Theorem prarloclemarch 6516

Description: A version of the Archimedean property. This variation is "stronger" than archnqq 6515 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 e. Q. /\ B e. Q. ) -> E. x e. N. A <Q ( [ <. x , 1o >. ] ~Q .Q B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem prarloclemarch

Step	Hyp	Ref	Expression
1		recclnq 6490	. . . 4 |- ( B e. Q. -> ( *Q ` B ) e. Q. )
2		mulclnq 6474	. . . 4 |- ( ( A e. Q. /\ ( *Q ` B ) e. Q. ) -> ( A .Q ( *Q ` B ) ) e. Q. )
3	1, 2	sylan2 270	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q ( *Q ` B ) ) e. Q. )
4		archnqq 6515	. . 3 |- ( ( A .Q ( *Q ` B ) ) e. Q. -> E. x e. N. ( A .Q ( *Q ` B ) ) <Q [ <. x , 1o >. ] ~Q )
5	3, 4	syl 14	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> E. x e. N. ( A .Q ( *Q ` B ) ) <Q [ <. x , 1o >. ] ~Q )
6		simpll 481	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> A e. Q. )
7		1pi 6413	. . . . . . . . . . 11 |- 1o e. N.
8		opelxpi 4376	. . . . . . . . . . 11 |- ( ( x e. N. /\ 1o e. N. ) -> <. x , 1o >. e. ( N. X. N. ) )
9	7, 8	mpan2 401	. . . . . . . . . 10 |- ( x e. N. -> <. x , 1o >. e. ( N. X. N. ) )
10		enqex 6458	. . . . . . . . . . 11 |- ~Q e. _V
11	10	ecelqsi 6160	. . . . . . . . . 10 |- ( <. x , 1o >. e. ( N. X. N. ) -> [ <. x , 1o >. ] ~Q e. ( ( N. X. N. ) /. ~Q ) )
12	9, 11	syl 14	. . . . . . . . 9 |- ( x e. N. -> [ <. x , 1o >. ] ~Q e. ( ( N. X. N. ) /. ~Q ) )
13		df-nqqs 6446	. . . . . . . . 9 |- Q. = ( ( N. X. N. ) /. ~Q )
14	12, 13	syl6eleqr 2131	. . . . . . . 8 |- ( x e. N. -> [ <. x , 1o >. ] ~Q e. Q. )
15	14	adantl 262	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> [ <. x , 1o >. ] ~Q e. Q. )
16		simplr 482	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> B e. Q. )
17		mulclnq 6474	. . . . . . 7 |- ( ( [ <. x , 1o >. ] ~Q e. Q. /\ B e. Q. ) -> ( [ <. x , 1o >. ] ~Q .Q B ) e. Q. )
18	15, 16, 17	syl2anc 391	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( [ <. x , 1o >. ] ~Q .Q B ) e. Q. )
19	16, 1	syl 14	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( *Q ` B ) e. Q. )
20		ltmnqg 6499	. . . . . 6 |- ( ( A e. Q. /\ ( [ <. x , 1o >. ] ~Q .Q B ) e. Q. /\ ( *Q ` B ) e. Q. ) -> ( A <Q ( [ <. x , 1o >. ] ~Q .Q B ) <-> ( ( *Q ` B ) .Q A ) <Q ( ( *Q ` B ) .Q ( [ <. x , 1o >. ] ~Q .Q B ) ) ) )
21	6, 18, 19, 20	syl3anc 1135	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( A <Q ( [ <. x , 1o >. ] ~Q .Q B ) <-> ( ( *Q ` B ) .Q A ) <Q ( ( *Q ` B ) .Q ( [ <. x , 1o >. ] ~Q .Q B ) ) ) )
22		mulcomnqg 6481	. . . . . . 7 |- ( ( ( *Q ` B ) e. Q. /\ A e. Q. ) -> ( ( *Q ` B ) .Q A ) = ( A .Q ( *Q ` B ) ) )
23	19, 6, 22	syl2anc 391	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( ( *Q ` B ) .Q A ) = ( A .Q ( *Q ` B ) ) )
