Theorem ltexprlemfu 7809

Description: Lemma for ltexpri 7811. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.

Assertion

Ref	Expression
ltexprlemfu	|- ( A <P B -> ( 2nd ` ( A +P. C ) ) C_ ( 2nd ` B ) )

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

Proof of Theorem ltexprlemfu

Dummy variables z w u f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelpr 7703	. . . . . 6 |- <P C_ ( P. X. P. )
2	1	brel 4771	. . . . 5 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3	2	simpld 112	. . . 4 |- ( A <P B -> A e. P. )
4		ltexprlem.1	. . . . 5 |- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.
5	4	ltexprlempr 7806	. . . 4 |- ( A <P B -> C e. P. )
6		df-iplp 7666	. . . . 5 |- +P. = ( z e. P. , y e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` z ) /\ h e. ( 1st ` y ) /\ f = ( g +Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` z ) /\ h e. ( 2nd ` y ) /\ f = ( g +Q h ) ) } >. )
7		addclnq 7573	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
8	6, 7	genpelvu 7711	. . . 4 |- ( ( A e. P. /\ C e. P. ) -> ( z e. ( 2nd ` ( A +P. C ) ) <-> E. w e. ( 2nd ` A ) E. u e. ( 2nd ` C ) z = ( w +Q u ) ) )
9	3, 5, 8	syl2anc 411	. . 3 |- ( A <P B -> ( z e. ( 2nd ` ( A +P. C ) ) <-> E. w e. ( 2nd ` A ) E. u e. ( 2nd ` C ) z = ( w +Q u ) ) )
10		simprr 531	. . . . . 6 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> z = ( w +Q u ) )
11	4	ltexprlemelu 7797	. . . . . . . . . . 11 |- ( u e. ( 2nd ` C ) <-> ( u e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) )
12	11	biimpi 120	. . . . . . . . . 10 |- ( u e. ( 2nd ` C ) -> ( u e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) )
13	12	ad2antlr 489	. . . . . . . . 9 |- ( ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) -> ( u e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) )
14	13	simprd 114	. . . . . . . 8 |- ( ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) -> E. y ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) )
15	14	adantl 277	. . . . . . 7 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> E. y ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) )
16		prop 7673	. . . . . . . . . . . . . . 15 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
17	3, 16	syl 14	. . . . . . . . . . . . . 14 |- ( A <P B -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
18		prltlu 7685	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) /\ w e. ( 2nd ` A ) ) -> y <Q w )
19	17, 18	syl3an1 1304	. . . . . . . . . . . . 13 |- ( ( A <P B /\ y e. ( 1st ` A ) /\ w e. ( 2nd ` A ) ) -> y <Q w )
20	19	3com23 1233	. . . . . . . . . . . 12 |- ( ( A <P B /\ w e. ( 2nd ` A ) /\ y e. ( 1st ` A ) ) -> y <Q w )
21	20	3adant2r 1257	. . . . . . . . . . 11 |- ( ( A <P B /\ ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ y e. ( 1st ` A ) ) -> y <Q w )
22	21	3adant2r 1257	. . . . . . . . . 10 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ y e. ( 1st ` A ) ) -> y <Q w )
23	22	3adant3r 1259	. . . . . . . . 9 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> y <Q w )
24		ltanqg 7598	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
25	24	adantl 277	. . . . . . . . . . 11 |- ( ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
26		elprnql 7679	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
27	17, 26	sylan 283	. . . . . . . . . . . . 13 |- ( ( A <P B /\ y e. ( 1st ` A ) ) -> y e. Q. )
28	27	adantrr 479	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> y e. Q. )
29	28	3adant2 1040	. . . . . . . . . . 11 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> y e. Q. )
30		elprnqu 7680	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. ( 2nd ` A ) ) -> w e. Q. )
31	17, 30	sylan 283	. . . . . . . . . . . . . 14 |- ( ( A <P B /\ w e. ( 2nd ` A ) ) -> w e. Q. )
32	31	adantrr 479	. . . . . . . . . . . . 13 |- ( ( A <P B /\ ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) ) -> w e. Q. )
