Theorem recexprlemss1u 7846

Description: The upper cut of A .P. B is a subset of the upper cut of one. Lemma for recexpr 7848. (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 7842	. . . . 5 |- ( A e. P. -> B e. P. )
3		df-imp 7679	. . . . . 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 7586	. . . . . 6 |- ( ( f e. Q. /\ g e. Q. ) -> ( f .Q g ) e. Q. )
5	3, 4	genpelvu 7723	. . . . 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 7833	. . . . . . . 8 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
8		ltrelnq 7575	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
9	8	brel 4776	. . . . . . . . . . . . 13 |- ( y <Q q -> ( y e. Q. /\ q e. Q. ) )
10	9	simpld 112	. . . . . . . . . . . 12 |- ( y <Q q -> y e. Q. )
11		prop 7685	. . . . . . . . . . . . . . . . . 18 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12		elprnqu 7692	. . . . . . . . . . . . . . . . . 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 7613	. . . . . . . . . . . . . . . . . 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 7697	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( *Q ` y ) e. ( 1st ` A ) /\ z e. ( 2nd ` A ) ) -> ( *Q ` y ) <Q z )
19	11, 18	syl3an1 1304	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. P. /\ ( *Q ` y ) e. ( 1st ` A ) /\ z e. ( 2nd ` A ) ) -> ( *Q ` y ) <Q z )
20	19	3com23 1233	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ z e. ( 2nd ` A ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> ( *Q ` y ) <Q z )
21	20	3expia 1229	. . . . . . . . . . . . . . . . 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 7613	. . . . . . . . . . . . . . . . . . . . 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 7603	. . . . . . . . . . . . . . . . . . . . 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 7593	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
29	27, 28	breq12d 4099	. . . . . . . . . . . . . . . . . . 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 7608	. . . . . . . . . . . . . . . 16 |- <Q Or Q.
36	35, 8	sotri 5130	. . . . . . . . . . . . . . 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 1865	. . . . . . . 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 4090	. . . . . . . 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 2655	. . . 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 7766	. . . 4 |- ( 2nd ` 1P ) = { w | 1Q <Q w }
52	51	abeq2i 2340	. . 3 |- ( w e. ( 2nd ` 1P ) <-> 1Q <Q w )
53	50, 52	imbitrrdi 162	. 2 |- ( A e. P. -> ( w e. ( 2nd ` ( A .P. B ) ) -> w e. ( 2nd ` 1P ) ) )
54	53	ssrdv 3231	1 |- ( A e. P. -> ( 2nd ` ( A .P. B ) ) C_ ( 2nd ` 1P ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   E.wrex 2509   C_ wss 3198   <.cop 3670   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   1stc1st 6296   2ndc2nd 6297   Q.cnq 7490   1Qc1q 7491   .Q cmq 7493   *Qcrq 7494   <Q cltq 7495   P.cnp 7501   1Pc1p 7502   .P. cmp 7504

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-inp 7676  df-i1p 7677  df-imp 7679

This theorem is referenced by:  recexprlemex  7847