Theorem recexprlemss1l 7665

Description: The lower cut of A .P. B is a subset of the lower cut of one. Lemma for recexpr 7668. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.

Assertion

Ref	Expression
recexprlemss1l	|- ( A e. P. -> ( 1st ` ( A .P. B ) ) C_ ( 1st ` 1P ) )

Distinct variable groups:   x, y, A   x, B, y

Proof of Theorem recexprlemss1l

Dummy variables q z w u f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		recexpr.1	. . . . . 6 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
2	1	recexprlempr 7662	. . . . 5 |- ( A e. P. -> B e. P. )
3		df-imp 7499	. . . . . 6 |- .P. = ( y e. P. , w e. P. |-> <. { u e. Q. | E. f e. Q. E. g e. Q. ( f e. ( 1st ` y ) /\ g e. ( 1st ` w ) /\ u = ( f .Q g ) ) } , { u e. Q. | E. f e. Q. E. g e. Q. ( f e. ( 2nd ` y ) /\ g e. ( 2nd ` w ) /\ u = ( f .Q g ) ) } >. )
4		mulclnq 7406	. . . . . 6 |- ( ( f e. Q. /\ g e. Q. ) -> ( f .Q g ) e. Q. )
5	3, 4	genpelvl 7542	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( w e. ( 1st ` ( A .P. B ) ) <-> E. z e. ( 1st ` A ) E. q e. ( 1st ` B ) w = ( z .Q q ) ) )
6	2, 5	mpdan 421	. . . 4 |- ( A e. P. -> ( w e. ( 1st ` ( A .P. B ) ) <-> E. z e. ( 1st ` A ) E. q e. ( 1st ` B ) w = ( z .Q q ) ) )
7	1	recexprlemell 7652	. . . . . . . 8 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
8		ltrelnq 7395	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
9	8	brel 4696	. . . . . . . . . . . . 13 |- ( q <Q y -> ( q e. Q. /\ y e. Q. ) )
10	9	simprd 114	. . . . . . . . . . . 12 |- ( q <Q y -> y e. Q. )
11		prop 7505	. . . . . . . . . . . . . . . . . 18 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12		elprnql 7511	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
13	11, 12	sylan 283	. . . . . . . . . . . . . . . . 17 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
14		ltmnqi 7433	. . . . . . . . . . . . . . . . . 18 |- ( ( q <Q y /\ z e. Q. ) -> ( z .Q q ) <Q ( z .Q y ) )
15	14	expcom 116	. . . . . . . . . . . . . . . . 17 |- ( z e. Q. -> ( q <Q y -> ( z .Q q ) <Q ( z .Q y ) ) )
16	13, 15	syl 14	. . . . . . . . . . . . . . . 16 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( q <Q y -> ( z .Q q ) <Q ( z .Q y ) ) )
17	16	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ z e. ( 1st ` A ) ) /\ y e. Q. ) -> ( q <Q y -> ( z .Q q ) <Q ( z .Q y ) ) )
18		prltlu 7517	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> z <Q ( *Q ` y ) )
19	11, 18	syl3an1 1282	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ z e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> z <Q ( *Q ` y ) )
20	19	3expia 1207	. . . . . . . . . . . . . . . . 17 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( ( *Q ` y ) e. ( 2nd ` A ) -> z <Q ( *Q ` y ) ) )
21	20	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. P. /\ z e. ( 1st ` A ) ) /\ y e. Q. ) -> ( ( *Q ` y ) e. ( 2nd ` A ) -> z <Q ( *Q ` y ) ) )
22		ltmnqi 7433	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( z <Q ( *Q ` y ) /\ y e. Q. ) -> ( y .Q z ) <Q ( y .Q ( *Q ` y ) ) )
23	22	expcom 116	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. Q. -> ( z <Q ( *Q ` y ) -> ( y .Q z ) <Q ( y .Q ( *Q ` y ) ) ) )
24	23	adantr 276	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. Q. /\ z e. Q. ) -> ( z <Q ( *Q ` y ) -> ( y .Q z ) <Q ( y .Q ( *Q ` y ) ) ) )
25		mulcomnqg 7413	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
26		recidnq 7423	. . . . . . . . . . . . . . . . . . . . 21 |- ( y e. Q. -> ( y .Q ( *Q ` y ) ) = 1Q )
27	26	adantr 276	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q ( *Q ` y ) ) = 1Q )
