Theorem archrecpr 7975

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 7786	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prml 7788	. . . 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 7974	. . . . 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 538	. . . . . 6 |- ( ( ( A e. P. /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ j e. N. ) -> x e. ( 1st ` A ) )
8		prcdnql 7795	. . . . . 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 2644	. . . 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 2665	. 2 |- ( A e. P. -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) e. ( 1st ` A ) )
13		nnnq 7733	. . . . . 6 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
14		recclnq 7703	. . . . . 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 7862	. . . 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 2539	. 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 2203   {cab 2218   E.wrex 2521   <.cop 3691   class class class wbr 4108   ` cfv 5351   1stc1st 6331   2ndc2nd 6332   1oc1o 6639   [cec 6764   N.cnpi 7583   ~Q ceq 7590   Q.cnq 7591   *Qcrq 7595   <Q cltq 7596   P.cnp 7602   <P cltp 7606

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7615  df-pli 7616  df-mi 7617  df-lti 7618  df-plpq 7655  df-mpq 7656  df-enq 7658  df-nqqs 7659  df-plqqs 7660  df-mqqs 7661  df-1nqqs 7662  df-rq 7663  df-ltnqqs 7664  df-inp 7777  df-iltp 7781

This theorem is referenced by:  caucvgprprlemlim  8022