Theorem addnqprlemfl 7890

Description: Lemma for addnqpr 7892. 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 7889	. . . . . 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 7729	. . . . . . . . 9 |- <Q Or Q.
3		addclnq 7706	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
4		sonr 4443	. . . . . . . . 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 7696	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
7	6	brel 4807	. . . . . . . . . . 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 2827	. . . . . . . . . 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 4118	. . . . . . . . 9 |- ( u = ( A +Q B ) -> ( ( A +Q B ) <Q u <-> ( A +Q B ) <Q ( A +Q B ) ) )
12	10, 11	elab3 2972	. . . . . . . 8 |- ( ( A +Q B ) e. { u | ( A +Q B ) <Q u } <-> ( A +Q B ) <Q ( A +Q B ) )
13	5, 12	sylnibr 684	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) e. { u | ( A +Q B ) <Q u } )
14		ltnqex 7880	. . . . . . . . 9 |- { l | l <Q ( A +Q B ) } e. _V
15		gtnqex 7881	. . . . . . . . 9 |- { u | ( A +Q B ) <Q u } e. _V
16	14, 15	op2nd 6354	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = { u | ( A +Q B ) <Q u }
17	16	eleq2i 2301	. . . . . . 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 684	. . . . . 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 3245	. . . . 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 276	. . . 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 7878	. . . . . . 7 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
22		nqprlu 7878	. . . . . . 7 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
23		addclpr 7868	. . . . . . 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 289	. . . . . 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 7806	. . . . . 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 2818	. . . . . . 7 |- r e. _V
28		breq1 4117	. . . . . . 7 |- ( l = r -> ( l <Q ( A +Q B ) <-> r <Q ( A +Q B ) ) )
29	14, 15	op1st 6353	. . . . . . 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 2968	. . . . . 6 |- ( r e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) <-> r <Q ( A +Q B ) )
31	30	biimpi 120	. . . . 5 |- ( r e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) -> r <Q ( A +Q B ) )
32		prloc 7822	. . . . 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 289	. . . 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 1386	. . 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 115	. 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 3248	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 104   \/ wo 716   e. wcel 2205   {cab 2220   _Vcvv 2815   C_ wss 3214   <.cop 3697   class class class wbr 4114   Or wor 4421   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   Q.cnq 7611   +Q cplq 7613   <Q cltq 7616   P.cnp 7622   +P. cpp 7624

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-iplp 7799

This theorem is referenced by:  addnqpr  7892