Theorem ltexprlemfu 7610

Description: Lemma for ltexpri 7612. 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 7504	. . . . . 6 |- <P C_ ( P. X. P. )
2	1	brel 4679	. . . . 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 7607	. . . 4 |- ( A <P B -> C e. P. )
6		df-iplp 7467	. . . . 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 7374	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
8	6, 7	genpelvu 7512	. . . 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 7598	. . . . . . . . . . 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 7474	. . . . . . . . . . . . . . 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 7486	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) /\ w e. ( 2nd ` A ) ) -> y <Q w )
19	17, 18	syl3an1 1271	. . . . . . . . . . . . 13 |- ( ( A <P B /\ y e. ( 1st ` A ) /\ w e. ( 2nd ` A ) ) -> y <Q w )
20	19	3com23 1209	. . . . . . . . . . . 12 |- ( ( A <P B /\ w e. ( 2nd ` A ) /\ y e. ( 1st ` A ) ) -> y <Q w )
21	20	3adant2r 1233	. . . . . . . . . . 11 |- ( ( A <P B /\ ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ y e. ( 1st ` A ) ) -> y <Q w )
22	21	3adant2r 1233	. . . . . . . . . 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 1235	. . . . . . . . 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 7399	. . . . . . . . . . . 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 7480	. . . . . . . . . . . . . 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 1016	. . . . . . . . . . 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 7481	. . . . . . . . . . . . . . 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 1017	. . . . . . . . . . 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 7474	. . . . . . . . . . . . . . . 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 7481	. . . . . . . . . . . . . . 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 1017	. . . . . . . . . . 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 7380	. . . . . . . . . . . 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 6044	. . . . . . . . . 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 7474	. . . . . . . . . . . . . 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 7484	. . . . . . . . . . . . 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 1016	. . . . . . . . . 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 1203	. . . . . . 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 1898	. . . . . 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 2254	. . . . 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 2602	. . 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 3162	1 |- ( A <P B -> ( 2nd ` ( A +P. C ) ) C_ ( 2nd ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   {crab 2459   C_ wss 3130   <.cop 3596   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279   +Q cplq 7281   <Q cltq 7284   P.cnp 7290   +P. cpp 7292   <P cltp 7294

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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-iplp 7467  df-iltp 7469

This theorem is referenced by:  ltexpri  7612