Theorem nqprl 7611

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 7535	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prnmaxl 7548	. . . . . 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 7541	. . . . . . . . . 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 2763	. . . . . . . . . . . 12 |- x e. _V
8		breq2 4033	. . . . . . . . . . . 12 |- ( u = x -> ( A <Q u <-> A <Q x ) )
9	7, 8	elab 2904	. . . . . . . . . . 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 7609	. . . . . . . . . . . 12 |- { l | l <Q A } e. _V
12		gtnqex 7610	. . . . . . . . . . . 12 |- { u | A <Q u } e. _V
13	11, 12	op2nd 6200	. . . . . . . . . . 11 |- ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) = { u | A <Q u }
14	13	eleq2i 2260	. . . . . . . . . 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 1601	. . . . . . . 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 2478	. . . . . . 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 7541	. . . . . . . . 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 7607	. . . . . . . . 9 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
26		ltdfpr 7566	. . . . . . . . 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 2616	. . . 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 7544	. . . . . . 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 2616	. . 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 1503   e. wcel 2164   {cab 2179   E.wrex 2473   <.cop 3621   class class class wbr 4029   ` cfv 5254   1stc1st 6191   2ndc2nd 6192   Q.cnq 7340   <Q cltq 7345   P.cnp 7351   <P cltp 7355

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-inp 7526  df-iltp 7530

This theorem is referenced by:  caucvgprlemcanl  7704  cauappcvgprlem1  7719  archrecpr  7724  caucvgprlem1  7739  caucvgprprlemml  7754  caucvgprprlemopl  7757