . . . . 11 1oN
8 opelxpi 4571 . . . . . . . . . . 11 ((𝑥N ∧ 1oN) → ⟨𝑥, 1o⟩ ∈ (N × N))
97, 8mpan2 421 . . . . . . . . . 10 (𝑥N → ⟨𝑥, 1o⟩ ∈ (N × N))
10 enqex 7175 . . . . . . . . . . 11 ~Q ∈ V
1110ecelqsi 6483 . . . . . . . . . 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 7163 . . . . . . . . 9 Q = ((N × N) / ~Q )
1412, 13eleqtrrdi 2233 . . . . . . . 8 (𝑥N → [⟨𝑥, 1o⟩] ~QQ)
1514adantl 275 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → [⟨𝑥, 1o⟩] ~QQ)
16 simplr 519 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → 𝐵Q)
17 mulclnq 7191 . . . . . . 7 (([⟨𝑥, 1o⟩] ~QQ𝐵Q) → ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ∈ Q)
1815, 16, 17syl2anc 408 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ∈ Q)
1916, 1syl 14 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → (*Q𝐵) ∈ Q)
20 ltmnqg 7216 . . . . . 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 1216 . . . . 5 (((𝐴Q𝐵Q) ∧ 𝑥N) → (𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ↔ ((*Q𝐵) ·Q 𝐴) <Q ((*Q𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵))))
22 mulcomnqg 7198 . . . . . . 7 (((*Q𝐵) ∈ Q𝐴Q) → ((*Q𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q𝐵)))
2319, 6, 22syl2anc 408 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((*Q𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q𝐵)))
24 mulcomnqg 7198 . . . . . . . 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 408 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((*Q𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)) = (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)))
26 mulassnqg 7199 . . . . . . . . 9 (([⟨𝑥, 1o⟩] ~QQ𝐵Q ∧ (*Q𝐵) ∈ Q) → (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)) = ([⟨𝑥, 1o⟩] ~Q ·Q (𝐵 ·Q (*Q𝐵))))
2715, 16, 19, 26syl3anc 1216 . . . . . . . 8 (((𝐴Q𝐵Q) ∧ 𝑥N) → (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)) = ([⟨𝑥, 1o⟩] ~Q ·Q (𝐵 ·Q (*Q𝐵))))
28 recidnq 7208 . . . . . . . . . 10 (𝐵Q → (𝐵 ·Q (*Q𝐵)) = 1Q)
2928oveq2d 5790 . . . . . . . . 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 7204 . . . . . . . . 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 2176 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → (([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)) = [⟨𝑥, 1o⟩] ~Q )
3425, 33eqtrd 2172 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((*Q𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)) = [⟨𝑥, 1o⟩] ~Q )
3523, 34breq12d 3942 . . . . 5 (((𝐴Q𝐵Q) ∧ 𝑥N) → (((*Q𝐵) ·Q 𝐴) <Q ((*Q𝐵) ·Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)) ↔ (𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1o⟩] ~Q ))
3621, 35bitrd 187 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑥N) → (𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵) ↔ (𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1o⟩] ~Q ))
3736biimprd 157 . . 3 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1o⟩] ~Q𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵)))
3837reximdva 2534 . 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 103  wb 104   = wceq 1331  wcel 1480  wrex 2417  cop 3530   class class class wbr 3929   × cxp 4537  cfv 5123  (class class class)co 5774  1oc1o 6306  [cec 6427   / cqs 6428  Ncnpi 7087   ~Q ceq 7094  Qcnq 7095  1Qc1q 7096   ·Q cmq 7098  *Qcrq 7099   <Q cltq 7100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168
This theorem is referenced by:  prarloclemarch2  7234
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