Theorem prarloclemarch 7791

Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7790 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 7765	. . . 4 |- ( B e. Q. -> ( *Q ` B ) e. Q. )
2		mulclnq 7749	. . . 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 7790	. . 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 531	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> A e. Q. )
7		1pi 7688	. . . . . . . . . . 11 |- 1o e. N.
8		opelxpi 4808	. . . . . . . . . . 11 |- ( ( x e. N. /\ 1o e. N. ) -> <. x , 1o >. e. ( N. X. N. ) )
9	7, 8	mpan2 429	. . . . . . . . . 10 |- ( x e. N. -> <. x , 1o >. e. ( N. X. N. ) )
10		enqex 7733	. . . . . . . . . . 11 |- ~Q e. _V
11	10	ecelqsi 6867	. . . . . . . . . 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 7721	. . . . . . . . 9 |- Q. = ( ( N. X. N. ) /. ~Q )
14	12, 13	eleqtrrdi 2332	. . . . . . . 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 533	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> B e. Q. )
17		mulclnq 7749	. . . . . . 7 |- ( ( [ <. x , 1o >. ] ~Q e. Q. /\ B e. Q. ) -> ( [ <. x , 1o >. ] ~Q .Q B ) e. Q. )
18	15, 16, 17	syl2anc 415	. . . . . 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 7774	. . . . . 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 1278	. . . . 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 7756	. . . . . . 7 |- ( ( ( *Q ` B ) e. Q. /\ A e. Q. ) -> ( ( *Q ` B ) .Q A ) = ( A .Q ( *Q ` B ) ) )
23	19, 6, 22	syl2anc 415	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ x e. N. ) -> ( ( *Q ` B ) .Q A ) = ( A .Q ( *Q ` B ) ) )
24		mulcomnqg 7756	. . . . . . . 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 415	. . . . . . 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 7757	. . . . . . . . 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 1278	. . . . . . . 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 7766	. . . . . . . . . 10 |- ( B e. Q. -> ( B .Q ( *Q ` B ) ) = 1Q )
29	28	oveq2d 6104	. . . . . . . . 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 7762	. . . . . . . . 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 2275	. . . . . . 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 2271	. . . . . 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 4144	. . . . 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 2652	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   <.cop 3712   class class class wbr 4131   X. cxp 4774   ` cfv 5380  (class class class)co 6088   1oc1o 6684   [cec 6809   /.cqs 6810   N.cnpi 7645   ~Q ceq 7652   Q.cnq 7653   1Qc1q 7654   .Q cmq 7656   *Qcrq 7657   <Q cltq 7658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726

This theorem is used by:  prarloclemarch2  7792