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Theorem prarloclemarch 6275
Description: A version of the Archimedean property. This variation is "stronger" than archnqq 6274 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 6251 . . . 4 (B Q → (*QB) Q)
2 mulclnq 6235 . . . 4 ((A Q (*QB) Q) → (A ·Q (*QB)) Q)
31, 2sylan2 270 . . 3 ((A Q B Q) → (A ·Q (*QB)) Q)
4 archnqq 6274 . . 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 469 . . . . . 6 (((A Q B Q) x N) → A Q)
7 1pi 6175 . . . . . . . . . . 11 1𝑜 N
8 opelxpi 4303 . . . . . . . . . . 11 ((x N 1𝑜 N) → ⟨x, 1𝑜 (N × N))
97, 8mpan2 403 . . . . . . . . . 10 (x N → ⟨x, 1𝑜 (N × N))
10 enqex 6219 . . . . . . . . . . 11 ~Q V
1110ecelqsi 6071 . . . . . . . . . 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 6207 . . . . . . . . 9 Q = ((N × N) / ~Q )
1412, 13syl6eleqr 2113 . . . . . . . 8 (x N → [⟨x, 1𝑜⟩] ~Q Q)
1514adantl 262 . . . . . . 7 (((A Q B Q) x N) → [⟨x, 1𝑜⟩] ~Q Q)
16 simplr 470 . . . . . . 7 (((A Q B Q) x N) → B Q)
17 mulclnq 6235 . . . . . . 7 (([⟨x, 1𝑜⟩] ~Q Q B Q) → ([⟨x, 1𝑜⟩] ~Q ·Q B) Q)
1815, 16, 17syl2anc 393 . . . . . 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 6260 . . . . . 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 1121 . . . . 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 6242 . . . . . . 7 (((*QB) Q A Q) → ((*QB) ·Q A) = (A ·Q (*QB)))
2319, 6, 22syl2anc 393 . . . . . 6 (((A Q B Q) x N) → ((*QB) ·Q A) = (A ·Q (*QB)))
24 mulcomnqg 6242 . . . . . . . 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 393 . . . . . . 7 (((A Q B Q) x N) → ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B)) = (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)))
26 mulassnqg 6243 . . . . . . . . 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 1121 . . . . . . . 8 (((A Q B Q) x N) → (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)) = ([⟨x, 1𝑜⟩] ~Q ·Q (B ·Q (*QB))))
28 recidnq 6252 . . . . . . . . . 10 (B Q → (B ·Q (*QB)) = 1Q)
2928oveq2d 5452 . . . . . . . . 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 6248 . . . . . . . . 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 2058 . . . . . . 7 (((A Q B Q) x N) → (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)) = [⟨x, 1𝑜⟩] ~Q )
3425, 33eqtrd 2054 . . . . . 6 (((A Q B Q) x N) → ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B)) = [⟨x, 1𝑜⟩] ~Q )
3523, 34breq12d 3751 . . . . 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 2399 . 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 1228   wcel 1374  wrex 2285  cop 3353   class class class wbr 3738   × cxp 4270  cfv 4829  (class class class)co 5436  1𝑜c1o 5909  [cec 6015   / cqs 6016  Ncnpi 6130   ~Q ceq 6137  Qcnq 6138  1Qc1q 6139   ·Q cmq 6141  *Qcrq 6142   <Q cltq 6143
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-dc 734  df-3or 874  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-eprel 4000  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-1o 5916  df-oadd 5920  df-omul 5921  df-er 6017  df-ec 6019  df-qs 6023  df-ni 6164  df-pli 6165  df-mi 6166  df-lti 6167  df-mpq 6204  df-enq 6206  df-nqqs 6207  df-mqqs 6209  df-1nqqs 6210  df-rq 6211  df-ltnqqs 6212
This theorem is referenced by:  prarloclemarch2  6276
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