Theorem nqprl 7352

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 7276	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prnmaxl 7289	. . . . . 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 7282	. . . . . . . . . 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 500	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> x e. Q. )
7		vex 2684	. . . . . . . . . . . 12 |- x e. _V
8		breq2 3928	. . . . . . . . . . . 12 |- ( u = x -> ( A <Q u <-> A <Q x ) )
9	7, 8	elab 2823	. . . . . . . . . . 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 7350	. . . . . . . . . . . 12 |- { l | l <Q A } e. _V
12		gtnqex 7351	. . . . . . . . . . . 12 |- { u | A <Q u } e. _V
13	11, 12	op2nd 6038	. . . . . . . . . . 11 |- ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) = { u | A <Q u }
14	13	eleq2i 2204	. . . . . . . . . 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 482	. . . . . . . 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 520	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> x e. ( 1st ` B ) )
18		19.8a 1569	. . . . . . . 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 1214	. . . . . . 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 2420	. . . . . . 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 7282	. . . . . . . . 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 7348	. . . . . . . . 9 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
26		ltdfpr 7307	. . . . . . . . 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 408	. . . . . . 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 2552	. . . 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 481	. . . . . 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 523	. . . . . 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 529	. . . . . 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 7285	. . . . . . 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 408	. . . . 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 2552	. . 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 1468   e. wcel 1480   {cab 2123   E.wrex 2415   <.cop 3525   class class class wbr 3924   ` cfv 5118   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   <Q cltq 7086   P.cnp 7092   <P cltp 7096

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267  df-iltp 7271

This theorem is referenced by:  caucvgprlemcanl  7445  cauappcvgprlem1  7460  archrecpr  7465  caucvgprlem1  7480  caucvgprprlemml  7495  caucvgprprlemopl  7498