Theorem nqpru 7700

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 7623	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prnminu 7637	. . . . . 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 7630	. . . . . . . . . 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 509	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> x e. Q. )
7		simprl 529	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> x e. ( 2nd ` B ) )
8		vex 2779	. . . . . . . . . . . 12 |- x e. _V
9		breq1 4062	. . . . . . . . . . . 12 |- ( l = x -> ( l <Q A <-> x <Q A ) )
10	8, 9	elab 2924	. . . . . . . . . . 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 7697	. . . . . . . . . . . 12 |- { l | l <Q A } e. _V
13		gtnqex 7698	. . . . . . . . . . . 12 |- { u | A <Q u } e. _V
14	12, 13	op1st 6255	. . . . . . . . . . 11 |- ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) = { l | l <Q A }
15	14	eleq2i 2274	. . . . . . . . . 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 491	. . . . . . . 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 1614	. . . . . . . 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 1248	. . . . . . 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 2492	. . . . . . 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 7630	. . . . . . . . 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 7695	. . . . . . . . 9 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
25		ltdfpr 7654	. . . . . . . . 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 2630	. . . 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 491	. . . . . 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 534	. . . . . 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 539	. . . . . 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 7633	. . . . . . 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 411	. . . . 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 2630	. . 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 1516   e. wcel 2178   {cab 2193   E.wrex 2487   <.cop 3646   class class class wbr 4059   ` cfv 5290   1stc1st 6247   2ndc2nd 6248   Q.cnq 7428   <Q cltq 7433   P.cnp 7439   <P cltp 7443

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-inp 7614  df-iltp 7618

This theorem is referenced by:  prplnqu  7768  caucvgprprlemmu  7843  caucvgprprlemopu  7847  caucvgprprlemexbt  7854  caucvgprprlem2  7858  suplocexprlemloc  7869