Theorem recexprlemss1u 7634

Description: The upper cut of A .P. B is a subset of the upper cut of one. Lemma for recexpr 7636. (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
recexprlemss1u	|- ( A e. P. -> ( 2nd ` ( A .P. B ) ) C_ ( 2nd ` 1P ) )

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

Proof of Theorem recexprlemss1u

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 7630	. . . . 5 |- ( A e. P. -> B e. P. )
3		df-imp 7467	. . . . . 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 7374	. . . . . 6 |- ( ( f e. Q. /\ g e. Q. ) -> ( f .Q g ) e. Q. )
5	3, 4	genpelvu 7511	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( w e. ( 2nd ` ( A .P. B ) ) <-> E. z e. ( 2nd ` A ) E. q e. ( 2nd ` B ) w = ( z .Q q ) ) )
6	2, 5	mpdan 421	. . . 4 |- ( A e. P. -> ( w e. ( 2nd ` ( A .P. B ) ) <-> E. z e. ( 2nd ` A ) E. q e. ( 2nd ` B ) w = ( z .Q q ) ) )
7	1	recexprlemelu 7621	. . . . . . . 8 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
8		ltrelnq 7363	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
9	8	brel 4678	. . . . . . . . . . . . 13 |- ( y <Q q -> ( y e. Q. /\ q e. Q. ) )
10	9	simpld 112	. . . . . . . . . . . 12 |- ( y <Q q -> y e. Q. )
11		prop 7473	. . . . . . . . . . . . . . . . . 18 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12		elprnqu 7480	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 2nd ` A ) ) -> z e. Q. )
13	11, 12	sylan 283	. . . . . . . . . . . . . . . . 17 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> z e. Q. )
14		ltmnqi 7401	. . . . . . . . . . . . . . . . . 18 |- ( ( y <Q q /\ z e. Q. ) -> ( z .Q y ) <Q ( z .Q q ) )
15	14	expcom 116	. . . . . . . . . . . . . . . . 17 |- ( z e. Q. -> ( y <Q q -> ( z .Q y ) <Q ( z .Q q ) ) )
16	13, 15	syl 14	. . . . . . . . . . . . . . . 16 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( y <Q q -> ( z .Q y ) <Q ( z .Q q ) ) )
17	16	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ z e. ( 2nd ` A ) ) /\ y e. Q. ) -> ( y <Q q -> ( z .Q y ) <Q ( z .Q q ) ) )
18		prltlu 7485	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( *Q ` y ) e. ( 1st ` A ) /\ z e. ( 2nd ` A ) ) -> ( *Q ` y ) <Q z )
19	11, 18	syl3an1 1271	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. P. /\ ( *Q ` y ) e. ( 1st ` A ) /\ z e. ( 2nd ` A ) ) -> ( *Q ` y ) <Q z )
20	19	3com23 1209	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ z e. ( 2nd ` A ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> ( *Q ` y ) <Q z )
21	20	3expia 1205	. . . . . . . . . . . . . . . . 17 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( ( *Q ` y ) e. ( 1st ` A ) -> ( *Q ` y ) <Q z ) )
22	21	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. P. /\ z e. ( 2nd ` A ) ) /\ y e. Q. ) -> ( ( *Q ` y ) e. ( 1st ` A ) -> ( *Q ` y ) <Q z ) )
23		ltmnqi 7401	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( *Q ` y ) <Q z /\ y e. Q. ) -> ( y .Q ( *Q ` y ) ) <Q ( y .Q z ) )
24	23	expcom 116	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. Q. -> ( ( *Q ` y ) <Q z -> ( y .Q ( *Q ` y ) ) <Q ( y .Q z ) ) )
25	24	adantr 276	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. Q. /\ z e. Q. ) -> ( ( *Q ` y ) <Q z -> ( y .Q ( *Q ` y ) ) <Q ( y .Q z ) ) )
26		recidnq 7391	. . . . . . . . . . . . . . . . . . . . 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		mulcomnqg 7381	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
29	27, 28	breq12d 4016	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. Q. /\ z e. Q. ) -> ( ( y .Q ( *Q ` y ) ) <Q ( y .Q z ) <-> 1Q <Q ( z .Q y ) ) )
