Theorem archrecpr 7874

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 7685	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prml 7687	. . . 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 7873	. . . . 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 7694	. . . . . 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 7632	. . . . . 6 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
14		recclnq 7602	. . . . . 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 7761	. . . 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 3670   class class class wbr 4086   ` cfv 5324   1stc1st 6296   2ndc2nd 6297   1oc1o 6570   [cec 6695   N.cnpi 7482   ~Q ceq 7489   Q.cnq 7490   *Qcrq 7494   <Q cltq 7495   P.cnp 7501   <P cltp 7505

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-inp 7676  df-iltp 7680

This theorem is referenced by:  caucvgprprlemlim  7921