Theorem ltexprlemfl 7571

Description: Lemma for ltexpri 7575. One direction 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 7467	. . . . . 6 |- <P C_ ( P. X. P. )
2	1	brel 4663	. . . . 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 7570	. . . 4 |- ( A <P B -> C e. P. )
6		df-iplp 7430	. . . . 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 7337	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
8	6, 7	genpelvl 7474	. . . 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 409	. . 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 527	. . . . . 6 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> z = ( w +Q u ) )
11	4	ltexprlemell 7560	. . . . . . . . . . 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 486	. . . . . . . . 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 275	. . . . . . . 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 7437	. . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
19	17, 18	syl3an1 1266	. . . . . . . . . . . 12 |- ( ( A <P B /\ w e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
20	19	3adant2r 1228	. . . . . . . . . . 11 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
21	20	3adant2r 1228	. . . . . . . . . 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 1230	. . . . . . . . 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 7362	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
24	23	adantl 275	. . . . . . . . . . 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 7327	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
26	25	brel 4663	. . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . . . 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 7443	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ u e. ( 1st ` C ) ) -> u e. Q. )
33	31, 32	sylan 281	. . . . . . . . . . . . . 14 |- ( ( A <P B /\ u e. ( 1st ` C ) ) -> u e. Q. )
34	33	adantrl 475	. . . . . . . . . . . . 13 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) ) -> u e. Q. )
35	34	adantrr 476	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> u e. Q. )
36	35	3adant3 1012	. . . . . . . . . . 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 7343	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
38	37	adantl 275	. . . . . . . . . . 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 6022	. . . . . . . . . 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 7437	. . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . 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 281	. . . . . . . . . . . 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 475	. . . . . . . . . . 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 1011	. . . . . . . . . 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 1198	. . . . . . 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 1891	. . . . . 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 2247	. . . . 5 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> z e. ( 1st ` B ) )
52	51	expr 373	. . . 4 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) ) -> ( z = ( w +Q u ) -> z e. ( 1st ` B ) ) )
53	52	rexlimdvva 2595	. . 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 3153	1 |- ( A <P B -> ( 1st ` ( A +P. C ) ) C_ ( 1st ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   E.wex 1485   e. wcel 2141   E.wrex 2449   {crab 2452   C_ wss 3121   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   +Q cplq 7244   <Q cltq 7247   P.cnp 7253   +P. cpp 7255   <P cltp 7257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432

This theorem is referenced by:  ltexpri  7575