Theorem recexprlemss1l 7576

Description: The lower cut of A .P. B is a subset of the lower cut of one. Lemma for recexpr 7579. (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 7573	. . . . 5 |- ( A e. P. -> B e. P. )
3		df-imp 7410	. . . . . 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 7317	. . . . . 6 |- ( ( f e. Q. /\ g e. Q. ) -> ( f .Q g ) e. Q. )
5	3, 4	genpelvl 7453	. . . . 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 418	. . . 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 7563	. . . . . . . 8 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
8		ltrelnq 7306	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
9	8	brel 4656	. . . . . . . . . . . . 13 |- ( q <Q y -> ( q e. Q. /\ y e. Q. ) )
10	9	simprd 113	. . . . . . . . . . . 12 |- ( q <Q y -> y e. Q. )
11		prop 7416	. . . . . . . . . . . . . . . . . 18 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12		elprnql 7422	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
13	11, 12	sylan 281	. . . . . . . . . . . . . . . . 17 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
14		ltmnqi 7344	. . . . . . . . . . . . . . . . . 18 |- ( ( q <Q y /\ z e. Q. ) -> ( z .Q q ) <Q ( z .Q y ) )
15	14	expcom 115	. . . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ z e. ( 1st ` A ) ) /\ y e. Q. ) -> ( q <Q y -> ( z .Q q ) <Q ( z .Q y ) ) )
18		prltlu 7428	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> z <Q ( *Q ` y ) )
19	11, 18	syl3an1 1261	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ z e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> z <Q ( *Q ` y ) )
20	19	3expia 1195	. . . . . . . . . . . . . . . . 17 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( ( *Q ` y ) e. ( 2nd ` A ) -> z <Q ( *Q ` y ) ) )
21	20	adantr 274	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. P. /\ z e. ( 1st ` A ) ) /\ y e. Q. ) -> ( ( *Q ` y ) e. ( 2nd ` A ) -> z <Q ( *Q ` y ) ) )
22		ltmnqi 7344	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( z <Q ( *Q ` y ) /\ y e. Q. ) -> ( y .Q z ) <Q ( y .Q ( *Q ` y ) ) )
23	22	expcom 115	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. Q. -> ( z <Q ( *Q ` y ) -> ( y .Q z ) <Q ( y .Q ( *Q ` y ) ) ) )
24	23	adantr 274	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. Q. /\ z e. Q. ) -> ( z <Q ( *Q ` y ) -> ( y .Q z ) <Q ( y .Q ( *Q ` y ) ) ) )
25		mulcomnqg 7324	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
26		recidnq 7334	. . . . . . . . . . . . . . . . . . . . 21 |- ( y e. Q. -> ( y .Q ( *Q ` y ) ) = 1Q )
27	26	adantr 274	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q ( *Q ` y ) ) = 1Q )
28	25, 27	breq12d 3995	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. Q. /\ z e. Q. ) -> ( ( y .Q z ) <Q ( y .Q ( *Q ` y ) ) <-> ( z .Q y ) <Q 1Q ) )
29	24, 28	sylibd 148	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. Q. /\ z e. Q. ) -> ( z <Q ( *Q ` y ) -> ( z .Q y ) <Q 1Q ) )
30	29	ancoms 266	. . . . . . . . . . . . . . . . 17 |- ( ( z e. Q. /\ y e. Q. ) -> ( z <Q ( *Q ` y ) -> ( z .Q y ) <Q 1Q ) )
31	13, 30	sylan 281	. . . . . . . . . . . . . . . 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 333	. . . . . . . . . . . . . 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 7339	. . . . . . . . . . . . . . 15 |- <Q Or Q.
35	34, 8	sotri 4999	. . . . . . . . . . . . . 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 365	. . . . . . . . . . . 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 252	. . . . . . . . 9 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( z .Q q ) <Q 1Q ) )
41	40	exlimdv 1807	. . . . . . . 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	syl5bi 151	. . . . . . 7 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> ( q e. ( 1st ` B ) -> ( z .Q q ) <Q 1Q ) )
43		breq1 3985	. . . . . . . 8 |- ( w = ( z .Q q ) -> ( w <Q 1Q <-> ( z .Q q ) <Q 1Q ) )
44	43	biimprcd 159	. . . . . . 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 361	. . . . 5 |- ( A e. P. -> ( ( z e. ( 1st ` A ) /\ q e. ( 1st ` B ) ) -> ( w = ( z .Q q ) -> w <Q 1Q ) ) )
47	46	rexlimdvv 2590	. . . 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 149	. . 3 |- ( A e. P. -> ( w e. ( 1st ` ( A .P. B ) ) -> w <Q 1Q ) )
49		1prl 7496	. . . 4 |- ( 1st ` 1P ) = { w | w <Q 1Q }
50	49	abeq2i 2277	. . 3 |- ( w e. ( 1st ` 1P ) <-> w <Q 1Q )
51	48, 50	syl6ibr 161	. 2 |- ( A e. P. -> ( w e. ( 1st ` ( A .P. B ) ) -> w e. ( 1st ` 1P ) ) )
52	51	ssrdv 3148	1 |- ( A e. P. -> ( 1st ` ( A .P. B ) ) C_ ( 1st ` 1P ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   E.wrex 2445   C_ wss 3116   <.cop 3579   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   1Qc1q 7222   .Q cmq 7224   *Qcrq 7225   <Q cltq 7226   P.cnp 7232   1Pc1p 7233   .P. cmp 7235

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407  df-i1p 7408  df-imp 7410

This theorem is referenced by:  recexprlemex  7578