24		mulcomnqg 6481	. . . . . . . 8 |- ( ( ( *Q ` B ) e. Q. /\ ( [ <. x , 1o >. ] ~Q .Q B ) e. Q. ) -> ( ( *Q ` B ) .Q ( [ <. x , 1o >. ] ~Q .Q B ) ) = ( ( [ <. x , 1o >. ] ~Q .Q B ) .Q ( *Q ` B ) ) )
25	19, 18, 24	syl2anc 391	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( ( *Q ` B ) .Q ( [ <. x , 1o >. ] ~Q .Q B ) ) = ( ( [ <. x , 1o >. ] ~Q .Q B ) .Q ( *Q ` B ) ) )
26		mulassnqg 6482	. . . . . . . . 9 |- ( ( [ <. x , 1o >. ] ~Q e. Q. /\ B e. Q. /\ ( *Q ` B ) e. Q. ) -> ( ( [ <. x , 1o >. ] ~Q .Q B ) .Q ( *Q ` B ) ) = ( [ <. x , 1o >. ] ~Q .Q ( B .Q ( *Q ` B ) ) ) )
27	15, 16, 19, 26	syl3anc 1135	. . . . . . . 8 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( ( [ <. x , 1o >. ] ~Q .Q B ) .Q ( *Q ` B ) ) = ( [ <. x , 1o >. ] ~Q .Q ( B .Q ( *Q ` B ) ) ) )
28		recidnq 6491	. . . . . . . . . 10 |- ( B e. Q. -> ( B .Q ( *Q ` B ) ) = 1Q )
29	28	oveq2d 5528	. . . . . . . . 9 |- ( B e. Q. -> ( [ <. x , 1o >. ] ~Q .Q ( B .Q ( *Q ` B ) ) ) = ( [ <. x , 1o >. ] ~Q .Q 1Q ) )
30	16, 29	syl 14	. . . . . . . 8 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( [ <. x , 1o >. ] ~Q .Q ( B .Q ( *Q ` B ) ) ) = ( [ <. x , 1o >. ] ~Q .Q 1Q ) )
31		mulidnq 6487	. . . . . . . . 9 |- ( [ <. x , 1o >. ] ~Q e. Q. -> ( [ <. x , 1o >. ] ~Q .Q 1Q ) = [ <. x , 1o >. ] ~Q )
32	15, 31	syl 14	. . . . . . . 8 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( [ <. x , 1o >. ] ~Q .Q 1Q ) = [ <. x , 1o >. ] ~Q )
33	27, 30, 32	3eqtrd 2076	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( ( [ <. x , 1o >. ] ~Q .Q B ) .Q ( *Q ` B ) ) = [ <. x , 1o >. ] ~Q )
34	25, 33	eqtrd 2072	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( ( *Q ` B ) .Q ( [ <. x , 1o >. ] ~Q .Q B ) ) = [ <. x , 1o >. ] ~Q )
35	23, 34	breq12d 3777	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( ( ( *Q ` B ) .Q A ) <Q ( ( *Q ` B ) .Q ( [ <. x , 1o >. ] ~Q .Q B ) ) <-> ( A .Q ( *Q ` B ) ) <Q [ <. x , 1o >. ] ~Q ) )
36	21, 35	bitrd 177	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( A <Q ( [ <. x , 1o >. ] ~Q .Q B ) <-> ( A .Q ( *Q ` B ) ) <Q [ <. x , 1o >. ] ~Q ) )
37	36	biimprd 147	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( ( A .Q ( *Q ` B ) ) <Q [ <. x , 1o >. ] ~Q -> A <Q ( [ <. x , 1o >. ] ~Q .Q B ) ) )
38	37	reximdva 2421	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( E. x e. N. ( A .Q ( *Q ` B ) ) <Q [ <. x , 1o >. ] ~Q -> E. x e. N. A <Q ( [ <. x , 1o >. ] ~Q .Q B ) ) )
39	5, 38	mpd 13	1 |- ( ( A e. Q. /\ B e. Q. ) -> E. x e. N. A <Q ( [ <. x , 1o >. ] ~Q .Q B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   E.wrex 2307   <.cop 3378   class class class wbr 3764   X. cxp 4343   ` cfv 4902  (class class class)co 5512   1oc1o 5994   [cec 6104   /.cqs 6105   N.cnpi 6370   ~Q ceq 6377   Q.cnq 6378   1Qc1q 6379   .Q cmq 6381   *Qcrq 6382   <Q cltq 6383

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451

This theorem is referenced by:  prarloclemarch2  6517