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Theorem prarloclemarch 6264
Description: A version of the Archimedean property. This variation is "stronger" than archnqq 6263 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 6240 . . . 4 (B Q → (*QB) Q)
2 mulclnq 6224 . . . 4 ((A Q (*QB) Q) → (A ·Q (*QB)) Q)
31, 2sylan2 270 . . 3 ((A Q B Q) → (A ·Q (*QB)) Q)
4 archnqq 6263 . . 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 466 . . . . . 6 (((A Q B Q) x N) → A Q)
7 1pi 6164 . . . . . . . . . . 11 1𝑜 N
8 opelxpi 4294 . . . . . . . . . . 11 ((x N 1𝑜 N) → ⟨x, 1𝑜 (N × N))
97, 8mpan2 401 . . . . . . . . . 10 (x N → ⟨x, 1𝑜 (N × N))
10 enqex 6208 . . . . . . . . . . 11 ~Q V
1110ecelqsi 6062 . . . . . . . . . 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 6196 . . . . . . . . 9 Q = ((N × N) / ~Q )
1412, 13syl6eleqr 2107 . . . . . . . 8 (x N → [⟨x, 1𝑜⟩] ~Q Q)
1514adantl 262 . . . . . . 7 (((A Q B Q) x N) → [⟨x, 1𝑜⟩] ~Q Q)
16 simplr 467 . . . . . . 7 (((A Q B Q) x N) → B Q)
17 mulclnq 6224 . . . . . . 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 6249 . . . . . 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 1118 . . . . 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 6231 . . . . . . 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 6231 . . . . . . . 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 6232 . . . . . . . . 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 1118 . . . . . . . 8 (((A Q B Q) x N) → (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)) = ([⟨x, 1𝑜⟩] ~Q ·Q (B ·Q (*QB))))
28 recidnq 6241 . . . . . . . . . 10 (B Q → (B ·Q (*QB)) = 1Q)
2928oveq2d 5443 . . . . . . . . 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 6237 . . . . . . . . 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 2052 . . . . . . 7 (((A Q B Q) x N) → (([⟨x, 1𝑜⟩] ~Q ·Q B) ·Q (*QB)) = [⟨x, 1𝑜⟩] ~Q )
3425, 33eqtrd 2048 . . . . . 6 (((A Q B Q) x N) → ((*QB) ·Q ([⟨x, 1𝑜⟩] ~Q ·Q B)) = [⟨x, 1𝑜⟩] ~Q )
3523, 34breq12d 3743 . . . . 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 2393 . 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 1226   wcel 1369  wrex 2279  cop 3345   class class class wbr 3730   × cxp 4261  cfv 4820  (class class class)co 5427  1𝑜c1o 5900  [cec 6006   / cqs 6007  Ncnpi 6121   ~Q ceq 6128  Qcnq 6129  1Qc1q 6130   ·Q cmq 6132  *Qcrq 6133   <Q cltq 6134
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 529  ax-in2 530  ax-io 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-13 1380  ax-14 1381  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998  ax-coll 3838  ax-sep 3841  ax-nul 3849  ax-pow 3893  ax-pr 3910  ax-un 4111  ax-setind 4195  ax-iinf 4229
This theorem depends on definitions:  df-bi 110  df-dc 727  df-3or 870  df-3an 871  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1622  df-eu 1879  df-mo 1880  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-ne 2182  df-ral 2283  df-rex 2284  df-reu 2285  df-rab 2287  df-v 2531  df-sbc 2736  df-csb 2824  df-dif 2891  df-un 2893  df-in 2895  df-ss 2902  df-nul 3196  df-pw 3328  df-sn 3348  df-pr 3349  df-op 3351  df-uni 3547  df-int 3582  df-iun 3625  df-br 3731  df-opab 3785  df-mpt 3786  df-tr 3821  df-eprel 3992  df-id 3996  df-iord 4044  df-on 4046  df-suc 4049  df-iom 4232  df-xp 4269  df-rel 4270  df-cnv 4271  df-co 4272  df-dm 4273  df-rn 4274  df-res 4275  df-ima 4276  df-iota 4785  df-fun 4822  df-fn 4823  df-f 4824  df-f1 4825  df-fo 4826  df-f1o 4827  df-fv 4828  df-ov 5430  df-oprab 5431  df-mpt2 5432  df-1st 5681  df-2nd 5682  df-recs 5833  df-irdg 5869  df-1o 5907  df-oadd 5911  df-omul 5912  df-er 6008  df-ec 6010  df-qs 6014  df-ni 6153  df-pli 6154  df-mi 6155  df-lti 6156  df-mpq 6193  df-enq 6195  df-nqqs 6196  df-mqqs 6198  df-1nqqs 6199  df-rq 6200  df-ltnqqs 6201
This theorem is referenced by:  prarloclemarch2  6265
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