Theorem archrecpr 7677

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 7488	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prml 7490	. . . 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 7676	. . . . 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 7497	. . . . . 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 2589	. . . 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 2609	. 2 |- ( A e. P. -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) e. ( 1st ` A ) )
13		nnnq 7435	. . . . . 6 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
14		recclnq 7405	. . . . . 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 7564	. . . 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 2484	. 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 2158   {cab 2173   E.wrex 2466   <.cop 3607   class class class wbr 4015   ` cfv 5228   1stc1st 6153   2ndc2nd 6154   1oc1o 6424   [cec 6547   N.cnpi 7285   ~Q ceq 7292   Q.cnq 7293   *Qcrq 7297   <Q cltq 7298   P.cnp 7304   <P cltp 7308

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-1o 6431  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-pli 7318  df-mi 7319  df-lti 7320  df-plpq 7357  df-mpq 7358  df-enq 7360  df-nqqs 7361  df-plqqs 7362  df-mqqs 7363  df-1nqqs 7364  df-rq 7365  df-ltnqqs 7366  df-inp 7479  df-iltp 7483

This theorem is referenced by:  caucvgprprlemlim  7724