Theorem ltexprlemfl 7757

Description: Lemma for ltexpri 7761. 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 7653	. . . . . 6 |- <P C_ ( P. X. P. )
2	1	brel 4745	. . . . 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 7756	. . . 4 |- ( A <P B -> C e. P. )
6		df-iplp 7616	. . . . 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 7523	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
8	6, 7	genpelvl 7660	. . . 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 411	. . 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 531	. . . . . 6 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> z = ( w +Q u ) )
11	4	ltexprlemell 7746	. . . . . . . . . . 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 489	. . . . . . . . 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 7623	. . . . . . . . . . . . . 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 7635	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
19	17, 18	syl3an1 1283	. . . . . . . . . . . 12 |- ( ( A <P B /\ w e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
20	19	3adant2r 1236	. . . . . . . . . . 11 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ y e. ( 2nd ` A ) ) -> w <Q y )
21	20	3adant2r 1236	. . . . . . . . . 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 1238	. . . . . . . . 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 7548	. . . . . . . . . . . 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 7513	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
26	25	brel 4745	. . . . . . . . . . . . 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 7623	. . . . . . . . . . . . . . . 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 7629	. . . . . . . . . . . . . . 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 478	. . . . . . . . . . . . 13 |- ( ( A <P B /\ ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) ) -> u e. Q. )
35	34	adantrr 479	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( ( w e. ( 1st ` A ) /\ u e. ( 1st ` C ) ) /\ z = ( w +Q u ) ) ) -> u e. Q. )
36	35	3adant3 1020	. . . . . . . . . . 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 7529	. . . . . . . . . . . 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 6139	. . . . . . . . . 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 7623	. . . . . . . . . . . . . 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 7632	. . . . . . . . . . . . 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 478	. . . . . . . . . . 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 1019	. . . . . . . . . 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 1206	. . . . . . 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 1923	. . . . . 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 2284	. . . . 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 2633	. . 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 3207	1 |- ( A <P B -> ( 1st ` ( A +P. C ) ) C_ ( 1st ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   E.wex 1516   e. wcel 2178   E.wrex 2487   {crab 2490   C_ wss 3174   <.cop 3646   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   1stc1st 6247   2ndc2nd 6248   Q.cnq 7428   +Q cplq 7430   <Q cltq 7433   P.cnp 7439   +P. cpp 7441   <P cltp 7443

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614  df-iplp 7616  df-iltp 7618

This theorem is referenced by:  ltexpri  7761