Theorem addnqprlemfu 7501

Description: Lemma for addnqpr 7502. 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 7498	. . . . . 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 7339	. . . . . . . . 9 |- <Q Or Q.
3		addclnq 7316	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
4		sonr 4295	. . . . . . . . 9 |- ( ( <Q Or Q. /\ ( A +Q B ) e. Q. ) -> -. ( A +Q B ) <Q ( A +Q B ) )
5	2, 3, 4	sylancr 411	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) <Q ( A +Q B ) )
6		ltrelnq 7306	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
7	6	brel 4656	. . . . . . . . . . 11 |- ( ( A +Q B ) <Q ( A +Q B ) -> ( ( A +Q B ) e. Q. /\ ( A +Q B ) e. Q. ) )
8	7	simpld 111	. . . . . . . . . 10 |- ( ( A +Q B ) <Q ( A +Q B ) -> ( A +Q B ) e. Q. )
9		elex 2737	. . . . . . . . . 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 3985	. . . . . . . . 9 |- ( l = ( A +Q B ) -> ( l <Q ( A +Q B ) <-> ( A +Q B ) <Q ( A +Q B ) ) )
12	10, 11	elab3 2878	. . . . . . . 8 |- ( ( A +Q B ) e. { l | l <Q ( A +Q B ) } <-> ( A +Q B ) <Q ( A +Q B ) )
13	5, 12	sylnibr 667	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) e. { l | l <Q ( A +Q B ) } )
14		ltnqex 7490	. . . . . . . . 9 |- { l | l <Q ( A +Q B ) } e. _V
15		gtnqex 7491	. . . . . . . . 9 |- { u | ( A +Q B ) <Q u } e. _V
16	14, 15	op1st 6114	. . . . . . . 8 |- ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = { l | l <Q ( A +Q B ) }
17	16	eleq2i 2233	. . . . . . 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 667	. . . . . 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 3145	. . . . 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 274	. . . 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 7488	. . . . . . . 8 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
22		nqprlu 7488	. . . . . . . 8 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
23		addclpr 7478	. . . . . . . 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 287	. . . . . . 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 7416	. . . . . . 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 2729	. . . . . . . 8 |- r e. _V
28		breq2 3986	. . . . . . . 8 |- ( u = r -> ( ( A +Q B ) <Q u <-> ( A +Q B ) <Q r ) )
29	14, 15	op2nd 6115	. . . . . . . 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 2874	. . . . . . 7 |- ( r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) <-> ( A +Q B ) <Q r )
31	30	biimpi 119	. . . . . 6 |- ( r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) -> ( A +Q B ) <Q r )
32		prloc 7432	. . . . . 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 287	. . . . 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 719	. . . 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 1339	. . 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 114	. 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 3148	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 103   \/ wo 698   e. wcel 2136   {cab 2151   _Vcvv 2726   C_ wss 3116   <.cop 3579   class class class wbr 3982   Or wor 4273   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226   P.cnp 7232   +P. cpp 7234

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409

This theorem is referenced by:  addnqpr  7502