Theorem nqpru 7909

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 7832	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prnminu 7846	. . . . . 6 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 2nd ` B ) ) -> E. x e. ( 2nd ` B ) x <Q A )
3	1, 2	sylan 283	. . . . 5 |- ( ( B e. P. /\ A e. ( 2nd ` B ) ) -> E. x e. ( 2nd ` B ) x <Q A )
4		elprnqu 7839	. . . . . . . . . 10 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 2nd ` B ) ) -> x e. Q. )
5	1, 4	sylan 283	. . . . . . . . 9 |- ( ( B e. P. /\ x e. ( 2nd ` B ) ) -> x e. Q. )
6	5	ad2ant2r 513	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> x e. Q. )
7		simprl 535	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> x e. ( 2nd ` B ) )
8		vex 2824	. . . . . . . . . . . 12 |- x e. _V
9		breq1 4128	. . . . . . . . . . . 12 |- ( l = x -> ( l <Q A <-> x <Q A ) )
10	8, 9	elab 2970	. . . . . . . . . . 11 |- ( x e. { l | l <Q A } <-> x <Q A )
11	10	biimpri 133	. . . . . . . . . 10 |- ( x <Q A -> x e. { l | l <Q A } )
12		ltnqex 7906	. . . . . . . . . . . 12 |- { l | l <Q A } e. _V
13		gtnqex 7907	. . . . . . . . . . . 12 |- { u | A <Q u } e. _V
14	12, 13	op1st 6370	. . . . . . . . . . 11 |- ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) = { l | l <Q A }
15	14	eleq2i 2305	. . . . . . . . . 10 |- ( x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> x e. { l | l <Q A } )
16	11, 15	sylibr 134	. . . . . . . . 9 |- ( x <Q A -> x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) )
17	16	ad2antll 495	. . . . . . . 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 1643	. . . . . . . 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 1276	. . . . . . 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 2534	. . . . . . 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 134	. . . . . 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 7839	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 2nd ` B ) ) -> A e. Q. )
23	1, 22	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ A e. ( 2nd ` B ) ) -> A e. Q. )
24		nqprlu 7904	. . . . . . . . 9 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
25		ltdfpr 7863	. . . . . . . . 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 286	. . . . . . . 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 282	. . . . . . 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 276	. . . . . 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 167	. . . . 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 2673	. . . 4 |- ( ( B e. P. /\ A e. ( 2nd ` B ) ) -> B <P <. { l | l <Q A } , { u | A <Q u } >. )
31	30	ex 115	. . 3 |- ( B e. P. -> ( A e. ( 2nd ` B ) -> B <P <. { l | l <Q A } , { u | A <Q u } >. ) )
32	31	adantl 277	. 2 |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 2nd ` B ) -> B <P <. { l | l <Q A } , { u | A <Q u } >. ) )
33	26	ancoms 268	. . . . 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 296	. . . 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 184	. . . . . . . 8 |- ( x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> x <Q A )
36	35	biimpi 120	. . . . . . 7 |- ( x e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) -> x <Q A )
37	36	ad2antll 495	. . . . . 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 277	. . . . 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 540	. . . . . 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 545	. . . . . 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 7842	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 2nd ` B ) ) -> ( x <Q A -> A e. ( 2nd ` B ) ) )
42	1, 41	sylan 283	. . . . . 6 |- ( ( B e. P. /\ x e. ( 2nd ` B ) ) -> ( x <Q A -> A e. ( 2nd ` B ) ) )
43	39, 40, 42	syl2anc 415	. . . . 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 2673	. . 3 |- ( ( ( A e. Q. /\ B e. P. ) /\ B <P <. { l | l <Q A } , { u | A <Q u } >. ) -> A e. ( 2nd ` B ) )
46	45	ex 115	. 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 129	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 104   <-> wb 105   E.wex 1545   e. wcel 2209   {cab 2224   E.wrex 2529   <.cop 3708   class class class wbr 4125   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   <Q cltq 7642   P.cnp 7648   <P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-inp 7823  df-iltp 7827

This theorem is referenced by:  prplnqu  7977  caucvgprprlemmu  8052  caucvgprprlemopu  8056  caucvgprprlemexbt  8063  caucvgprprlem2  8067  suplocexprlemloc  8078