Theorem recexprlemss1l 7915

Description: The lower cut of A .P. B is a subset of the lower cut of one. Lemma for recexpr 7918. (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 7912	. . . . 5 |- ( A e. P. -> B e. P. )
3		df-imp 7749	. . . . . 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 7656	. . . . . 6 |- ( ( f e. Q. /\ g e. Q. ) -> ( f .Q g ) e. Q. )
5	3, 4	genpelvl 7792	. . . . 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 7902	. . . . . . . 8 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
8		ltrelnq 7645	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
9	8	brel 4784	. . . . . . . . . . . . 13 |- ( q <Q y -> ( q e. Q. /\ y e. Q. ) )
10	9	simprd 114	. . . . . . . . . . . 12 |- ( q <Q y -> y e. Q. )
11		prop 7755	. . . . . . . . . . . . . . . . . 18 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12		elprnql 7761	. . . . . . . . . . . . . . . . . 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 7683	. . . . . . . . . . . . . . . . . 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 7767	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> z <Q ( *Q ` y ) )
19	11, 18	syl3an1 1307	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ z e. ( 1st ` A ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> z <Q ( *Q ` y ) )
20	19	3expia 1232	. . . . . . . . . . . . . . . . 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 7683	. . . . . . . . . . . . . . . . . . . . 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 7663	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
26		recidnq 7673	. . . . . . . . . . . . . . . . . . . . 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 4106	. . . . . . . . . . . . . . . . . . 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 7678	. . . . . . . . . . . . . . 15 |- <Q Or Q.
35	34, 8	sotri 5139	. . . . . . . . . . . . . 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 1867	. . . . . . . 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 4096	. . . . . . . 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 2658	. . . 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 7835	. . . 4 |- ( 1st ` 1P ) = { w | w <Q 1Q }
50	49	abeq2i 2342	. . 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 3234	1 |- ( A e. P. -> ( 1st ` ( A .P. B ) ) C_ ( 1st ` 1P ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   {cab 2217   E.wrex 2512   C_ wss 3201   <.cop 3676   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   1stc1st 6310   2ndc2nd 6311   Q.cnq 7560   1Qc1q 7561   .Q cmq 7563   *Qcrq 7564   <Q cltq 7565   P.cnp 7571   1Pc1p 7572   .P. cmp 7574

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-inp 7746  df-i1p 7747  df-imp 7749

This theorem is referenced by:  recexprlemex  7917