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Theorem prarloclemarch 6670
Description: A version of the Archimedean property. This variation is "stronger" than archnqq 6669 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 6644 . . . 4 (𝐵Q → (*Q𝐵) ∈ Q)
2 mulclnq 6628 . . . 4 ((𝐴Q ∧ (*Q𝐵) ∈ Q) → (𝐴 ·Q (*Q𝐵)) ∈ Q)
31, 2sylan2 280 . . 3 ((𝐴Q𝐵Q) → (𝐴 ·Q (*Q𝐵)) ∈ Q)
4 archnqq 6669 . . 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 496 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → 𝐴Q)
7 1pi 6567 . . . . . . . . . . 11 1𝑜N
8 opelxpi 4402 . . . . . . . . . . 11 ((𝑥N ∧ 1𝑜N) → ⟨𝑥, 1𝑜⟩ ∈ (N × N))
97, 8mpan2 416 . . . . . . . . . 10 (𝑥N → ⟨𝑥, 1𝑜⟩ ∈ (N × N))
10 enqex 6612 . . . . . . . . . . 11 ~Q ∈ V
1110ecelqsi 6226 . . . . . . . . . 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 6600 . . . . . . . . 9 Q = ((N × N) / ~Q )
1412, 13syl6eleqr 2173 . . . . . . . 8 (𝑥N → [⟨𝑥, 1𝑜⟩] ~QQ)
1514adantl 271 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → [⟨𝑥, 1𝑜⟩] ~QQ)
16 simplr 497 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → 𝐵Q)
17 mulclnq 6628 . . . . . . 7 (([⟨𝑥, 1𝑜⟩] ~QQ𝐵Q) → ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ∈ Q)
1815, 16, 17syl2anc 403 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ∈ Q)
1916, 1syl 14 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → (*Q𝐵) ∈ Q)
20 ltmnqg 6653 . . . . . 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 1170 . . . . 5 (((𝐴Q𝐵Q) ∧ 𝑥N) → (𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ↔ ((*Q𝐵) ·Q 𝐴) <Q ((*Q𝐵) ·Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵))))
22 mulcomnqg 6635 . . . . . . 7 (((*Q𝐵) ∈ Q𝐴Q) → ((*Q𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q𝐵)))
2319, 6, 22syl2anc 403 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((*Q𝐵) ·Q 𝐴) = (𝐴 ·Q (*Q𝐵)))
24 mulcomnqg 6635 . . . . . . . 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 403 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((*Q𝐵) ·Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵)) = (([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)))
26 mulassnqg 6636 . . . . . . . . 9 (([⟨𝑥, 1𝑜⟩] ~QQ𝐵Q ∧ (*Q𝐵) ∈ Q) → (([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)) = ([⟨𝑥, 1𝑜⟩] ~Q ·Q (𝐵 ·Q (*Q𝐵))))
2715, 16, 19, 26syl3anc 1170 . . . . . . . 8 (((𝐴Q𝐵Q) ∧ 𝑥N) → (([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)) = ([⟨𝑥, 1𝑜⟩] ~Q ·Q (𝐵 ·Q (*Q𝐵))))
28 recidnq 6645 . . . . . . . . . 10 (𝐵Q → (𝐵 ·Q (*Q𝐵)) = 1Q)
2928oveq2d 5559 . . . . . . . . 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 6641 . . . . . . . . 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 2118 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑥N) → (([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ·Q (*Q𝐵)) = [⟨𝑥, 1𝑜⟩] ~Q )
3425, 33eqtrd 2114 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((*Q𝐵) ·Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵)) = [⟨𝑥, 1𝑜⟩] ~Q )
3523, 34breq12d 3806 . . . . 5 (((𝐴Q𝐵Q) ∧ 𝑥N) → (((*Q𝐵) ·Q 𝐴) <Q ((*Q𝐵) ·Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵)) ↔ (𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1𝑜⟩] ~Q ))
3621, 35bitrd 186 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑥N) → (𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵) ↔ (𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1𝑜⟩] ~Q ))
3736biimprd 156 . . 3 (((𝐴Q𝐵Q) ∧ 𝑥N) → ((𝐴 ·Q (*Q𝐵)) <Q [⟨𝑥, 1𝑜⟩] ~Q𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵)))
3837reximdva 2464 . 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 102  wb 103   = wceq 1285  wcel 1434  wrex 2350  cop 3409   class class class wbr 3793   × cxp 4369  cfv 4932  (class class class)co 5543  1𝑜c1o 6058  [cec 6170   / cqs 6171  Ncnpi 6524   ~Q ceq 6531  Qcnq 6532  1Qc1q 6533   ·Q cmq 6535  *Qcrq 6536   <Q cltq 6537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605
This theorem is referenced by:  prarloclemarch2  6671
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