Theorem nqpru 7514

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 7437	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prnminu 7451	. . . . . 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 7444	. . . . . . . . . 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 506	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> x e. Q. )
7		simprl 526	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 2nd ` B ) ) /\ ( x e. ( 2nd ` B ) /\ x <Q A ) ) -> x e. ( 2nd ` B ) )
8		vex 2733	. . . . . . . . . . . 12 |- x e. _V
9		breq1 3992	. . . . . . . . . . . 12 |- ( l = x -> ( l <Q A <-> x <Q A ) )
10	8, 9	elab 2874	. . . . . . . . . . 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 7511	. . . . . . . . . . . 12 |- { l | l <Q A } e. _V
13		gtnqex 7512	. . . . . . . . . . . 12 |- { u | A <Q u } e. _V
14	12, 13	op1st 6125	. . . . . . . . . . 11 |- ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) = { l | l <Q A }
15	14	eleq2i 2237	. . . . . . . . . 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 488	. . . . . . . 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 1583	. . . . . . . 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 1231	. . . . . . 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 2454	. . . . . . 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 7444	. . . . . . . . 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 7509	. . . . . . . . 9 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
25		ltdfpr 7468	. . . . . . . . 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 2592	. . . 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 488	. . . . . 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 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 } >. ) ) ) ) -> B e. P. )
40		simprrl 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 } >. ) ) ) ) -> x e. ( 2nd ` B ) )
41		prcunqu 7447	. . . . . . 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 2592	. . 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 1485   e. wcel 2141   {cab 2156   E.wrex 2449   <.cop 3586   class class class wbr 3989   ` cfv 5198   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   <Q cltq 7247   P.cnp 7253   <P cltp 7257

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-iltp 7432

This theorem is referenced by:  prplnqu  7582  caucvgprprlemmu  7657  caucvgprprlemopu  7661  caucvgprprlemexbt  7668  caucvgprprlem2  7672  suplocexprlemloc  7683