Theorem recexprlemss1l 7538

Description: The lower cut of A .P. B is a subset of the lower cut of one. Lemma for recexpr 7541. (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 7535	. . . . 5 |- ( A e. P. -> B e. P. )
3		df-imp 7372	. . . . . 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 7279	. . . . . 6 |- ( ( f e. Q. /\ g e. Q. ) -> ( f .Q g ) e. Q. )
5	3, 4	genpelvl 7415	. . . . 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 7525	. . . . . . . 8 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
8		ltrelnq 7268	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
9	8	brel 4635	. . . . . . . . . . . . 13 |- ( q <Q y -> ( q e. Q. /\ y e. Q. ) )
10	9	simprd 113	. . . . . . . . . . . 12 |- ( q <Q y -> y e. Q. )
11		prop 7378	. . . . . . . . . . . . . . . . . 18 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12		elprnql 7384	. . . . . . . . . . . . . . . . . 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 7306	. . . . . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> z <Q ( *Q ` y ) )
19	11, 18	syl3an1 1253	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ z e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> z <Q ( *Q ` y ) )
20	19	3expia 1187	. . . . . . . . . . . . . . . . 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 7306	. . . . . . . . . . . . . . . . . . . . 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 7286	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
26		recidnq 7296	. . . . . . . . . . . . . . . . . . . . 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 3978	. . . . . . . . . . . . . . . . . . 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 7301	. . . . . . . . . . . . . . 15 |- <Q Or Q.
35	34, 8	sotri 4978	. . . . . . . . . . . . . 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 1799	. . . . . . . 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 3968	. . . . . . . 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 2581	. . . 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 7458	. . . 4 |- ( 1st ` 1P ) = { w | w <Q 1Q }
50	49	abeq2i 2268	. . 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 3134	1 |- ( A e. P. -> ( 1st ` ( A .P. B ) ) C_ ( 1st ` 1P ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   E.wex 1472   e. wcel 2128   {cab 2143   E.wrex 2436   C_ wss 3102   <.cop 3563   class class class wbr 3965   ` cfv 5167  (class class class)co 5818   1stc1st 6080   2ndc2nd 6081   Q.cnq 7183   1Qc1q 7184   .Q cmq 7186   *Qcrq 7187   <Q cltq 7188   P.cnp 7194   1Pc1p 7195   .P. cmp 7197

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4248  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-1o 6357  df-oadd 6361  df-omul 6362  df-er 6473  df-ec 6475  df-qs 6479  df-ni 7207  df-pli 7208  df-mi 7209  df-lti 7210  df-plpq 7247  df-mpq 7248  df-enq 7250  df-nqqs 7251  df-plqqs 7252  df-mqqs 7253  df-1nqqs 7254  df-rq 7255  df-ltnqqs 7256  df-inp 7369  df-i1p 7370  df-imp 7372

This theorem is referenced by:  recexprlemex  7540