Theorem nqprl 7492

Description: Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by <P. (Contributed by Jim Kingdon, 8-Jul-2020.)

Assertion

Ref	Expression
nqprl	|- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 1st ` B ) <-> <. { l | l <Q A } , { u | A <Q u } >. <P B ) )

Distinct variable group:   A, l, u

Allowed substitution hints:   B( u, l)

Proof of Theorem nqprl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7416	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prnmaxl 7429	. . . . . 6 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 1st ` B ) ) -> E. x e. ( 1st ` B ) A <Q x )
3	1, 2	sylan 281	. . . . 5 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> E. x e. ( 1st ` B ) A <Q x )
4		elprnql 7422	. . . . . . . . . 10 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 1st ` B ) ) -> x e. Q. )
5	1, 4	sylan 281	. . . . . . . . 9 |- ( ( B e. P. /\ x e. ( 1st ` B ) ) -> x e. Q. )
6	5	ad2ant2r 501	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> x e. Q. )
7		vex 2729	. . . . . . . . . . . 12 |- x e. _V
8		breq2 3986	. . . . . . . . . . . 12 |- ( u = x -> ( A <Q u <-> A <Q x ) )
9	7, 8	elab 2870	. . . . . . . . . . 11 |- ( x e. { u | A <Q u } <-> A <Q x )
10	9	biimpri 132	. . . . . . . . . 10 |- ( A <Q x -> x e. { u | A <Q u } )
11		ltnqex 7490	. . . . . . . . . . . 12 |- { l | l <Q A } e. _V
12		gtnqex 7491	. . . . . . . . . . . 12 |- { u | A <Q u } e. _V
13	11, 12	op2nd 6115	. . . . . . . . . . 11 |- ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) = { u | A <Q u }
14	13	eleq2i 2233	. . . . . . . . . 10 |- ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> x e. { u | A <Q u } )
15	10, 14	sylibr 133	. . . . . . . . 9 |- ( A <Q x -> x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) )
16	15	ad2antll 483	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) )
17		simprl 521	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> x e. ( 1st ` B ) )
18		19.8a 1578	. . . . . . . 8 |- ( ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) -> E. x ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
19	6, 16, 17, 18	syl12anc 1226	. . . . . . 7 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> E. x ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
20		df-rex 2450	. . . . . . 7 |- ( E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) <-> E. x ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
21	19, 20	sylibr 133	. . . . . 6 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) )
22		elprnql 7422	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 1st ` B ) ) -> A e. Q. )
23	1, 22	sylan 281	. . . . . . . 8 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> A e. Q. )
24		simpl 108	. . . . . . . 8 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> B e. P. )
25		nqprlu 7488	. . . . . . . . 9 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
26		ltdfpr 7447	. . . . . . . . 9 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ B e. P. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. <P B <-> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
27	25, 26	sylan 281	. . . . . . . 8 |- ( ( A e. Q. /\ B e. P. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. <P B <-> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
28	23, 24, 27	syl2anc 409	. . . . . . 7 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> ( <. { l | l <Q A } , { u | A <Q u } >. <P B <-> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
29	28	adantr 274	. . . . . 6 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> ( <. { l | l <Q A } , { u | A <Q u } >. <P B <-> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
30	21, 29	mpbird 166	. . . . 5 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> <. { l | l <Q A } , { u | A <Q u } >. <P B )
31	3, 30	rexlimddv 2588	. . . 4 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> <. { l | l <Q A } , { u | A <Q u } >. <P B )
32	31	ex 114	. . 3 |- ( B e. P. -> ( A e. ( 1st ` B ) -> <. { l | l <Q A } , { u | A <Q u } >. <P B ) )
33	32	adantl 275	. 2 |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 1st ` B ) -> <. { l | l <Q A } , { u | A <Q u } >. <P B ) )
34	27	biimpa 294	. . . 4 |- ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) -> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) )
35	14, 9	bitri 183	. . . . . . . 8 |- ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> A <Q x )
36	35	biimpi 119	. . . . . . 7 |- ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) -> A <Q x )
37	36	ad2antrl 482	. . . . . 6 |- ( ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) -> A <Q x )
38	37	adantl 275	. . . . 5 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) /\ ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) ) -> A <Q x )
39		simpllr 524	. . . . . 6 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) /\ ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) ) -> B e. P. )
40		simprrr 530	. . . . . 6 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) /\ ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) ) -> x e. ( 1st ` B ) )
41		prcdnql 7425	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 1st ` B ) ) -> ( A <Q x -> A e. ( 1st ` B ) ) )
42	1, 41	sylan 281	. . . . . 6 |- ( ( B e. P. /\ x e. ( 1st ` B ) ) -> ( A <Q x -> A e. ( 1st ` B ) ) )
43	39, 40, 42	syl2anc 409	. . . . 5 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) /\ ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) ) -> ( A <Q x -> A e. ( 1st ` B ) ) )
44	38, 43	mpd 13	. . . 4 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) /\ ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) ) -> A e. ( 1st ` B ) )
45	34, 44	rexlimddv 2588	. . 3 |- ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) -> A e. ( 1st ` B ) )
46	45	ex 114	. 2 |- ( ( A e. Q. /\ B e. P. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. <P B -> A e. ( 1st ` B ) ) )
47	33, 46	impbid 128	1 |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 1st ` B ) <-> <. { l | l <Q A } , { u | A <Q u } >. <P B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E.wex 1480   e. wcel 2136   {cab 2151   E.wrex 2445   <.cop 3579   class class class wbr 3982   ` cfv 5188   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   <Q cltq 7226   P.cnp 7232   <P cltp 7236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407  df-iltp 7411

This theorem is referenced by:  caucvgprlemcanl  7585  cauappcvgprlem1  7600  archrecpr  7605  caucvgprlem1  7620  caucvgprprlemml  7635  caucvgprprlemopl  7638