Theorem recexprlemss1l 8002

Description: The lower cut of A .P. B is a subset of the lower cut of one. Lemma for recexpr 8005. (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 7999	. . . . 5 |- ( A e. P. -> B e. P. )
3		df-imp 7836	. . . . . 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 7743	. . . . . 6 |- ( ( f e. Q. /\ g e. Q. ) -> ( f .Q g ) e. Q. )
5	3, 4	genpelvl 7879	. . . . 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 425	. . . 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 7989	. . . . . . . 8 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
8		ltrelnq 7732	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
9	8	brel 4827	. . . . . . . . . . . . 13 |- ( q <Q y -> ( q e. Q. /\ y e. Q. ) )
10	9	simprd 114	. . . . . . . . . . . 12 |- ( q <Q y -> y e. Q. )
11		prop 7842	. . . . . . . . . . . . . . . . . 18 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12		elprnql 7848	. . . . . . . . . . . . . . . . . 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 7770	. . . . . . . . . . . . . . . . . 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 7854	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> z <Q ( *Q ` y ) )
19	11, 18	syl3an1 1311	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ z e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> z <Q ( *Q ` y ) )
20	19	3expia 1236	. . . . . . . . . . . . . . . . 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 7770	. . . . . . . . . . . . . . . . . . . . 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 7750	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
26		recidnq 7760	. . . . . . . . . . . . . . . . . . . . 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 4143	. . . . . . . . . . . . . . . . . . 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 7765	. . . . . . . . . . . . . . 15 |- <Q Or Q.
35	34, 8	sotri 5183	. . . . . . . . . . . . . 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 1872	. . . . . . . 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 4133	. . . . . . . 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 2675	. . . 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 7922	. . . 4 |- ( 1st ` 1P ) = { w | w <Q 1Q }
50	49	abeq2i 2349	. . 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 3254	1 |- ( A e. P. -> ( 1st ` ( A .P. B ) ) C_ ( 1st ` 1P ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   E.wrex 2529   C_ wss 3220   <.cop 3712   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   1Qc1q 7648   .Q cmq 7650   *Qcrq 7651   <Q cltq 7652   P.cnp 7658   1Pc1p 7659   .P. cmp 7661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-inp 7833  df-i1p 7834  df-imp 7836

This theorem is used by:  recexprlemex  8004