Theorem archrecpr 7731

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 7542	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prml 7544	. . . 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 7730	. . . . 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 7551	. . . . . 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 2599	. . . 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 2619	. 2 |- ( A e. P. -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) e. ( 1st ` A ) )
13		nnnq 7489	. . . . . 6 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
14		recclnq 7459	. . . . . 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 7618	. . . 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 2494	. 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 2167   {cab 2182   E.wrex 2476   <.cop 3625   class class class wbr 4033   ` cfv 5258   1stc1st 6196   2ndc2nd 6197   1oc1o 6467   [cec 6590   N.cnpi 7339   ~Q ceq 7346   Q.cnq 7347   *Qcrq 7351   <Q cltq 7352   P.cnp 7358   <P cltp 7362

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-inp 7533  df-iltp 7537

This theorem is referenced by:  caucvgprprlemlim  7778