28	25, 27	breq12d 4031	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. Q. /\ z e. Q. ) -> ( ( y .Q z ) <Q ( y .Q ( *Q ` y ) ) <-> ( z .Q y ) <Q 1Q ) )
29	24, 28	sylibd 149	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. Q. /\ z e. Q. ) -> ( z <Q ( *Q ` y ) -> ( z .Q y ) <Q 1Q ) )
30	29	ancoms 268	. . . . . . . . . . . . . . . . 17 |- ( ( z e. Q. /\ y e. Q. ) -> ( z <Q ( *Q ` y ) -> ( z .Q y ) <Q 1Q ) )
31	13, 30	sylan 283	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. P. /\ z e. ( 1st ` A ) ) /\ y e. Q. ) -> ( z <Q ( *Q ` y ) -> ( z .Q y ) <Q 1Q ) )
32	21, 31	syld 45	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ z e. ( 1st ` A ) ) /\ y e. Q. ) -> ( ( *Q ` y ) e. ( 2nd ` A ) -> ( z .Q y ) <Q 1Q ) )
33	17, 32	anim12d 335	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ z e. ( 1st ` A ) ) /\ y e. Q. ) -> ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( ( z .Q q ) <Q ( z .Q y ) /\ ( z .Q y ) <Q 1Q ) ) )
34		ltsonq 7428	. . . . . . . . . . . . . . 15 |- <Q Or Q.
35	34, 8	sotri 5042	. . . . . . . . . . . . . 14 |- ( ( ( z .Q q ) <Q ( z .Q y ) /\ ( z .Q y ) <Q 1Q ) -> ( z .Q q ) <Q 1Q )
36	33, 35	syl6 33	. . . . . . . . . . . . 13 |- ( ( ( A e. P. /\ z e. ( 1st ` A ) ) /\ y e. Q. ) -> ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( z .Q q ) <Q 1Q ) )
37	36	exp4b 367	. . . . . . . . . . . 12 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( y e. Q. -> ( q <Q y -> ( ( *Q ` y ) e. ( 2nd ` A ) -> ( z .Q q ) <Q 1Q ) ) ) )
38	10, 37	syl5 32	. . . . . . . . . . 11 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( q <Q y -> ( q <Q y -> ( ( *Q ` y ) e. ( 2nd ` A ) -> ( z .Q q ) <Q 1Q ) ) ) )
39	38	pm2.43d 50	. . . . . . . . . 10 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( q <Q y -> ( ( *Q ` y ) e. ( 2nd ` A ) -> ( z .Q q ) <Q 1Q ) ) )
40	39	impd 254	. . . . . . . . 9 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( z .Q q ) <Q 1Q ) )
41	40	exlimdv 1830	. . . . . . . 8 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( z .Q q ) <Q 1Q ) )
42	7, 41	biimtrid 152	. . . . . . 7 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( q e. ( 1st ` B ) -> ( z .Q q ) <Q 1Q ) )
43		breq1 4021	. . . . . . . 8 |- ( w = ( z .Q q ) -> ( w <Q 1Q <-> ( z .Q q ) <Q 1Q ) )
44	43	biimprcd 160	. . . . . . 7 |- ( ( z .Q q ) <Q 1Q -> ( w = ( z .Q q ) -> w <Q 1Q ) )
45	42, 44	syl6 33	. . . . . 6 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( q e. ( 1st ` B ) -> ( w = ( z .Q q ) -> w <Q 1Q ) ) )
46	45	expimpd 363	. . . . 5 |- ( A e. P. -> ( ( z e. ( 1st ` A ) /\ q e. ( 1st ` B ) ) -> ( w = ( z .Q q ) -> w <Q 1Q ) ) )
47	46	rexlimdvv 2614	. . . 4 |- ( A e. P. -> ( E. z e. ( 1st ` A ) E. q e. ( 1st ` B ) w = ( z .Q q ) -> w <Q 1Q ) )
48	6, 47	sylbid 150	. . 3 |- ( A e. P. -> ( w e. ( 1st ` ( A .P. B ) ) -> w <Q 1Q ) )
49		1prl 7585	. . . 4 |- ( 1st ` 1P ) = { w | w <Q 1Q }
50	49	abeq2i 2300	. . 3 |- ( w e. ( 1st ` 1P ) <-> w <Q 1Q )
51	48, 50	imbitrrdi 162	. 2 |- ( A e. P. -> ( w e. ( 1st ` ( A .P. B ) ) -> w e. ( 1st ` 1P ) ) )
52	51	ssrdv 3176	1 |- ( A e. P. -> ( 1st ` ( A .P. B ) ) C_ ( 1st ` 1P ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2160   {cab 2175   E.wrex 2469   C_ wss 3144   <.cop 3610   class class class wbr 4018   ` cfv 5235  (class class class)co 5897   1stc1st 6164   2ndc2nd 6165   Q.cnq 7310   1Qc1q 7311   .Q cmq 7313   *Qcrq 7314   <Q cltq 7315   P.cnp 7321   1Pc1p 7322   .P. cmp 7324

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-inp 7496  df-i1p 7497  df-imp 7499

This theorem is referenced by:  recexprlemex  7667