Theorem addnqprlemfl 6540

Description: Lemma for addnqpr 6542. 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 6539	. . . . . 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 6382	. . . . . . . . 9 |- <Q Or Q.
3		addclnq 6359	. . . . . . . . 9 |- ((A e. Q. /\ B e. Q.) -> (A +Q B) e. Q.)
4		sonr 4045	. . . . . . . . 9 |- (( <Q Or Q. /\ (A +Q B) e. Q.) -> -. (A +Q B) <Q (A +Q B))
5	2, 3, 4	sylancr 393	. . . . . . . 8 |- ((A e. Q. /\ B e. Q.) -> -. (A +Q B) <Q (A +Q B))
6		ltrelnq 6349	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q.)
7	6	brel 4335	. . . . . . . . . . 11 |- ((A +Q B) <Q (A +Q B) -> ((A +Q B) e. Q. /\ (A +Q B) e. Q.))
8	7	simpld 105	. . . . . . . . . 10 |- ((A +Q B) <Q (A +Q B) -> (A +Q B) e. Q.)
9		elex 2560	. . . . . . . . . 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 3759	. . . . . . . . 9 |- (u = (A +Q B) -> ((A +Q B) <Q u <-> (A +Q B) <Q (A +Q B)))
12	10, 11	elab3 2688	. . . . . . . 8 |- ((A +Q B) e. {u | (A +Q B) <Q u } <-> (A +Q B) <Q (A +Q B))
13	5, 12	sylnibr 601	. . . . . . 7 |- ((A e. Q. /\ B e. Q.) -> -. (A +Q B) e. {u | (A +Q B) <Q u })
14		ltnqex 6531	. . . . . . . . 9 |- { l | l <Q (A +Q B) } e. _V
15		gtnqex 6532	. . . . . . . . 9 |- {u | (A +Q B) <Q u } e. _V
16	14, 15	op2nd 5716	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q (A +Q B) } , {u | (A +Q B) <Q u } >.) = {u | (A +Q B) <Q u }
17	16	eleq2i 2101	. . . . . . 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 601	. . . . . 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 2942	. . . . 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 261	. . . 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 6530	. . . . . . 7 |- (A e. Q. -> <. { l | l <Q A } , {u | A <Q u } >. e. P.)
22		nqprlu 6530	. . . . . . 7 |- (B e. Q. -> <. { l | l <Q B } , {u | B <Q u } >. e. P.)
23		addclpr 6520	. . . . . . 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 273	. . . . . 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 6458	. . . . . 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 2554	. . . . . . 7 |- r e. _V
28		breq1 3758	. . . . . . 7 |- ( l = r -> ( l <Q (A +Q B) <-> r <Q (A +Q B)))
29	14, 15	op1st 5715	. . . . . . 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 2684	. . . . . 6 |- ( r e. ( 1st ` <. { l | l <Q (A +Q B) } , {u | (A +Q B) <Q u } >.) <-> r <Q (A +Q B))
31	30	biimpi 113	. . . . 5 |- ( r e. ( 1st ` <. { l | l <Q (A +Q B) } , {u | (A +Q B) <Q u } >.) -> r <Q (A +Q B))
32		prloc 6474	. . . . 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 273	. . . 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 1238	. . 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 108	. 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 2945	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 97   \/ wo 628   e. wcel 1390   {cab 2023   _Vcvv 2551   C_ wss 2911   <.cop 3370   class class class wbr 3755   Or wor 4023   ` cfv 4845  (class class class)co 5455   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   +Q cplq 6266   <Q cltq 6269   P.cnp 6275   +P. cpp 6277

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-iplp 6451

This theorem is referenced by:  addnqpr  6542