Theorem nqprl 7635

Description: Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by <P. (Contributed by Jim Kingdon, 8-Jul-2020.)

Assertion

Ref	Expression
nqprl	|- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 1st ` B ) <-> <. { l | l <Q A } , { u | A <Q u } >. <P B ) )

Distinct variable group:   A, l, u

Allowed substitution hints:   B( u, l)

Proof of Theorem nqprl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7559	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prnmaxl 7572	. . . . . 6 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 1st ` B ) ) -> E. x e. ( 1st ` B ) A <Q x )
3	1, 2	sylan 283	. . . . 5 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> E. x e. ( 1st ` B ) A <Q x )
4		elprnql 7565	. . . . . . . . . 10 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 1st ` B ) ) -> x e. Q. )
5	1, 4	sylan 283	. . . . . . . . 9 |- ( ( B e. P. /\ x e. ( 1st ` B ) ) -> x e. Q. )
6	5	ad2ant2r 509	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> x e. Q. )
7		vex 2766	. . . . . . . . . . . 12 |- x e. _V
8		breq2 4038	. . . . . . . . . . . 12 |- ( u = x -> ( A <Q u <-> A <Q x ) )
9	7, 8	elab 2908	. . . . . . . . . . 11 |- ( x e. { u | A <Q u } <-> A <Q x )
10	9	biimpri 133	. . . . . . . . . 10 |- ( A <Q x -> x e. { u | A <Q u } )
11		ltnqex 7633	. . . . . . . . . . . 12 |- { l | l <Q A } e. _V
12		gtnqex 7634	. . . . . . . . . . . 12 |- { u | A <Q u } e. _V
13	11, 12	op2nd 6214	. . . . . . . . . . 11 |- ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) = { u | A <Q u }
14	13	eleq2i 2263	. . . . . . . . . 10 |- ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> x e. { u | A <Q u } )
15	10, 14	sylibr 134	. . . . . . . . 9 |- ( A <Q x -> x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) )
16	15	ad2antll 491	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) )
17		simprl 529	. . . . . . . 8 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> x e. ( 1st ` B ) )
18		19.8a 1604	. . . . . . . 8 |- ( ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) -> E. x ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
19	6, 16, 17, 18	syl12anc 1247	. . . . . . 7 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> E. x ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
20		df-rex 2481	. . . . . . 7 |- ( E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) <-> E. x ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
21	19, 20	sylibr 134	. . . . . 6 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) )
22		elprnql 7565	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ A e. ( 1st ` B ) ) -> A e. Q. )
23	1, 22	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> A e. Q. )
24		simpl 109	. . . . . . . 8 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> B e. P. )
25		nqprlu 7631	. . . . . . . . 9 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
26		ltdfpr 7590	. . . . . . . . 9 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ B e. P. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. <P B <-> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
27	25, 26	sylan 283	. . . . . . . 8 |- ( ( A e. Q. /\ B e. P. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. <P B <-> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
28	23, 24, 27	syl2anc 411	. . . . . . 7 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> ( <. { l | l <Q A } , { u | A <Q u } >. <P B <-> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
29	28	adantr 276	. . . . . 6 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> ( <. { l | l <Q A } , { u | A <Q u } >. <P B <-> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) )
30	21, 29	mpbird 167	. . . . 5 |- ( ( ( B e. P. /\ A e. ( 1st ` B ) ) /\ ( x e. ( 1st ` B ) /\ A <Q x ) ) -> <. { l | l <Q A } , { u | A <Q u } >. <P B )
31	3, 30	rexlimddv 2619	. . . 4 |- ( ( B e. P. /\ A e. ( 1st ` B ) ) -> <. { l | l <Q A } , { u | A <Q u } >. <P B )
32	31	ex 115	. . 3 |- ( B e. P. -> ( A e. ( 1st ` B ) -> <. { l | l <Q A } , { u | A <Q u } >. <P B ) )
33	32	adantl 277	. 2 |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 1st ` B ) -> <. { l | l <Q A } , { u | A <Q u } >. <P B ) )
34	27	biimpa 296	. . . 4 |- ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) -> E. x e. Q. ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) )
35	14, 9	bitri 184	. . . . . . . 8 |- ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> A <Q x )
36	35	biimpi 120	. . . . . . 7 |- ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) -> A <Q x )
37	36	ad2antrl 490	. . . . . 6 |- ( ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) -> A <Q x )
38	37	adantl 277	. . . . 5 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) /\ ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) ) -> A <Q x )
39		simpllr 534	. . . . . 6 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) /\ ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) ) -> B e. P. )
40		simprrr 540	. . . . . 6 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) /\ ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) ) -> x e. ( 1st ` B ) )
41		prcdnql 7568	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 1st ` B ) ) -> ( A <Q x -> A e. ( 1st ` B ) ) )
42	1, 41	sylan 283	. . . . . 6 |- ( ( B e. P. /\ x e. ( 1st ` B ) ) -> ( A <Q x -> A e. ( 1st ` B ) ) )
43	39, 40, 42	syl2anc 411	. . . . 5 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) /\ ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) ) -> ( A <Q x -> A e. ( 1st ` B ) ) )
44	38, 43	mpd 13	. . . 4 |- ( ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) /\ ( x e. Q. /\ ( x e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ x e. ( 1st ` B ) ) ) ) -> A e. ( 1st ` B ) )
45	34, 44	rexlimddv 2619	. . 3 |- ( ( ( A e. Q. /\ B e. P. ) /\ <. { l | l <Q A } , { u | A <Q u } >. <P B ) -> A e. ( 1st ` B ) )
46	45	ex 115	. 2 |- ( ( A e. Q. /\ B e. P. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. <P B -> A e. ( 1st ` B ) ) )
47	33, 46	impbid 129	1 |- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 1st ` B ) <-> <. { l | l <Q A } , { u | A <Q u } >. <P B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1506   e. wcel 2167   {cab 2182   E.wrex 2476   <.cop 3626   class class class wbr 4034   ` cfv 5259   1stc1st 6205   2ndc2nd 6206   Q.cnq 7364   <Q cltq 7369   P.cnp 7375   <P cltp 7379

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-inp 7550  df-iltp 7554

This theorem is referenced by:  caucvgprlemcanl  7728  cauappcvgprlem1  7743  archrecpr  7748  caucvgprlem1  7763  caucvgprprlemml  7778  caucvgprprlemopl  7781