30	25, 29	sylibd 149	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. Q. /\ z e. Q. ) -> ( ( *Q ` y ) <Q z -> 1Q <Q ( z .Q y ) ) )
31	30	ancoms 268	. . . . . . . . . . . . . . . . 17 |- ( ( z e. Q. /\ y e. Q. ) -> ( ( *Q ` y ) <Q z -> 1Q <Q ( z .Q y ) ) )
32	13, 31	sylan 283	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. P. /\ z e. ( 2nd ` A ) ) /\ y e. Q. ) -> ( ( *Q ` y ) <Q z -> 1Q <Q ( z .Q y ) ) )
33	22, 32	syld 45	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ z e. ( 2nd ` A ) ) /\ y e. Q. ) -> ( ( *Q ` y ) e. ( 1st ` A ) -> 1Q <Q ( z .Q y ) ) )
34	17, 33	anim12d 335	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ z e. ( 2nd ` A ) ) /\ y e. Q. ) -> ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> ( ( z .Q y ) <Q ( z .Q q ) /\ 1Q <Q ( z .Q y ) ) ) )
35		ltsonq 7396	. . . . . . . . . . . . . . . 16 |- <Q Or Q.
36	35, 8	sotri 5024	. . . . . . . . . . . . . . 15 |- ( ( 1Q <Q ( z .Q y ) /\ ( z .Q y ) <Q ( z .Q q ) ) -> 1Q <Q ( z .Q q ) )
37	36	ancoms 268	. . . . . . . . . . . . . 14 |- ( ( ( z .Q y ) <Q ( z .Q q ) /\ 1Q <Q ( z .Q y ) ) -> 1Q <Q ( z .Q q ) )
38	34, 37	syl6 33	. . . . . . . . . . . . 13 |- ( ( ( A e. P. /\ z e. ( 2nd ` A ) ) /\ y e. Q. ) -> ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> 1Q <Q ( z .Q q ) ) )
39	38	exp4b 367	. . . . . . . . . . . 12 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( y e. Q. -> ( y <Q q -> ( ( *Q ` y ) e. ( 1st ` A ) -> 1Q <Q ( z .Q q ) ) ) ) )
40	10, 39	syl5 32	. . . . . . . . . . 11 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( y <Q q -> ( y <Q q -> ( ( *Q ` y ) e. ( 1st ` A ) -> 1Q <Q ( z .Q q ) ) ) ) )
41	40	pm2.43d 50	. . . . . . . . . 10 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( y <Q q -> ( ( *Q ` y ) e. ( 1st ` A ) -> 1Q <Q ( z .Q q ) ) ) )
42	41	impd 254	. . . . . . . . 9 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> 1Q <Q ( z .Q q ) ) )
43	42	exlimdv 1819	. . . . . . . 8 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> 1Q <Q ( z .Q q ) ) )
44	7, 43	biimtrid 152	. . . . . . 7 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( q e. ( 2nd ` B ) -> 1Q <Q ( z .Q q ) ) )
45		breq2 4007	. . . . . . . 8 |- ( w = ( z .Q q ) -> ( 1Q <Q w <-> 1Q <Q ( z .Q q ) ) )
46	45	biimprcd 160	. . . . . . 7 |- ( 1Q <Q ( z .Q q ) -> ( w = ( z .Q q ) -> 1Q <Q w ) )
47	44, 46	syl6 33	. . . . . 6 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( q e. ( 2nd ` B ) -> ( w = ( z .Q q ) -> 1Q <Q w ) ) )
48	47	expimpd 363	. . . . 5 |- ( A e. P. -> ( ( z e. ( 2nd ` A ) /\ q e. ( 2nd ` B ) ) -> ( w = ( z .Q q ) -> 1Q <Q w ) ) )
49	48	rexlimdvv 2601	. . . 4 |- ( A e. P. -> ( E. z e. ( 2nd ` A ) E. q e. ( 2nd ` B ) w = ( z .Q q ) -> 1Q <Q w ) )
50	6, 49	sylbid 150	. . 3 |- ( A e. P. -> ( w e. ( 2nd ` ( A .P. B ) ) -> 1Q <Q w ) )
51		1pru 7554	. . . 4 |- ( 2nd ` 1P ) = { w | 1Q <Q w }
52	51	abeq2i 2288	. . 3 |- ( w e. ( 2nd ` 1P ) <-> 1Q <Q w )
53	50, 52	syl6ibr 162	. 2 |- ( A e. P. -> ( w e. ( 2nd ` ( A .P. B ) ) -> w e. ( 2nd ` 1P ) ) )
54	53	ssrdv 3161	1 |- ( A e. P. -> ( 2nd ` ( A .P. B ) ) C_ ( 2nd ` 1P ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   E.wrex 2456   C_ wss 3129   <.cop 3595   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   1stc1st 6138   2ndc2nd 6139   Q.cnq 7278   1Qc1q 7279   .Q cmq 7281   *Qcrq 7282   <Q cltq 7283   P.cnp 7289   1Pc1p 7290   .P. cmp 7292

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-inp 7464  df-i1p 7465  df-imp 7467

This theorem is referenced by:  recexprlemex  7635