Theorem ltexprlemfl 7229

Description: Lemma for ltexpri 7233. One directon of our result for lower 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
ltexprlemfl	|- ( A <P B -> ( 1st ` ( A +P. C ) ) C_ ( 1st ` B ) )

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

Proof of Theorem ltexprlemfl

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

Step	Hyp	Ref	Expression
1		ltrelpr 7125	. . . . . 6 |- <P C_ ( P. X. P. )
2	1	brel 4503	. . . . 5 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3	2	simpld 111	. . . 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 7228	. . . 4 |- ( A <P B -> C e. P. )
6		df-iplp 7088	. . . . 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 6995	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
8	6, 7	genpelvl 7132	. . . 4 |- ( ( A e. P. /\ C e. P. ) -> ( z e. ( 1st ` ( A +P. C ) ) <-> E. w e. ( 1st ` A ) E. u e. ( 1st ` C ) z = ( w +Q u ) ) )
9	3, 5, 8	syl2anc 404	. . 3 |- ( A <P B -> ( z e. ( 1st ` ( A +P. C ) ) <-> E. w e. ( 1st ` A ) E. u e. ( 1st ` C ) z = ( w +Q u ) ) )
10		simprr 500	. . . . . 6 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> z = ( w +Q u ) )
11	4	ltexprlemell 7218	. . . . . . . . . . 11 |- ( u e. ( 1st ` C ) <-> ( u e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) )
12	11	biimpi 119	. . . . . . . . . 10 |- ( u e. ( 1st ` C ) -> ( u e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) )
13	12	ad2antlr 474	. . . . . . . . 9 |- ( ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) -> ( u e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) )
14	13	adantl 272	. . . . . . . 8 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> ( u e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) )
15	14	simprd 113	. . . . . . 7 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> E. y ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) )
16		prop 7095	. . . . . . . . . . . . . 14 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
17	3, 16	syl 14	. . . . . . . . . . . . 13 |- ( A <P B -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
18		prltlu 7107	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
19	17, 18	syl3an1 1208	. . . . . . . . . . . 12 |- ( ( A <P B /\ w e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
20	19	3adant2r 1170	. . . . . . . . . . 11 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
21	20	3adant2r 1170	. . . . . . . . . 10 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
22	21	3adant3r 1172	. . . . . . . . 9 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) -> w <Q y )
23		ltanqg 7020	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
24	23	adantl 272	. . . . . . . . . . 11 |- ( ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
25		ltrelnq 6985	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
26	25	brel 4503	. . . . . . . . . . . . 13 |- ( w <Q y -> ( w e. Q. /\ y e. Q. ) )
27	22, 26	syl 14	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) -> ( w e. Q. /\ y e. Q. ) )
28	27	simpld 111	. . . . . . . . . . 11 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) -> w e. Q. )
29	27	simprd 113	. . . . . . . . . . 11 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) -> y e. Q. )
30		prop 7095	. . . . . . . . . . . . . . . 16 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
31	5, 30	syl 14	. . . . . . . . . . . . . . 15 |- ( A <P B -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
32		elprnql 7101	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ u e. ( 1st ` C ) ) -> u e. Q. )
33	31, 32	sylan 278	. . . . . . . . . . . . . 14 |- ( ( A <P B /\ u e. ( 1st ` C ) ) -> u e. Q. )
34	33	adantrl 463	. . . . . . . . . . . . 13 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) ) -> u e. Q. )
35	34	adantrr 464	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> u e. Q. )
36	35	3adant3 964	. . . . . . . . . . 11 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) -> u e. Q. )
37		addcomnqg 7001	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
38	37	adantl 272	. . . . . . . . . . 11 |- ( ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
39	24, 28, 29, 36, 38	caovord2d 5828	. . . . . . . . . 10 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) -> ( w <Q y <-> ( w +Q u ) <Q ( y +Q u ) ) )
40	2	simprd 113	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
41		prop 7095	. . . . . . . . . . . . . 14 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
42	40, 41	syl 14	. . . . . . . . . . . . 13 |- ( A <P B -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
43		prcdnql 7104	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q u ) e. ( 1st ` B ) ) -> ( ( w +Q u ) <Q ( y +Q u ) -> ( w +Q u ) e. ( 1st ` B ) ) )
44	42, 43	sylan 278	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( y +Q u ) e. ( 1st ` B ) ) -> ( ( w +Q u ) <Q ( y +Q u ) -> ( w +Q u ) e. ( 1st ` B ) ) )
45	44	adantrl 463	. . . . . . . . . . 11 |- ( ( A <P B /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) -> ( ( w +Q u ) <Q ( y +Q u ) -> ( w +Q u ) e. ( 1st ` B ) ) )
46	45	3adant2 963	. . . . . . . . . 10 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) -> ( ( w +Q u ) <Q ( y +Q u ) -> ( w +Q u ) e. ( 1st ` B ) ) )
47	39, 46	sylbid 149	. . . . . . . . 9 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) -> ( w <Q y -> ( w +Q u ) e. ( 1st ` B ) ) )
48	22, 47	mpd 13	. . . . . . . 8 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) -> ( w +Q u ) e. ( 1st ` B ) )
49	48	3expa 1144	. . . . . . 7 |- ( ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) -> ( w +Q u ) e. ( 1st ` B ) )
50	15, 49	exlimddv 1827	. . . . . 6 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> ( w +Q u ) e. ( 1st ` B ) )
51	10, 50	eqeltrd 2165	. . . . 5 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> z e. ( 1st ` B ) )
52	51	expr 368	. . . 4 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) ) -> ( z = ( w +Q u ) -> z e. ( 1st ` B ) ) )
53	52	rexlimdvva 2497	. . 3 |- ( A <P B -> ( E. w e. ( 1st ` A ) E. u e. ( 1st ` C ) z = ( w +Q u ) -> z e. ( 1st ` B ) ) )
54	9, 53	sylbid 149	. 2 |- ( A <P B -> ( z e. ( 1st ` ( A +P. C ) ) -> z e. ( 1st ` B ) ) )
55	54	ssrdv 3032	1 |- ( A <P B -> ( 1st ` ( A +P. C ) ) C_ ( 1st ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 925   = wceq 1290   E.wex 1427   e. wcel 1439   E.wrex 2361   {crab 2364   C_ wss 3000   <.cop 3453   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   1stc1st 5923   2ndc2nd 5924   Q.cnq 6900   +Q cplq 6902   <Q cltq 6905   P.cnp 6911   +P. cpp 6913   <P cltp 6915

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 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

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 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-2o 6196  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-enq0 7044  df-nq0 7045  df-0nq0 7046  df-plq0 7047  df-mq0 7048  df-inp 7086  df-iplp 7088  df-iltp 7090

This theorem is referenced by:  ltexpri  7233