Theorem ltexprlemfl 7441

Description: Lemma for ltexpri 7445. 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 7337	. . . . . 6 |- <P C_ ( P. X. P. )
2	1	brel 4599	. . . . 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 7440	. . . 4 |- ( A <P B -> C e. P. )
6		df-iplp 7300	. . . . 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 7207	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
8	6, 7	genpelvl 7344	. . . 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 522	. . . . . 6 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> z = ( w +Q u ) )
11	4	ltexprlemell 7430	. . . . . . . . . . 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 481	. . . . . . . . 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 7307	. . . . . . . . . . . . . 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 7319	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
19	17, 18	syl3an1 1250	. . . . . . . . . . . 12 |- ( ( A <P B /\ w e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
20	19	3adant2r 1212	. . . . . . . . . . 11 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
21	20	3adant2r 1212	. . . . . . . . . 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 1214	. . . . . . . . 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 7232	. . . . . . . . . . . 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 7197	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
26	25	brel 4599	. . . . . . . . . . . . 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 7307	. . . . . . . . . . . . . . . 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 7313	. . . . . . . . . . . . . . 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 470	. . . . . . . . . . . . 13 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) ) -> u e. Q. )
35	34	adantrr 471	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> u e. Q. )
36	35	3adant3 1002	. . . . . . . . . . 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 7213	. . . . . . . . . . . 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 5948	. . . . . . . . . 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 7307	. . . . . . . . . . . . . 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 7316	. . . . . . . . . . . . 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 470	. . . . . . . . . . 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 1001	. . . . . . . . . 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 1182	. . . . . . 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 1871	. . . . . 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 2217	. . . . 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 2560	. . 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 3108	1 |- ( A <P B -> ( 1st ` ( A +P. C ) ) C_ ( 1st ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1332   E.wex 1469   e. wcel 1481   E.wrex 2418   {crab 2421   C_ wss 3076   <.cop 3535   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   1stc1st 6044   2ndc2nd 6045   Q.cnq 7112   +Q cplq 7114   <Q cltq 7117   P.cnp 7123   +P. cpp 7125   <P cltp 7127

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300  df-iltp 7302

This theorem is referenced by:  ltexpri  7445