Theorem addnqprlemfl 7391

Description: Lemma for addnqpr 7393. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)

Assertion

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

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		addnqprlemru 7390	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )
2		ltsonq 7230	. . . . . . . . 9 |- <Q Or Q.
3		addclnq 7207	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
4		sonr 4247	. . . . . . . . 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 7197	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
7	6	brel 4599	. . . . . . . . . . 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 2700	. . . . . . . . . 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		breq2 3941	. . . . . . . . 9 |- ( u = ( A +Q B ) -> ( ( A +Q B ) <Q u <-> ( A +Q B ) <Q ( A +Q B ) ) )
12	10, 11	elab3 2840	. . . . . . . 8 |- ( ( A +Q B ) e. { u | ( A +Q B ) <Q u } <-> ( A +Q B ) <Q ( A +Q B ) )
13	5, 12	sylnibr 667	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) e. { u | ( A +Q B ) <Q u } )
14		ltnqex 7381	. . . . . . . . 9 |- { l | l <Q ( A +Q B ) } e. _V
15		gtnqex 7382	. . . . . . . . 9 |- { u | ( A +Q B ) <Q u } e. _V
16	14, 15	op2nd 6053	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = { u | ( A +Q B ) <Q u }
17	16	eleq2i 2207	. . . . . . 7 |- ( ( A +Q B ) e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) <-> ( A +Q B ) e. { u | ( A +Q B ) <Q u } )
18	13, 17	sylnibr 667	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )
19	1, 18	ssneldd 3105	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) e. ( 2nd ` ( <. { 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. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) -> -. ( A +Q B ) e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
21		nqprlu 7379	. . . . . . 7 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
22		nqprlu 7379	. . . . . . 7 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
23		addclpr 7369	. . . . . . 7 |- ( ( <. { 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	. . . . . 6 |- ( ( 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 7307	. . . . . 6 |- ( ( <. { 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	. . . . 5 |- ( ( 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 2692	. . . . . . 7 |- r e. _V
28		breq1 3940	. . . . . . 7 |- ( l = r -> ( l <Q ( A +Q B ) <-> r <Q ( A +Q B ) ) )
29	14, 15	op1st 6052	. . . . . . 7 |- ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = { l | l <Q ( A +Q B ) }
30	27, 28, 29	elab2 2836	. . . . . 6 |- ( r e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) <-> r <Q ( A +Q B ) )
31	30	biimpi 119	. . . . 5 |- ( r e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) -> r <Q ( A +Q B ) )
32		prloc 7323	. . . . 5 |- ( ( <. ( 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. /\ r <Q ( A +Q B ) ) -> ( r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) \/ ( A +Q B ) 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	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) -> ( r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) \/ ( A +Q B ) e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
34	20, 33	ecased 1328	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) -> r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
35	34	ex 114	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) -> r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
36	35	ssrdv 3108	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) C_ ( 1st ` ( <. { 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 1481   {cab 2126   _Vcvv 2689   C_ wss 3076   <.cop 3535   class class class wbr 3937   Or wor 4225   ` cfv 5131  (class class class)co 5782   1stc1st 6044   2ndc2nd 6045   Q.cnq 7112   +Q cplq 7114   <Q cltq 7117   P.cnp 7123   +P. cpp 7125

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300

This theorem is referenced by:  addnqpr  7393