Theorem nqpru 6742

Description: Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by <P. (Contributed by Jim Kingdon, 29-Nov-2020.)

Assertion

Ref	Expression
nqpru	|- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 2nd ` B ) <-> B <P <. { l | l <Q A } , { u | A <Q u } >. ) )

Distinct variable group:   A, l, u

Allowed substitution hints:   B( u, l)

Proof of Theorem nqpru

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 6665	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prnminu 6679	. . . . . 6 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 2nd ` B ) ) -> E. x e. ( 2nd ` B ) x <Q A )
3	1, 2	sylan 277	. . . . 5 |- ( ( B e. P. /\ A e. ( 2nd ` B ) ) -> E. x e. ( 2nd ` B ) x <Q A )
4		elprnqu 6672	. . . . . . . . . 10 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 2nd ` B ) ) -> x e. Q. )
5	1, 4	sylan 277	. . . . . . . . 9 |- ( ( B e. P. /\ x e. ( 2nd ` B ) ) -> x e. Q. )
6	5	ad2ant2r 492	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> x e. Q. )
7		simprl 497	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> x e. ( 2nd ` B ) )
8		vex 2604	. . . . . . . . . . . 12 |- x e. _V
9		breq1 3788	. . . . . . . . . . . 12 |- ( l = x -> ( l <Q A <-> x <Q A ) )
10	8, 9	elab 2738	. . . . . . . . . . 11 |- ( x e. { l | l <Q A } <-> x <Q A )
11	10	biimpri 131	. . . . . . . . . 10 |- ( x <Q A -> x e. { l | l <Q A } )
12		ltnqex 6739	. . . . . . . . . . . 12 |- { l | l <Q A } e. _V
13		gtnqex 6740	. . . . . . . . . . . 12 |- { u | A <Q u } e. _V
14	12, 13	op1st 5793	. . . . . . . . . . 11 |- ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) = { l | l <Q A }
15	14	eleq2i 2145	. . . . . . . . . 10 |- ( x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> x e. { l | l <Q A } )
16	11, 15	sylibr 132	. . . . . . . . 9 |- ( x <Q A -> x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) )
17	16	ad2antll 474	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) )
18		19.8a 1522	. . . . . . . 8 |- ( ( x e. Q. /\ ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) -> E. x ( x e. Q. /\ ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) )
19	6, 7, 17, 18	syl12anc 1167	. . . . . . 7 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> E. x ( x e. Q. /\ ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) )
20		df-rex 2354	. . . . . . 7 |- ( E. x e. Q. ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) <-> E. x ( x e. Q. /\ ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) )
21	19, 20	sylibr 132	. . . . . 6 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> E. x e. Q. ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) )
22		elprnqu 6672	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 2nd ` B ) ) -> A e. Q. )
23	1, 22	sylan 277	. . . . . . . 8 |- ( ( B e. P. /\ A e. ( 2nd ` B ) ) -> A e. Q. )
24		nqprlu 6737	. . . . . . . . 9 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
25		ltdfpr 6696	. . . . . . . . 9 |- ( ( B e. P. /\ <. { l | l <Q A } , { u | A <Q u } >. e. P. ) -> ( B <P <. { l | l <Q A } , { u | A <Q u } >. <-> E. x e. Q. ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) )
26	24, 25	sylan2 280	. . . . . . . 8 |- ( ( B e. P. /\ A e. Q. ) -> ( B <P <. { l | l <Q A } , { u | A <Q u } >. <-> E. x e. Q. ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) )
27	23, 26	syldan 276	. . . . . . 7 |- ( ( B e. P. /\ A e. ( 2nd ` B ) ) -> ( B <P <. { l | l <Q A } , { u | A <Q u } >. <-> E. x e. Q. ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) )
