Theorem nqprl 7749

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 7673	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prnmaxl 7686	. . . . . 6 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 1st ` B ) ) -> E. x e. ( 1st ` B ) A <Q x )
3	1, 2	sylan 283	. . . . 5 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> E. x e. ( 1st ` B ) A <Q x )
4		elprnql 7679	. . . . . . . . . 10 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 1st ` B ) ) -> x e. Q. )
5	1, 4	sylan 283	. . . . . . . . 9 |- ( ( B e. P. /\ x e. ( 1st ` B ) ) -> x e. Q. )
6	5	ad2ant2r 509	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> x e. Q. )
7		vex 2802	. . . . . . . . . . . 12 |- x e. _V
8		breq2 4087	. . . . . . . . . . . 12 |- ( u = x -> ( A <Q u <-> A <Q x ) )
9	7, 8	elab 2947	. . . . . . . . . . 11 |- ( x e. { u | A <Q u } <-> A <Q x )
10	9	biimpri 133	. . . . . . . . . 10 |- ( A <Q x -> x e. { u | A <Q u } )
11		ltnqex 7747	. . . . . . . . . . . 12 |- { l | l <Q A } e. _V
12		gtnqex 7748	. . . . . . . . . . . 12 |- { u | A <Q u } e. _V
13	11, 12	op2nd 6299	. . . . . . . . . . 11 |- ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) = { u | A <Q u }
14	13	eleq2i 2296	. . . . . . . . . 10 |- ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> x e. { u | A <Q u } )
15	10, 14	sylibr 134	. . . . . . . . 9 |- ( A <Q x -> x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) )
16	15	ad2antll 491	. . . . . . . 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 529	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> x e. ( 1st ` B ) )
18		19.8a 1636	. . . . . . . 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 1269	. . . . . . 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 2514	. . . . . . 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 134	. . . . . 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 7679	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 1st ` B ) ) -> A e. Q. )
23	1, 22	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> A e. Q. )
24		simpl 109	. . . . . . . 8 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> B e. P. )
25		nqprlu 7745	. . . . . . . . 9 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
26		ltdfpr 7704	. . . . . . . . 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 283	. . . . . . . 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 411	. . . . . . 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 276	. . . . . 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 167	. . . . 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 2653	. . . 4 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> <. { l | l <Q A } , { u | A <Q u } >. <P B )
32	31	ex 115	. . 3 |- ( B e. P. -> ( A e. ( 1st ` B ) -> <. { l | l <Q A } , { u | A <Q u } >. <P B ) )
33	32	adantl 277	. 2 |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 1st ` B ) -> <. { l | l <Q A } , { u | A <Q u } >. <P B ) )
34	27	biimpa 296	. . . 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 184	. . . . . . . 8 |- ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> A <Q x )
36	35	biimpi 120	. . . . . . 7 |- ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) -> A <Q x )
37	36	ad2antrl 490	. . . . . 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 277	. . . . 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 534	. . . . . 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 540	. . . . . 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 7682	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 1st ` B ) ) -> ( A <Q x -> A e. ( 1st ` B ) ) )
42	1, 41	sylan 283	. . . . . 6 |- ( ( B e. P. /\ x e. ( 1st ` B ) ) -> ( A <Q x -> A e. ( 1st ` B ) ) )
43	39, 40, 42	syl2anc 411	. . . . 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 2653	. . 3 |- ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) -> A e. ( 1st ` B ) )
46	45	ex 115	. 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 129	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 104   <-> wb 105   E.wex 1538   e. wcel 2200   {cab 2215   E.wrex 2509   <.cop 3669   class class class wbr 4083   ` cfv 5318   1stc1st 6290   2ndc2nd 6291   Q.cnq 7478   <Q cltq 7483   P.cnp 7489   <P cltp 7493

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-pli 7503  df-mi 7504  df-lti 7505  df-plpq 7542  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-plqqs 7547  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-inp 7664  df-iltp 7668

This theorem is referenced by:  caucvgprlemcanl  7842  cauappcvgprlem1  7857  archrecpr  7862  caucvgprlem1  7877  caucvgprprlemml  7892  caucvgprprlemopl  7895