Theorem mulnqprlemfu 7575

Description: Lemma for mulnqpr 7576. 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 7572	. . . . . 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 7397	. . . . . . . . 9 |- <Q Or Q.
3		mulclnq 7375	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) e. Q. )
4		sonr 4318	. . . . . . . . 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 7364	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
7	6	brel 4679	. . . . . . . . . . 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 2749	. . . . . . . . . 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 4007	. . . . . . . . 9 |- ( l = ( A .Q B ) -> ( l <Q ( A .Q B ) <-> ( A .Q B ) <Q ( A .Q B ) ) )
12	10, 11	elab3 2890	. . . . . . . 8 |- ( ( A .Q B ) e. { l | l <Q ( A .Q B ) } <-> ( A .Q B ) <Q ( A .Q B ) )
13	5, 12	sylnibr 677	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A .Q B ) e. { l | l <Q ( A .Q B ) } )
14		ltnqex 7548	. . . . . . . . 9 |- { l | l <Q ( A .Q B ) } e. _V
15		gtnqex 7549	. . . . . . . . 9 |- { u | ( A .Q B ) <Q u } e. _V
16	14, 15	op1st 6147	. . . . . . . 8 |- ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) = { l | l <Q ( A .Q B ) }
17	16	eleq2i 2244	. . . . . . 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 677	. . . . . 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 3159	. . . . 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 276	. . . 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 7546	. . . . . . . 8 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
22		nqprlu 7546	. . . . . . . 8 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
23		mulclpr 7571	. . . . . . . 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 289	. . . . . . 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 7474	. . . . . . 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 2741	. . . . . . . 8 |- r e. _V
28		breq2 4008	. . . . . . . 8 |- ( u = r -> ( ( A .Q B ) <Q u <-> ( A .Q B ) <Q r ) )
29	14, 15	op2nd 6148	. . . . . . . 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 2886	. . . . . . 7 |- ( r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) <-> ( A .Q B ) <Q r )
31	30	biimpi 120	. . . . . 6 |- ( r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) -> ( A .Q B ) <Q r )
32		prloc 7490	. . . . . 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 289	. . . . 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 729	. . . 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 1349	. . 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 115	. 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 3162	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 104   \/ wo 708   e. wcel 2148   {cab 2163   _Vcvv 2738   C_ wss 3130   <.cop 3596   class class class wbr 4004   Or wor 4296   ` cfv 5217  (class class class)co 5875   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279   .Q cmq 7282   <Q cltq 7284   P.cnp 7290   .P. cmp 7293

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-imp 7468

This theorem is referenced by:  mulnqpr  7576