Theorem addnqprlemfu 7763

Description: Lemma for addnqpr 7764. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)

Assertion

Ref	Expression
addnqprlemfu	|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )

Distinct variable groups:   A, l, u   B, l, u

Proof of Theorem addnqprlemfu

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		addnqprlemrl 7760	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )
2		ltsonq 7601	. . . . . . . . 9 |- <Q Or Q.
3		addclnq 7578	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
4		sonr 4409	. . . . . . . . 9 |- ( ( <Q Or Q. /\ ( A +Q B ) e. Q. ) -> -. ( A +Q B ) <Q ( A +Q B ) )
5	2, 3, 4	sylancr 414	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) <Q ( A +Q B ) )
6		ltrelnq 7568	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
7	6	brel 4773	. . . . . . . . . . 11 |- ( ( A +Q B ) <Q ( A +Q B ) -> ( ( A +Q B ) e. Q. /\ ( A +Q B ) e. Q. ) )
8	7	simpld 112	. . . . . . . . . 10 |- ( ( A +Q B ) <Q ( A +Q B ) -> ( A +Q B ) e. Q. )
9		elex 2811	. . . . . . . . . 10 |- ( ( A +Q B ) e. Q. -> ( A +Q B ) e. _V )
10	8, 9	syl 14	. . . . . . . . 9 |- ( ( A +Q B ) <Q ( A +Q B ) -> ( A +Q B ) e. _V )
11		breq1 4086	. . . . . . . . 9 |- ( l = ( A +Q B ) -> ( l <Q ( A +Q B ) <-> ( A +Q B ) <Q ( A +Q B ) ) )
12	10, 11	elab3 2955	. . . . . . . 8 |- ( ( A +Q B ) e. { l | l <Q ( A +Q B ) } <-> ( A +Q B ) <Q ( A +Q B ) )
13	5, 12	sylnibr 681	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) e. { l | l <Q ( A +Q B ) } )
14		ltnqex 7752	. . . . . . . . 9 |- { l | l <Q ( A +Q B ) } e. _V
15		gtnqex 7753	. . . . . . . . 9 |- { u | ( A +Q B ) <Q u } e. _V
16	14, 15	op1st 6301	. . . . . . . 8 |- ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = { l | l <Q ( A +Q B ) }
17	16	eleq2i 2296	. . . . . . 7 |- ( ( A +Q B ) e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) <-> ( A +Q B ) e. { l | l <Q ( A +Q B ) } )
18	13, 17	sylnibr 681	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )
19	1, 18	ssneldd 3227	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
20	19	adantr 276	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) -> -. ( A +Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
21		nqprlu 7750	. . . . . . . 8 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
22		nqprlu 7750	. . . . . . . 8 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
23		addclpr 7740	. . . . . . . 8 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ <. { l | l <Q B } , { u | B <Q u } >. e. P. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. )
24	21, 22, 23	syl2an 289	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. )
25		prop 7678	. . . . . . 7 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. -> <. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) , ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) >. e. P. )
26	24, 25	syl 14	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> <. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) , ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) >. e. P. )
27		vex 2802	. . . . . . . 8 |- r e. _V
28		breq2 4087	. . . . . . . 8 |- ( u = r -> ( ( A +Q B ) <Q u <-> ( A +Q B ) <Q r ) )
29	14, 15	op2nd 6302	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = { u | ( A +Q B ) <Q u }
30	27, 28, 29	elab2 2951	. . . . . . 7 |- ( r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) <-> ( A +Q B ) <Q r )
31	30	biimpi 120	. . . . . 6 |- ( r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) -> ( A +Q B ) <Q r )
32		prloc 7694	. . . . . 6 |- ( ( <. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) , ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) >. e. P. /\ ( A +Q B ) <Q r ) -> ( ( A +Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) \/ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
33	26, 31, 32	syl2an 289	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) -> ( ( A +Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) \/ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
34	33	orcomd 734	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) \/ ( A +Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
35	20, 34	ecased 1383	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) -> r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
36	35	ex 115	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) -> r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
37	36	ssrdv 3230	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   e. wcel 2200   {cab 2215   _Vcvv 2799   C_ wss 3197   <.cop 3669   class class class wbr 4083   Or wor 4387   ` cfv 5321  (class class class)co 6010   1stc1st 6293   2ndc2nd 6294   Q.cnq 7483   +Q cplq 7485   <Q cltq 7488   P.cnp 7494   +P. cpp 7496

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4381  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-1o 6573  df-2o 6574  df-oadd 6577  df-omul 6578  df-er 6693  df-ec 6695  df-qs 6699  df-ni 7507  df-pli 7508  df-mi 7509  df-lti 7510  df-plpq 7547  df-mpq 7548  df-enq 7550  df-nqqs 7551  df-plqqs 7552  df-mqqs 7553  df-1nqqs 7554  df-rq 7555  df-ltnqqs 7556  df-enq0 7627  df-nq0 7628  df-0nq0 7629  df-plq0 7630  df-mq0 7631  df-inp 7669  df-iplp 7671

This theorem is referenced by:  addnqpr  7764