Theorem ltexprlemfl 7976

Description: Lemma for ltexpri 7980. 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 7872	. . . . . 6 |- <P C_ ( P. X. P. )
2	1	brel 4827	. . . . 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 7975	. . . 4 |- ( A <P B -> C e. P. )
6		df-iplp 7835	. . . . 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 7742	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
8	6, 7	genpelvl 7879	. . . 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 415	. . 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 537	. . . . . 6 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> z = ( w +Q u ) )
11	4	ltexprlemell 7965	. . . . . . . . . . 11 |- ( u e. ( 1st ` C ) <-> ( u e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) )
12	11	biimpi 120	. . . . . . . . . 10 |- ( u e. ( 1st ` C ) -> ( u e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q u ) e. ( 1st ` B ) ) ) )
13	12	ad2antlr 493	. . . . . . . . 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 277	. . . . . . . 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 114	. . . . . . 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 7842	. . . . . . . . . . . . . 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 7854	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
19	17, 18	syl3an1 1311	. . . . . . . . . . . 12 |- ( ( A <P B /\ w e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
20	19	3adant2r 1264	. . . . . . . . . . 11 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
21	20	3adant2r 1264	. . . . . . . . . 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 1266	. . . . . . . . 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 7767	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
24	23	adantl 277	. . . . . . . . . . 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 7732	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
26	25	brel 4827	. . . . . . . . . . . . 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 112	. . . . . . . . . . 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 114	. . . . . . . . . . 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 7842	. . . . . . . . . . . . . . . 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 7848	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ u e. ( 1st ` C ) ) -> u e. Q. )
33	31, 32	sylan 283	. . . . . . . . . . . . . 14 |- ( ( A <P B /\ u e. ( 1st ` C ) ) -> u e. Q. )
34	33	adantrl 482	. . . . . . . . . . . . 13 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) ) -> u e. Q. )
35	34	adantrr 483	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> u e. Q. )
36	35	3adant3 1048	. . . . . . . . . . 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 7748	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
38	37	adantl 277	. . . . . . . . . . 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 6259	. . . . . . . . . 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 114	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
41		prop 7842	. . . . . . . . . . . . . 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 7851	. . . . . . . . . . . . 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 283	. . . . . . . . . . . 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 482	. . . . . . . . . . 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 1047	. . . . . . . . . 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 150	. . . . . . . . 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 1234	. . . . . . 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 1954	. . . . . 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 2315	. . . . 5 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> z e. ( 1st ` B ) )
52	51	expr 375	. . . 4 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) ) -> ( z = ( w +Q u ) -> z e. ( 1st ` B ) ) )
53	52	rexlimdvva 2676	. . 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 150	. 2 |- ( A <P B -> ( z e. ( 1st ` ( A +P. C ) ) -> z e. ( 1st ` B ) ) )
55	54	ssrdv 3254	1 |- ( A <P B -> ( 1st ` ( A +P. C ) ) C_ ( 1st ` B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   {crab 2532   C_ wss 3220   <.cop 3712   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   +Q cplq 7649   <Q cltq 7652   P.cnp 7658   +P. cpp 7660   <P cltp 7662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-iplp 7835  df-iltp 7837

This theorem is used by:  ltexpri  7980