Theorem archrecpr 7851

Description: Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)

Assertion

Ref	Expression
archrecpr	|- ( A e. P. -> E. j e. N. <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P A )

Distinct variable groups:   A, j   j, l, u

Allowed substitution hints:   A( u, l)

Proof of Theorem archrecpr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7662	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prml 7664	. . . 4 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 1st ` A ) )
3	1, 2	syl 14	. . 3 |- ( A e. P. -> E. x e. Q. x e. ( 1st ` A ) )
4		archrecnq 7850	. . . . 5 |- ( x e. Q. -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x )
5	4	ad2antrl 490	. . . 4 |- ( ( A e. P. /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x )
6	1	ad2antrr 488	. . . . . 6 |- ( ( ( A e. P. /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ j e. N. ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
7		simplrr 536	. . . . . 6 |- ( ( ( A e. P. /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ j e. N. ) -> x e. ( 1st ` A ) )
8		prcdnql 7671	. . . . . 6 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 1st ` A ) ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. ( 1st ` A ) ) )
9	6, 7, 8	syl2anc 411	. . . . 5 |- ( ( ( A e. P. /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ j e. N. ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. ( 1st ` A ) ) )
10	9	reximdva 2632	. . . 4 |- ( ( A e. P. /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) -> ( E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) e. ( 1st ` A ) ) )
11	5, 10	mpd 13	. . 3 |- ( ( A e. P. /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) e. ( 1st ` A ) )
12	3, 11	rexlimddv 2653	. 2 |- ( A e. P. -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) e. ( 1st ` A ) )
13		nnnq 7609	. . . . . 6 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
14		recclnq 7579	. . . . . 6 |- ( [ <. j , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
15	13, 14	syl 14	. . . . 5 |- ( j e. N. -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
16	15	adantl 277	. . . 4 |- ( ( A e. P. /\ j e. N. ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
17		simpl 109	. . . 4 |- ( ( A e. P. /\ j e. N. ) -> A e. P. )
18		nqprl 7738	. . . 4 |- ( ( ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. /\ A e. P. ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) e. ( 1st ` A ) <-> <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P A ) )
19	16, 17, 18	syl2anc 411	. . 3 |- ( ( A e. P. /\ j e. N. ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) e. ( 1st ` A ) <-> <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P A ) )
20	19	rexbidva 2527	. 2 |- ( A e. P. -> ( E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) e. ( 1st ` A ) <-> E. j e. N. <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P A ) )
21	12, 20	mpbid 147	1 |- ( A e. P. -> E. j e. N. <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   {cab 2215   E.wrex 2509   <.cop 3669   class class class wbr 4083   ` cfv 5318   1stc1st 6284   2ndc2nd 6285   1oc1o 6555   [cec 6678   N.cnpi 7459   ~Q ceq 7466   Q.cnq 7467   *Qcrq 7471   <Q cltq 7472   P.cnp 7478   <P cltp 7482

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-mpq 7532  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-mqqs 7537  df-1nqqs 7538  df-rq 7539  df-ltnqqs 7540  df-inp 7653  df-iltp 7657

This theorem is referenced by:  caucvgprprlemlim  7898