Theorem mulnqprlemfu 6766

Description: Lemma for mulnqpr 6767. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)

Assertion

Ref	Expression
mulnqprlemfu	|- ( ( 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 mulnqprlemfu

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulnqprlemrl 6763	. . . . . 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 6588	. . . . . . . . 9 |- <Q Or Q.
3		mulclnq 6566	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) e. Q. )
4		sonr 4072	. . . . . . . . 9 |- ( ( <Q Or Q. /\ ( A .Q B ) e. Q. ) -> -. ( A .Q B ) <Q ( A .Q B ) )
5	2, 3, 4	sylancr 405	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A .Q B ) <Q ( A .Q B ) )
6		ltrelnq 6555	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
7	6	brel 4410	. . . . . . . . . . 11 |- ( ( A .Q B ) <Q ( A .Q B ) -> ( ( A .Q B ) e. Q. /\ ( A .Q B ) e. Q. ) )
8	7	simpld 110	. . . . . . . . . 10 |- ( ( A .Q B ) <Q ( A .Q B ) -> ( A .Q B ) e. Q. )
9		elex 2610	. . . . . . . . . 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 3788	. . . . . . . . 9 |- ( l = ( A .Q B ) -> ( l <Q ( A .Q B ) <-> ( A .Q B ) <Q ( A .Q B ) ) )
12	10, 11	elab3 2745	. . . . . . . 8 |- ( ( A .Q B ) e. { l | l <Q ( A .Q B ) } <-> ( A .Q B ) <Q ( A .Q B ) )
13	5, 12	sylnibr 634	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A .Q B ) e. { l | l <Q ( A .Q B ) } )
14		ltnqex 6739	. . . . . . . . 9 |- { l | l <Q ( A .Q B ) } e. _V
15		gtnqex 6740	. . . . . . . . 9 |- { u | ( A .Q B ) <Q u } e. _V
16	14, 15	op1st 5793	. . . . . . . 8 |- ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) = { l | l <Q ( A .Q B ) }
17	16	eleq2i 2145	. . . . . . 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 634	. . . . . 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 3002	. . . . 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 270	. . . 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 6737	. . . . . . . 8 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
22		nqprlu 6737	. . . . . . . 8 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
23		mulclpr 6762	. . . . . . . 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 283	. . . . . . 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 6665	. . . . . . 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 2604	. . . . . . . 8 |- r e. _V
28		breq2 3789	. . . . . . . 8 |- ( u = r -> ( ( A .Q B ) <Q u <-> ( A .Q B ) <Q r ) )
29	14, 15	op2nd 5794	. . . . . . . 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 2741	. . . . . . 7 |- ( r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) <-> ( A .Q B ) <Q r )
31	30	biimpi 118	. . . . . 6 |- ( r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) -> ( A .Q B ) <Q r )
32		prloc 6681	. . . . . 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 283	. . . . 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 680	. . . 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 1280	. . 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 113	. 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 3005	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 102   \/ wo 661   e. wcel 1433   {cab 2067   _Vcvv 2601   C_ wss 2973   <.cop 3401   class class class wbr 3785   Or wor 4050   ` cfv 4922  (class class class)co 5532   1stc1st 5785   2ndc2nd 5786   Q.cnq 6470   .Q cmq 6473   <Q cltq 6475   P.cnp 6481   .P. cmp 6484

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-imp 6659

This theorem is referenced by:  mulnqpr  6767