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Theorem prarloclemarch 6401
Description: A version of the Archimedean property. This variation is "stronger" than archnqq 6400 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 ((A Q B Q) → x N A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B))
Distinct variable groups:   x,A   x,B

Proof of Theorem prarloclemarch
StepHypRef Expression
1 recclnq 6376 . . . 4 (B Q → (*QB) Q)
2 mulclnq 6360 . . . 4 ((A Q (*QB) Q) → (A ·Q (*QB)) Q)
31, 2sylan2 270 . . 3 ((A Q B Q) → (A ·Q (*QB)) Q)
4 archnqq 6400 . . 3 ((A ·Q (*QB)) Qx N (A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~Q )
53, 4syl 14 . 2 ((A Q B Q) → x N (A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~Q )
6 simpll 481 . . . . . 6 (((A Q B Q) x N) → A Q)
7 1pi 6299 . . . . . . . . . . 11 1𝑜 N
8 opelxpi 4319 . . . . . . . . . . 11 ((x N 1𝑜 N) → ⟨x, 1𝑜 (N × N))
97, 8mpan2 401 . . . . . . . . . 10 (x N → ⟨x, 1𝑜 (N × N))
10 enqex 6344 . . . . . . . . . . 11 ~Q V
1110ecelqsi 6096 . . . . . . . . . 10 (⟨x, 1𝑜 (N × N) → [⟨x, 1𝑜⟩] ~Q ((N × N) / ~Q ))
129, 11syl 14 . . . . . . . . 9 (x N → [⟨x, 1𝑜⟩] ~Q ((N × N) / ~Q ))
13 df-nqqs 6332 . . . . . . . . 9 Q = ((N × N) / ~Q )
1412, 13syl6eleqr 2128 . . . . . . . 8 (x N → [⟨x, 1𝑜⟩] ~Q Q)
1514adantl 262 . . . . . . 7 (((A Q B Q) x N) → [⟨x, 1𝑜⟩] ~Q Q)
16 simplr 482 . . . . . . 7 (((A Q B Q) x N) → B Q)
17 mulclnq 6360 . . . . . . 7 (([⟨x, 1𝑜⟩] ~Q Q B Q) → ([⟨x, 1𝑜⟩] ~Q ·Q B) Q)
1815, 16, 17syl2anc 391 . . . . . 6 (((A Q B Q) x N) → ([⟨x, 1𝑜⟩] ~Q ·Q B) Q)
1916, 1syl 14 . . . . . 6 (((A Q B Q) x N) → (*QB) Q)
20 ltmnqg 6385 . . . . . 6 ((A Q ([⟨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 1134 . . . . 5 (((A Q B Q) x N) → (A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B) ↔ ((*QB) ·Q A) <Q ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B))))
22 mulcomnqg 6367 . . . . . . 7 (((*QB) Q A Q) → ((*QB) ·Q A) = (A ·Q (*QB)))
2319, 6, 22syl2anc 391 . . . . . 6 (((A Q B Q) x N) → ((*QB) ·Q A) = (A ·Q (*QB)))
24 mulcomnqg 6367 . . . . . . . 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 391 . . . . . . 7 (((A Q B Q) x N) → ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B)) = (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)))
26 mulassnqg 6368 . . . . . . . . 9 (([⟨x, 1𝑜⟩] ~Q Q B Q (*QB) Q) → (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)) = ([⟨x, 1𝑜⟩] ~Q ·Q (B ·Q (*QB))))
2715, 16, 19, 26syl3anc 1134 . . . . . . . 8 (((A Q B Q) x N) → (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)) = ([⟨x, 1𝑜⟩] ~Q ·Q (B ·Q (*QB))))
28 recidnq 6377 . . . . . . . . . 10 (B Q → (B ·Q (*QB)) = 1Q)
2928oveq2d 5471 . . . . . . . . 9 (B Q → ([⟨x, 1𝑜⟩] ~Q ·Q (B ·Q (*QB))) = ([⟨x, 1𝑜⟩] ~Q ·Q 1Q))
3016, 29syl 14 . . . . . . . 8 (((A Q B Q) x N) → ([⟨x, 1𝑜⟩] ~Q ·Q (B ·Q (*QB))) = ([⟨x, 1𝑜⟩] ~Q ·Q 1Q))
31 mulidnq 6373 . . . . . . . . 9 ([⟨x, 1𝑜⟩] ~Q Q → ([⟨x, 1𝑜⟩] ~Q ·Q 1Q) = [⟨x, 1𝑜⟩] ~Q )
3215, 31syl 14 . . . . . . . 8 (((A Q B Q) x N) → ([⟨x, 1𝑜⟩] ~Q ·Q 1Q) = [⟨x, 1𝑜⟩] ~Q )
3327, 30, 323eqtrd 2073 . . . . . . 7 (((A Q B Q) x N) → (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)) = [⟨x, 1𝑜⟩] ~Q )
3425, 33eqtrd 2069 . . . . . 6 (((A Q B Q) x N) → ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B)) = [⟨x, 1𝑜⟩] ~Q )
3523, 34breq12d 3768 . . . . 5 (((A Q B Q) x N) → (((*QB) ·Q A) <Q ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B)) ↔ (A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~Q ))
3621, 35bitrd 177 . . . 4 (((A Q B Q) x N) → (A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B) ↔ (A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~Q ))
3736biimprd 147 . . 3 (((A Q B Q) x N) → ((A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~QA <Q ([⟨x, 1𝑜⟩] ~Q ·Q B)))
3837reximdva 2415 . 2 ((A Q B Q) → (x N (A ·Q (*QB)) <Q [⟨x, 1𝑜⟩] ~Qx N A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B)))
395, 38mpd 13 1 ((A Q B Q) → x N A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wrex 2301  cop 3370   class class class wbr 3755   × cxp 4286  cfv 4845  (class class class)co 5455  1𝑜c1o 5933  [cec 6040   / cqs 6041  Ncnpi 6256   ~Q ceq 6263  Qcnq 6264  1Qc1q 6265   ·Q cmq 6267  *Qcrq 6268   <Q cltq 6269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337
This theorem is referenced by:  prarloclemarch2  6402
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