Theorem mulnqprlemfl 7331

Description: Lemma for mulnqpr 7333. 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 7330	. . . . . 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 7154	. . . . . . . . 9 |- <Q Or Q.
3		mulclnq 7132	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) e. Q. )
4		sonr 4199	. . . . . . . . 9 |- ( ( <Q Or Q. /\ ( A .Q B ) e. Q. ) -> -. ( A .Q B ) <Q ( A .Q B ) )
5	2, 3, 4	sylancr 408	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A .Q B ) <Q ( A .Q B ) )
6		ltrelnq 7121	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
7	6	brel 4551	. . . . . . . . . . 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 2668	. . . . . . . . . 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 3899	. . . . . . . . 9 |- ( u = ( A .Q B ) -> ( ( A .Q B ) <Q u <-> ( A .Q B ) <Q ( A .Q B ) ) )
12	10, 11	elab3 2805	. . . . . . . 8 |- ( ( A .Q B ) e. { u | ( A .Q B ) <Q u } <-> ( A .Q B ) <Q ( A .Q B ) )
13	5, 12	sylnibr 649	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A .Q B ) e. { u | ( A .Q B ) <Q u } )
14		ltnqex 7305	. . . . . . . . 9 |- { l | l <Q ( A .Q B ) } e. _V
15		gtnqex 7306	. . . . . . . . 9 |- { u | ( A .Q B ) <Q u } e. _V
16	14, 15	op2nd 5999	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) = { u | ( A .Q B ) <Q u }
17	16	eleq2i 2181	. . . . . . 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 649	. . . . . 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 3066	. . . . 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 272	. . . 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 7303	. . . . . . 7 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
22		nqprlu 7303	. . . . . . 7 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
23		mulclpr 7328	. . . . . . 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 285	. . . . . 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 7231	. . . . . 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 2660	. . . . . . 7 |- r e. _V
28		breq1 3898	. . . . . . 7 |- ( l = r -> ( l <Q ( A .Q B ) <-> r <Q ( A .Q B ) ) )
29	14, 15	op1st 5998	. . . . . . 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 2801	. . . . . 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 7247	. . . . 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 285	. . . 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 1310	. . 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 3069	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 680   e. wcel 1463   {cab 2101   _Vcvv 2657   C_ wss 3037   <.cop 3496   class class class wbr 3895   Or wor 4177   ` cfv 5081  (class class class)co 5728   1stc1st 5990   2ndc2nd 5991   Q.cnq 7036   .Q cmq 7039   <Q cltq 7041   P.cnp 7047   .P. cmp 7050

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-1o 6267  df-2o 6268  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-mpq 7101  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-mqqs 7106  df-1nqqs 7107  df-rq 7108  df-ltnqqs 7109  df-enq0 7180  df-nq0 7181  df-0nq0 7182  df-plq0 7183  df-mq0 7184  df-inp 7222  df-imp 7225

This theorem is referenced by:  mulnqpr  7333