Theorem prarloclemarch 7340

Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7339 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 7314	. . . 4 |- ( B e. Q. -> ( *Q ` B ) e. Q. )
2		mulclnq 7298	. . . 4 |- ( ( A e. Q. /\ ( *Q ` B ) e. Q. ) -> ( A .Q ( *Q ` B ) ) e. Q. )
3	1, 2	sylan2 284	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q ( *Q ` B ) ) e. Q. )
4		archnqq 7339	. . 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 519	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> A e. Q. )
7		1pi 7237	. . . . . . . . . . 11 |- 1o e. N.
8		opelxpi 4620	. . . . . . . . . . 11 |- ( ( x e. N. /\ 1o e. N. ) -> <. x , 1o >. e. ( N. X. N. ) )
9	7, 8	mpan2 422	. . . . . . . . . 10 |- ( x e. N. -> <. x , 1o >. e. ( N. X. N. ) )
10		enqex 7282	. . . . . . . . . . 11 |- ~Q e. _V
11	10	ecelqsi 6536	. . . . . . . . . 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 7270	. . . . . . . . 9 |- Q. = ( ( N. X. N. ) /. ~Q )
14	12, 13	eleqtrrdi 2251	. . . . . . . 8 |- ( x e. N. -> [ <. x , 1o >. ] ~Q e. Q. )
15	14	adantl 275	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> [ <. x , 1o >. ] ~Q e. Q. )
16		simplr 520	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> B e. Q. )
17		mulclnq 7298	. . . . . . 7 |- ( ( [ <. x , 1o >. ] ~Q e. Q. /\ B e. Q. ) -> ( [ <. x , 1o >. ] ~Q .Q B ) e. Q. )
18	15, 16, 17	syl2anc 409	. . . . . 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 7323	. . . . . 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 1220	. . . . 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 7305	. . . . . . 7 |- ( ( ( *Q ` B ) e. Q. /\ A e. Q. ) -> ( ( *Q ` B ) .Q A ) = ( A .Q ( *Q ` B ) ) )
23	19, 6, 22	syl2anc 409	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( ( *Q ` B ) .Q A ) = ( A .Q ( *Q ` B ) ) )
24		mulcomnqg 7305	. . . . . . . 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 409	. . . . . . 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 7306	. . . . . . . . 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 1220	. . . . . . . 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 7315	. . . . . . . . . 10 |- ( B e. Q. -> ( B .Q ( *Q ` B ) ) = 1Q )
29	28	oveq2d 5842	. . . . . . . . 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 7311	. . . . . . . . 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 2194	. . . . . . 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 2190	. . . . . 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 3980	. . . . 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 187	. . . 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 157	. . 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 2559	. 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 103   <-> wb 104   = wceq 1335   e. wcel 2128   E.wrex 2436   <.cop 3564   class class class wbr 3967   X. cxp 4586   ` cfv 5172  (class class class)co 5826   1oc1o 6358   [cec 6480   /.cqs 6481   N.cnpi 7194   ~Q ceq 7201   Q.cnq 7202   1Qc1q 7203   .Q cmq 7205   *Qcrq 7206   <Q cltq 7207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-eprel 4251  df-id 4255  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-1o 6365  df-oadd 6369  df-omul 6370  df-er 6482  df-ec 6484  df-qs 6488  df-ni 7226  df-pli 7227  df-mi 7228  df-lti 7229  df-mpq 7267  df-enq 7269  df-nqqs 7270  df-mqqs 7272  df-1nqqs 7273  df-rq 7274  df-ltnqqs 7275

This theorem is referenced by:  prarloclemarch2  7341