Theorem prarloclemarch 6670

Description: A version of the Archimedean property. This variation is "stronger" than archnqq 6669 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 6644	. . . 4 |- ( B e. Q. -> ( *Q ` B ) e. Q. )
2		mulclnq 6628	. . . 4 |- ( ( A e. Q. /\ ( *Q ` B ) e. Q. ) -> ( A .Q ( *Q ` B ) ) e. Q. )
3	1, 2	sylan2 280	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q ( *Q ` B ) ) e. Q. )
4		archnqq 6669	. . 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 496	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> A e. Q. )
7		1pi 6567	. . . . . . . . . . 11 |- 1o e. N.
8		opelxpi 4402	. . . . . . . . . . 11 |- ( ( x e. N. /\ 1o e. N. ) -> <. x , 1o >. e. ( N. X. N. ) )
9	7, 8	mpan2 416	. . . . . . . . . 10 |- ( x e. N. -> <. x , 1o >. e. ( N. X. N. ) )
10		enqex 6612	. . . . . . . . . . 11 |- ~Q e. _V
11	10	ecelqsi 6226	. . . . . . . . . 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 6600	. . . . . . . . 9 |- Q. = ( ( N. X. N. ) /. ~Q )
14	12, 13	syl6eleqr 2173	. . . . . . . 8 |- ( x e. N. -> [ <. x , 1o >. ] ~Q e. Q. )
15	14	adantl 271	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> [ <. x , 1o >. ] ~Q e. Q. )
16		simplr 497	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> B e. Q. )
17		mulclnq 6628	. . . . . . 7 |- ( ( [ <. x , 1o >. ] ~Q e. Q. /\ B e. Q. ) -> ( [ <. x , 1o >. ] ~Q .Q B ) e. Q. )
18	15, 16, 17	syl2anc 403	. . . . . 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 6653	. . . . . 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 1170	. . . . 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 6635	. . . . . . 7 |- ( ( ( *Q ` B ) e. Q. /\ A e. Q. ) -> ( ( *Q ` B ) .Q A ) = ( A .Q ( *Q ` B ) ) )
23	19, 6, 22	syl2anc 403	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( ( *Q ` B ) .Q A ) = ( A .Q ( *Q ` B ) ) )
24		mulcomnqg 6635	. . . . . . . 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 403	. . . . . . 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 6636	. . . . . . . . 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 1170	. . . . . . . 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 6645	. . . . . . . . . 10 |- ( B e. Q. -> ( B .Q ( *Q ` B ) ) = 1Q )
29	28	oveq2d 5559	. . . . . . . . 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 6641	. . . . . . . . 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 2118	. . . . . . 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 2114	. . . . . 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 3806	. . . . 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 186	. . . 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 156	. . 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 2464	. 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 102   <-> wb 103   = wceq 1285   e. wcel 1434   E.wrex 2350   <.cop 3409   class class class wbr 3793   X. cxp 4369   ` cfv 4932  (class class class)co 5543   1oc1o 6058   [cec 6170   /.cqs 6171   N.cnpi 6524   ~Q ceq 6531   Q.cnq 6532   1Qc1q 6533   .Q cmq 6535   *Qcrq 6536   <Q cltq 6537

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605

This theorem is referenced by:  prarloclemarch2  6671