Theorem recexprlemss1u 7258

Description: The upper cut of A .P. B is a subset of the upper cut of one. Lemma for recexpr 7260. (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 7254	. . . . 5 |- ( A e. P. -> B e. P. )
3		df-imp 7091	. . . . . 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 6998	. . . . . 6 |- ( ( f e. Q. /\ g e. Q. ) -> ( f .Q g ) e. Q. )
5	3, 4	genpelvu 7135	. . . . 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 413	. . . 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 7245	. . . . . . . 8 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
8		ltrelnq 6987	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
9	8	brel 4505	. . . . . . . . . . . . 13 |- ( y <Q q -> ( y e. Q. /\ q e. Q. ) )
10	9	simpld 111	. . . . . . . . . . . 12 |- ( y <Q q -> y e. Q. )
11		prop 7097	. . . . . . . . . . . . . . . . . 18 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12		elprnqu 7104	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 2nd ` A ) ) -> z e. Q. )
13	11, 12	sylan 278	. . . . . . . . . . . . . . . . 17 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> z e. Q. )
14		ltmnqi 7025	. . . . . . . . . . . . . . . . . 18 |- ( ( y <Q q /\ z e. Q. ) -> ( z .Q y ) <Q ( z .Q q ) )
15	14	expcom 115	. . . . . . . . . . . . . . . . 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 271	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ z e. ( 2nd ` A ) ) /\ y e. Q. ) -> ( y <Q q -> ( z .Q y ) <Q ( z .Q q ) ) )
18		prltlu 7109	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( *Q ` y ) e. ( 1st ` A ) /\ z e. ( 2nd ` A ) ) -> ( *Q ` y ) <Q z )
19	11, 18	syl3an1 1208	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. P. /\ ( *Q ` y ) e. ( 1st ` A ) /\ z e. ( 2nd ` A ) ) -> ( *Q ` y ) <Q z )
20	19	3com23 1150	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ z e. ( 2nd ` A ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> ( *Q ` y ) <Q z )
21	20	3expia 1146	. . . . . . . . . . . . . . . . 17 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( ( *Q ` y ) e. ( 1st ` A ) -> ( *Q ` y ) <Q z ) )
22	21	adantr 271	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. P. /\ z e. ( 2nd ` A ) ) /\ y e. Q. ) -> ( ( *Q ` y ) e. ( 1st ` A ) -> ( *Q ` y ) <Q z ) )
23		ltmnqi 7025	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( *Q ` y ) <Q z /\ y e. Q. ) -> ( y .Q ( *Q ` y ) ) <Q ( y .Q z ) )
24	23	expcom 115	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. Q. -> ( ( *Q ` y ) <Q z -> ( y .Q ( *Q ` y ) ) <Q ( y .Q z ) ) )
25	24	adantr 271	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. Q. /\ z e. Q. ) -> ( ( *Q ` y ) <Q z -> ( y .Q ( *Q ` y ) ) <Q ( y .Q z ) ) )
26		recidnq 7015	. . . . . . . . . . . . . . . . . . . . 21 |- ( y e. Q. -> ( y .Q ( *Q ` y ) ) = 1Q )
27	26	adantr 271	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q ( *Q ` y ) ) = 1Q )
28		mulcomnqg 7005	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
29	27, 28	breq12d 3866	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. Q. /\ z e. Q. ) -> ( ( y .Q ( *Q ` y ) ) <Q ( y .Q z ) <-> 1Q <Q ( z .Q y ) ) )
30	25, 29	sylibd 148	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. Q. /\ z e. Q. ) -> ( ( *Q ` y ) <Q z -> 1Q <Q ( z .Q y ) ) )
31	30	ancoms 265	. . . . . . . . . . . . . . . . 17 |- ( ( z e. Q. /\ y e. Q. ) -> ( ( *Q ` y ) <Q z -> 1Q <Q ( z .Q y ) ) )
32	13, 31	sylan 278	. . . . . . . . . . . . . . . 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 329	. . . . . . . . . . . . . 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 7020	. . . . . . . . . . . . . . . 16 |- <Q Or Q.
36	35, 8	sotri 4842	. . . . . . . . . . . . . . 15 |- ( ( 1Q <Q ( z .Q y ) /\ ( z .Q y ) <Q ( z .Q q ) ) -> 1Q <Q ( z .Q q ) )
37	36	ancoms 265	. . . . . . . . . . . . . 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 360	. . . . . . . . . . . 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 252	. . . . . . . . 9 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> 1Q <Q ( z .Q q ) ) )
43	42	exlimdv 1748	. . . . . . . 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	syl5bi 151	. . . . . . 7 |- ( ( A e. P. /\ z e. ( 2nd ` A ) ) -> ( q e. ( 2nd ` B ) -> 1Q <Q ( z .Q q ) ) )
45		breq2 3857	. . . . . . . 8 |- ( w = ( z .Q q ) -> ( 1Q <Q w <-> 1Q <Q ( z .Q q ) ) )
46	45	biimprcd 159	. . . . . . 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 356	. . . . 5 |- ( A e. P. -> ( ( z e. ( 2nd ` A ) /\ q e. ( 2nd ` B ) ) -> ( w = ( z .Q q ) -> 1Q <Q w ) ) )
49	48	rexlimdvv 2498	. . . 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 149	. . 3 |- ( A e. P. -> ( w e. ( 2nd ` ( A .P. B ) ) -> 1Q <Q w ) )
51		1pru 7178	. . . 4 |- ( 2nd ` 1P ) = { w | 1Q <Q w }
52	51	abeq2i 2199	. . 3 |- ( w e. ( 2nd ` 1P ) <-> 1Q <Q w )
53	50, 52	syl6ibr 161	. 2 |- ( A e. P. -> ( w e. ( 2nd ` ( A .P. B ) ) -> w e. ( 2nd ` 1P ) ) )
54	53	ssrdv 3034	1 |- ( A e. P. -> ( 2nd ` ( A .P. B ) ) C_ ( 2nd ` 1P ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   E.wex 1427   e. wcel 1439   {cab 2075   E.wrex 2361   C_ wss 3002   <.cop 3455   class class class wbr 3853   ` cfv 5030  (class class class)co 5668   1stc1st 5925   2ndc2nd 5926   Q.cnq 6902   1Qc1q 6903   .Q cmq 6905   *Qcrq 6906   <Q cltq 6907   P.cnp 6913   1Pc1p 6914   .P. cmp 6916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-eprel 4127  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-irdg 6151  df-1o 6197  df-oadd 6201  df-omul 6202  df-er 6308  df-ec 6310  df-qs 6314  df-ni 6926  df-pli 6927  df-mi 6928  df-lti 6929  df-plpq 6966  df-mpq 6967  df-enq 6969  df-nqqs 6970  df-plqqs 6971  df-mqqs 6972  df-1nqqs 6973  df-rq 6974  df-ltnqqs 6975  df-inp 7088  df-i1p 7089  df-imp 7091

This theorem is referenced by:  recexprlemex  7259