Theorem prarloclemarch 7637

Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7636 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 7611	. . . 4 |- ( B e. Q. -> ( *Q ` B ) e. Q. )
2		mulclnq 7595	. . . 4 |- ( ( A e. Q. /\ ( *Q ` B ) e. Q. ) -> ( A .Q ( *Q ` B ) ) e. Q. )
3	1, 2	sylan2 286	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q ( *Q ` B ) ) e. Q. )
4		archnqq 7636	. . 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 527	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> A e. Q. )
7		1pi 7534	. . . . . . . . . . 11 |- 1o e. N.
8		opelxpi 4757	. . . . . . . . . . 11 |- ( ( x e. N. /\ 1o e. N. ) -> <. x , 1o >. e. ( N. X. N. ) )
9	7, 8	mpan2 425	. . . . . . . . . 10 |- ( x e. N. -> <. x , 1o >. e. ( N. X. N. ) )
10		enqex 7579	. . . . . . . . . . 11 |- ~Q e. _V
11	10	ecelqsi 6757	. . . . . . . . . 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 7567	. . . . . . . . 9 |- Q. = ( ( N. X. N. ) /. ~Q )
14	12, 13	eleqtrrdi 2325	. . . . . . . 8 |- ( x e. N. -> [ <. x , 1o >. ] ~Q e. Q. )
15	14	adantl 277	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> [ <. x , 1o >. ] ~Q e. Q. )
16		simplr 529	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> B e. Q. )
17		mulclnq 7595	. . . . . . 7 |- ( ( [ <. x , 1o >. ] ~Q e. Q. /\ B e. Q. ) -> ( [ <. x , 1o >. ] ~Q .Q B ) e. Q. )
18	15, 16, 17	syl2anc 411	. . . . . 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 7620	. . . . . 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 1273	. . . . 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 7602	. . . . . . 7 |- ( ( ( *Q ` B ) e. Q. /\ A e. Q. ) -> ( ( *Q ` B ) .Q A ) = ( A .Q ( *Q ` B ) ) )
23	19, 6, 22	syl2anc 411	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( ( *Q ` B ) .Q A ) = ( A .Q ( *Q ` B ) ) )
24		mulcomnqg 7602	. . . . . . . 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 411	. . . . . . 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 7603	. . . . . . . . 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 1273	. . . . . . . 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 7612	. . . . . . . . . 10 |- ( B e. Q. -> ( B .Q ( *Q ` B ) ) = 1Q )
29	28	oveq2d 6033	. . . . . . . . 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 7608	. . . . . . . . 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 2268	. . . . . . 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 2264	. . . . . 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 4101	. . . . 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 188	. . . 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 158	. . 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 2634	. 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 104   <-> wb 105   = wceq 1397   e. wcel 2202   E.wrex 2511   <.cop 3672   class class class wbr 4088   X. cxp 4723   ` cfv 5326  (class class class)co 6017   1oc1o 6574   [cec 6699   /.cqs 6700   N.cnpi 7491   ~Q ceq 7498   Q.cnq 7499   1Qc1q 7500   .Q cmq 7502   *Qcrq 7503   <Q cltq 7504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572

This theorem is referenced by:  prarloclemarch2  7638