Theorem mulnqprlemrl 7405

Description: Lemma for mulnqpr 7409. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)

Assertion

Ref	Expression
mulnqprlemrl	|- ( ( 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 } >. ) )

Distinct variable groups:   A, l, u   B, l, u

Proof of Theorem mulnqprlemrl

Dummy variables f g h r s t x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nqprlu 7379	. . . . . 6 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
2		nqprlu 7379	. . . . . 6 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
3		df-imp 7301	. . . . . . 7 |- .P. = ( x e. P. , y e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` x ) /\ h e. ( 1st ` y ) /\ f = ( g .Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` x ) /\ h e. ( 2nd ` y ) /\ f = ( g .Q h ) ) } >. )
4		mulclnq 7208	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
5	3, 4	genpelvl 7344	. . . . . 6 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ <. { l | l <Q B } , { u | B <Q u } >. e. P. ) -> ( r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) ) )
6	1, 2, 5	syl2an 287	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) ) )
7	6	biimpa 294	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> E. s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) )
8		vex 2692	. . . . . . . . . . . . 13 |- s e. _V
9		breq1 3940	. . . . . . . . . . . . 13 |- ( l = s -> ( l <Q A <-> s <Q A ) )
10		ltnqex 7381	. . . . . . . . . . . . . 14 |- { l | l <Q A } e. _V
11		gtnqex 7382	. . . . . . . . . . . . . 14 |- { u | A <Q u } e. _V
12	10, 11	op1st 6052	. . . . . . . . . . . . 13 |- ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) = { l | l <Q A }
13	8, 9, 12	elab2 2836	. . . . . . . . . . . 12 |- ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> s <Q A )
14	13	biimpi 119	. . . . . . . . . . 11 |- ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) -> s <Q A )
15	14	ad2antrl 482	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> s <Q A )
16	15	adantr 274	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> s <Q A )
17		vex 2692	. . . . . . . . . . . . 13 |- t e. _V
18		breq1 3940	. . . . . . . . . . . . 13 |- ( l = t -> ( l <Q B <-> t <Q B ) )
19		ltnqex 7381	. . . . . . . . . . . . . 14 |- { l | l <Q B } e. _V
20		gtnqex 7382	. . . . . . . . . . . . . 14 |- { u | B <Q u } e. _V
21	19, 20	op1st 6052	. . . . . . . . . . . . 13 |- ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) = { l | l <Q B }
22	17, 18, 21	elab2 2836	. . . . . . . . . . . 12 |- ( t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) <-> t <Q B )
23	22	biimpi 119	. . . . . . . . . . 11 |- ( t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) -> t <Q B )
24	23	ad2antll 483	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> t <Q B )
25	24	adantr 274	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> t <Q B )
26		ltrelnq 7197	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
27	26	brel 4599	. . . . . . . . . . 11 |- ( s <Q A -> ( s e. Q. /\ A e. Q. ) )
28	16, 27	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( s e. Q. /\ A e. Q. ) )
29	26	brel 4599	. . . . . . . . . . 11 |- ( t <Q B -> ( t e. Q. /\ B e. Q. ) )
30	25, 29	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( t e. Q. /\ B e. Q. ) )
31		lt2mulnq 7237	. . . . . . . . . 10 |- ( ( ( s e. Q. /\ A e. Q. ) /\ ( t e. Q. /\ B e. Q. ) ) -> ( ( s <Q A /\ t <Q B ) -> ( s .Q t ) <Q ( A .Q B ) ) )
32	28, 30, 31	syl2anc 409	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( ( s <Q A /\ t <Q B ) -> ( s .Q t ) <Q ( A .Q B ) ) )
33	16, 25, 32	mp2and 430	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( s .Q t ) <Q ( A .Q B ) )
34		breq1 3940	. . . . . . . . 9 |- ( r = ( s .Q t ) -> ( r <Q ( A .Q B ) <-> ( s .Q t ) <Q ( A .Q B ) ) )
35	34	adantl 275	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( r <Q ( A .Q B ) <-> ( s .Q t ) <Q ( A .Q B ) ) )
36	33, 35	mpbird 166	. . . . . . 7 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> r <Q ( A .Q B ) )
37		vex 2692	. . . . . . . 8 |- r e. _V
38		breq1 3940	. . . . . . . 8 |- ( l = r -> ( l <Q ( A .Q B ) <-> r <Q ( A .Q B ) ) )
39		ltnqex 7381	. . . . . . . . 9 |- { l | l <Q ( A .Q B ) } e. _V
40		gtnqex 7382	. . . . . . . . 9 |- { u | ( A .Q B ) <Q u } e. _V
41	39, 40	op1st 6052	. . . . . . . 8 |- ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) = { l | l <Q ( A .Q B ) }
42	37, 38, 41	elab2 2836	. . . . . . 7 |- ( r e. ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) <-> r <Q ( A .Q B ) )
43	36, 42	sylibr 133	. . . . . 6 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> r e. ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
44	43	ex 114	. . . . 5 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( r = ( s .Q t ) -> r e. ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
45	44	rexlimdvva 2560	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( E. s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) -> r e. ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
46	7, 45	mpd 13	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> r e. ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
47	46	ex 114	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) -> r e. ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
48	47	ssrdv 3108	1 |- ( ( 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 } >. ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   {cab 2126   E.wrex 2418   C_ wss 3076   <.cop 3535   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   1stc1st 6044   Q.cnq 7112   .Q cmq 7115   <Q cltq 7117   P.cnp 7123   .P. cmp 7126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-inp 7298  df-imp 7301

This theorem is referenced by:  mulnqprlemfu  7408  mulnqpr  7409