Theorem nqpru 7484

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 7407	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prnminu 7421	. . . . . 6 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 2nd ` B ) ) -> E. x e. ( 2nd ` B ) x <Q A )
3	1, 2	sylan 281	. . . . 5 |- ( ( B e. P. /\ A e. ( 2nd ` B ) ) -> E. x e. ( 2nd ` B ) x <Q A )
4		elprnqu 7414	. . . . . . . . . 10 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 2nd ` B ) ) -> x e. Q. )
5	1, 4	sylan 281	. . . . . . . . 9 |- ( ( B e. P. /\ x e. ( 2nd ` B ) ) -> x e. Q. )
6	5	ad2ant2r 501	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> x e. Q. )
7		simprl 521	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> x e. ( 2nd ` B ) )
8		vex 2724	. . . . . . . . . . . 12 |- x e. _V
9		breq1 3979	. . . . . . . . . . . 12 |- ( l = x -> ( l <Q A <-> x <Q A ) )
10	8, 9	elab 2865	. . . . . . . . . . 11 |- ( x e. { l | l <Q A } <-> x <Q A )
11	10	biimpri 132	. . . . . . . . . 10 |- ( x <Q A -> x e. { l | l <Q A } )
12		ltnqex 7481	. . . . . . . . . . . 12 |- { l | l <Q A } e. _V
13		gtnqex 7482	. . . . . . . . . . . 12 |- { u | A <Q u } e. _V
14	12, 13	op1st 6106	. . . . . . . . . . 11 |- ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) = { l | l <Q A }
15	14	eleq2i 2231	. . . . . . . . . 10 |- ( x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> x e. { l | l <Q A } )
16	11, 15	sylibr 133	. . . . . . . . 9 |- ( x <Q A -> x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) )
17	16	ad2antll 483	. . . . . . . 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 1577	. . . . . . . 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 1225	. . . . . . 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 2448	. . . . . . 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 133	. . . . . 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 7414	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 2nd ` B ) ) -> A e. Q. )
23	1, 22	sylan 281	. . . . . . . 8 |- ( ( B e. P. /\ A e. ( 2nd ` B ) ) -> A e. Q. )
24		nqprlu 7479	. . . . . . . . 9 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
25		ltdfpr 7438	. . . . . . . . 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 284	. . . . . . . 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 280	. . . . . . 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 274	. . . . . 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 166	. . . . 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 2586	. . . 4 |- ( ( B e. P. /\ A e. ( 2nd ` B ) ) -> B <P <. { l | l <Q A } , { u | A <Q u } >. )
31	30	ex 114	. . 3 |- ( B e. P. -> ( A e. ( 2nd ` B ) -> B <P <. { l | l <Q A } , { u | A <Q u } >. ) )
32	31	adantl 275	. 2 |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 2nd ` B ) -> B <P <. { l | l <Q A } , { u | A <Q u } >. ) )
33	26	ancoms 266	. . . . 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 294	. . . 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 183	. . . . . . . 8 |- ( x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> x <Q A )
36	35	biimpi 119	. . . . . . 7 |- ( x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) -> x <Q A )
37	36	ad2antll 483	. . . . . 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 275	. . . . 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 524	. . . . . 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 529	. . . . . 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 7417	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 2nd ` B ) ) -> ( x <Q A -> A e. ( 2nd ` B ) ) )
42	1, 41	sylan 281	. . . . . 6 |- ( ( B e. P. /\ x e. ( 2nd ` B ) ) -> ( x <Q A -> A e. ( 2nd ` B ) ) )
43	39, 40, 42	syl2anc 409	. . . . 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 2586	. . 3 |- ( ( ( A e. Q. /\ B e. P. ) /\ B <P <. { l | l <Q A } , { u | A <Q u } >. ) -> A e. ( 2nd ` B ) )
46	45	ex 114	. 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 128	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 103   <-> wb 104   E.wex 1479   e. wcel 2135   {cab 2150   E.wrex 2443   <.cop 3573   class class class wbr 3976   ` cfv 5182   1stc1st 6098   2ndc2nd 6099   Q.cnq 7212   <Q cltq 7217   P.cnp 7223   <P cltp 7227

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-eprel 4261  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-1o 6375  df-oadd 6379  df-omul 6380  df-er 6492  df-ec 6494  df-qs 6498  df-ni 7236  df-pli 7237  df-mi 7238  df-lti 7239  df-plpq 7276  df-mpq 7277  df-enq 7279  df-nqqs 7280  df-plqqs 7281  df-mqqs 7282  df-1nqqs 7283  df-rq 7284  df-ltnqqs 7285  df-inp 7398  df-iltp 7402

This theorem is referenced by:  prplnqu  7552  caucvgprprlemmu  7627  caucvgprprlemopu  7631  caucvgprprlemexbt  7638  caucvgprprlem2  7642  suplocexprlemloc  7653