28	27	adantr 270	. . . . . 6 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> ( B <P <. { l | l <Q A } , { u | A <Q u } >. <-> E. x e. Q. ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) )
29	21, 28	mpbird 165	. . . . 5 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> B <P <. { l | l <Q A } , { u | A <Q u } >. )
30	3, 29	rexlimddv 2481	. . . 4 |- ( ( B e. P. /\ A e. ( 2nd ` B ) ) -> B <P <. { l | l <Q A } , { u | A <Q u } >. )
31	30	ex 113	. . 3 |- ( B e. P. -> ( A e. ( 2nd ` B ) -> B <P <. { l | l <Q A } , { u | A <Q u } >. ) )
32	31	adantl 271	. 2 |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 2nd ` B ) -> B <P <. { l | l <Q A } , { u | A <Q u } >. ) )
33	26	ancoms 264	. . . . 5 |- ( ( A e. Q. /\ B e. P. ) -> ( B <P <. { l | l <Q A } , { u | A <Q u } >. <-> E. x e. Q. ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) )
34	33	biimpa 290	. . . 4 |- ( ( ( A e. Q. /\ B e. P. ) /\ B <P <. { l | l <Q A } , { u | A <Q u } >. ) -> E. x e. Q. ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) )
35	15, 10	bitri 182	. . . . . . . 8 |- ( x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> x <Q A )
36	35	biimpi 118	. . . . . . 7 |- ( x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) -> x <Q A )
37	36	ad2antll 474	. . . . . 6 |- ( ( x e. Q. /\ ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) -> x <Q A )
38	37	adantl 271	. . . . 5 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ B <P <. { l | l <Q A } , { u | A <Q u } >. ) /\ ( x e. Q. /\ ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) ) -> x <Q A )
39		simpllr 500	. . . . . 6 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ B <P <. { l | l <Q A } , { u | A <Q u } >. ) /\ ( x e. Q. /\ ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) ) -> B e. P. )
40		simprrl 505	. . . . . 6 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ B <P <. { l | l <Q A } , { u | A <Q u } >. ) /\ ( x e. Q. /\ ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) ) -> x e. ( 2nd ` B ) )
41		prcunqu 6675	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 2nd ` B ) ) -> ( x <Q A -> A e. ( 2nd ` B ) ) )
42	1, 41	sylan 277	. . . . . 6 |- ( ( B e. P. /\ x e. ( 2nd ` B ) ) -> ( x <Q A -> A e. ( 2nd ` B ) ) )
43	39, 40, 42	syl2anc 403	. . . . 5 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ B <P <. { l | l <Q A } , { u | A <Q u } >. ) /\ ( x e. Q. /\ ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) ) -> ( x <Q A -> A e. ( 2nd ` B ) ) )
44	38, 43	mpd 13	. . . 4 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ B <P <. { l | l <Q A } , { u | A <Q u } >. ) /\ ( x e. Q. /\ ( x e. ( 2nd ` B ) /\ x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) ) ) ) -> A e. ( 2nd ` B ) )
45	34, 44	rexlimddv 2481	. . 3 |- ( ( ( A e. Q. /\ B e. P. ) /\ B <P <. { l | l <Q A } , { u | A <Q u } >. ) -> A e. ( 2nd ` B ) )
46	45	ex 113	. 2 |- ( ( A e. Q. /\ B e. P. ) -> ( B <P <. { l | l <Q A } , { u | A <Q u } >. -> A e. ( 2nd ` B ) ) )
47	32, 46	impbid 127	1 |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 2nd ` B ) <-> B <P <. { l | l <Q A } , { u | A <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   E.wex 1421   e. wcel 1433   {cab 2067   E.wrex 2349   <.cop 3401   class class class wbr 3785   ` cfv 4922   1stc1st 5785   2ndc2nd 5786   Q.cnq 6470   <Q cltq 6475   P.cnp 6481   <P cltp 6485

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-inp 6656  df-iltp 6660

This theorem is referenced by:  prplnqu  6810  caucvgprprlemmu  6885  caucvgprprlemopu  6889  caucvgprprlemexbt  6896  caucvgprprlem2  6900