Theorem mulnqprlemfl 7587

Description: Lemma for mulnqpr 7589. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)

Assertion

Ref	Expression
mulnqprlemfl	|- ( ( 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 mulnqprlemfl

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulnqprlemru 7586	. . . . . 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 7410	. . . . . . . . 9 |- <Q Or Q.
3		mulclnq 7388	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) e. Q. )
4		sonr 4329	. . . . . . . . 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 7377	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
7	6	brel 4690	. . . . . . . . . . 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 2760	. . . . . . . . . 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 4019	. . . . . . . . 9 |- ( u = ( A .Q B ) -> ( ( A .Q B ) <Q u <-> ( A .Q B ) <Q ( A .Q B ) ) )
12	10, 11	elab3 2901	. . . . . . . 8 |- ( ( A .Q B ) e. { u | ( A .Q B ) <Q u } <-> ( A .Q B ) <Q ( A .Q B ) )
13	5, 12	sylnibr 678	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A .Q B ) e. { u | ( A .Q B ) <Q u } )
14		ltnqex 7561	. . . . . . . . 9 |- { l | l <Q ( A .Q B ) } e. _V
15		gtnqex 7562	. . . . . . . . 9 |- { u | ( A .Q B ) <Q u } e. _V
16	14, 15	op2nd 6161	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) = { u | ( A .Q B ) <Q u }
17	16	eleq2i 2254	. . . . . . 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 678	. . . . . 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 3170	. . . . 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 7559	. . . . . . 7 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
22		nqprlu 7559	. . . . . . 7 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
23		mulclpr 7584	. . . . . . 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 7487	. . . . . 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 2752	. . . . . . 7 |- r e. _V
28		breq1 4018	. . . . . . 7 |- ( l = r -> ( l <Q ( A .Q B ) <-> r <Q ( A .Q B ) ) )
29	14, 15	op1st 6160	. . . . . . 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 2897	. . . . . 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 7503	. . . . 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 1359	. . 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 3173	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 709   e. wcel 2158   {cab 2173   _Vcvv 2749   C_ wss 3141   <.cop 3607   class class class wbr 4015   Or wor 4307   ` cfv 5228  (class class class)co 5888   1stc1st 6152   2ndc2nd 6153   Q.cnq 7292   .Q cmq 7295   <Q cltq 7297   P.cnp 7303   .P. cmp 7306

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-2o 6431  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-enq0 7436  df-nq0 7437  df-0nq0 7438  df-plq0 7439  df-mq0 7440  df-inp 7478  df-imp 7481

This theorem is referenced by:  mulnqpr  7589