33	32	adantrr 479	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> w e. Q. )
34	33	3adant3 1041	. . . . . . . . . . 11 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> w e. Q. )
35		prop 7673	. . . . . . . . . . . . . . . 16 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
36	5, 35	syl 14	. . . . . . . . . . . . . . 15 |- ( A <P B -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
37		elprnqu 7680	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ u e. ( 2nd ` C ) ) -> u e. Q. )
38	36, 37	sylan 283	. . . . . . . . . . . . . 14 |- ( ( A <P B /\ u e. ( 2nd ` C ) ) -> u e. Q. )
39	38	adantrl 478	. . . . . . . . . . . . 13 |- ( ( A <P B /\ ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) ) -> u e. Q. )
40	39	adantrr 479	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> u e. Q. )
41	40	3adant3 1041	. . . . . . . . . . 11 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> u e. Q. )
42		addcomnqg 7579	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
43	42	adantl 277	. . . . . . . . . . 11 |- ( ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
44	25, 29, 34, 41, 43	caovord2d 6181	. . . . . . . . . 10 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( y <Q w <-> ( y +Q u ) <Q ( w +Q u ) ) )
45	2	simprd 114	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
46		prop 7673	. . . . . . . . . . . . . 14 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
47	45, 46	syl 14	. . . . . . . . . . . . 13 |- ( A <P B -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
48		prcunqu 7683	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q u ) e. ( 2nd ` B ) ) -> ( ( y +Q u ) <Q ( w +Q u ) -> ( w +Q u ) e. ( 2nd ` B ) ) )
49	47, 48	sylan 283	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( y +Q u ) e. ( 2nd ` B ) ) -> ( ( y +Q u ) <Q ( w +Q u ) -> ( w +Q u ) e. ( 2nd ` B ) ) )
50	49	adantrl 478	. . . . . . . . . . 11 |- ( ( A <P B /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( ( y +Q u ) <Q ( w +Q u ) -> ( w +Q u ) e. ( 2nd ` B ) ) )
51	50	3adant2 1040	. . . . . . . . . 10 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( ( y +Q u ) <Q ( w +Q u ) -> ( w +Q u ) e. ( 2nd ` B ) ) )
52	44, 51	sylbid 150	. . . . . . . . 9 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( y <Q w -> ( w +Q u ) e. ( 2nd ` B ) ) )
53	23, 52	mpd 13	. . . . . . . 8 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( w +Q u ) e. ( 2nd ` B ) )
54	53	3expa 1227	. . . . . . 7 |- ( ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( w +Q u ) e. ( 2nd ` B ) )
55	15, 54	exlimddv 1945	. . . . . 6 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> ( w +Q u ) e. ( 2nd ` B ) )
56	10, 55	eqeltrd 2306	. . . . 5 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> z e. ( 2nd ` B ) )
57	56	expr 375	. . . 4 |- ( ( A <P B /\ ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) ) -> ( z = ( w +Q u ) -> z e. ( 2nd ` B ) ) )
58	57	rexlimdvva 2656	. . 3 |- ( A <P B -> ( E. w e. ( 2nd ` A ) E. u e. ( 2nd ` C ) z = ( w +Q u ) -> z e. ( 2nd ` B ) ) )
59	9, 58	sylbid 150	. 2 |- ( A <P B -> ( z e. ( 2nd ` ( A +P. C ) ) -> z e. ( 2nd ` B ) ) )
60	59	ssrdv 3230	1 |- ( A <P B -> ( 2nd ` ( A +P. C ) ) C_ ( 2nd ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   {crab 2512   C_ wss 3197   <.cop 3669   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   1stc1st 6290   2ndc2nd 6291   Q.cnq 7478   +Q cplq 7480   <Q cltq 7483   P.cnp 7489   +P. cpp 7491   <P cltp 7493

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-pli 7503  df-mi 7504  df-lti 7505  df-plpq 7542  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-plqqs 7547  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-enq0 7622  df-nq0 7623  df-0nq0 7624  df-plq0 7625  df-mq0 7626  df-inp 7664  df-iplp 7666  df-iltp 7668

This theorem is referenced by:  ltexpri  7811