Theorem ltexprlemfu 7926

Description: Lemma for ltexpri 7928. One direction of our result for upper 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
ltexprlemfu	|- ( A <P B -> ( 2nd ` ( A +P. C ) ) C_ ( 2nd ` B ) )

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

Proof of Theorem ltexprlemfu

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

Step	Hyp	Ref	Expression
1		ltrelpr 7820	. . . . . 6 |- <P C_ ( P. X. P. )
2	1	brel 4802	. . . . 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 7923	. . . 4 |- ( A <P B -> C e. P. )
6		df-iplp 7783	. . . . 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 7690	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
8	6, 7	genpelvu 7828	. . . 4 |- ( ( A e. P. /\ C e. P. ) -> ( z e. ( 2nd ` ( A +P. C ) ) <-> E. w e. ( 2nd ` A ) E. u e. ( 2nd ` C ) z = ( w +Q u ) ) )
9	3, 5, 8	syl2anc 411	. . 3 |- ( A <P B -> ( z e. ( 2nd ` ( A +P. C ) ) <-> E. w e. ( 2nd ` A ) E. u e. ( 2nd ` C ) z = ( w +Q u ) ) )
10		simprr 533	. . . . . 6 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> z = ( w +Q u ) )
11	4	ltexprlemelu 7914	. . . . . . . . . . 11 |- ( u e. ( 2nd ` C ) <-> ( u e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) )
12	11	biimpi 120	. . . . . . . . . 10 |- ( u e. ( 2nd ` C ) -> ( u e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) )
13	12	ad2antlr 489	. . . . . . . . 9 |- ( ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) -> ( u e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) )
14	13	simprd 114	. . . . . . . 8 |- ( ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) -> E. y ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) )
15	14	adantl 277	. . . . . . 7 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> E. y ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) )
16		prop 7790	. . . . . . . . . . . . . . 15 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
17	3, 16	syl 14	. . . . . . . . . . . . . 14 |- ( A <P B -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
18		prltlu 7802	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) /\ w e. ( 2nd ` A ) ) -> y <Q w )
19	17, 18	syl3an1 1307	. . . . . . . . . . . . 13 |- ( ( A <P B /\ y e. ( 1st ` A ) /\ w e. ( 2nd ` A ) ) -> y <Q w )
20	19	3com23 1236	. . . . . . . . . . . 12 |- ( ( A <P B /\ w e. ( 2nd ` A ) /\ y e. ( 1st ` A ) ) -> y <Q w )
21	20	3adant2r 1260	. . . . . . . . . . 11 |- ( ( A <P B /\ ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ y e. ( 1st ` A ) ) -> y <Q w )
22	21	3adant2r 1260	. . . . . . . . . 10 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ y e. ( 1st ` A ) ) -> y <Q w )
23	22	3adant3r 1262	. . . . . . . . 9 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> y <Q w )
24		ltanqg 7715	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
25	24	adantl 277	. . . . . . . . . . 11 |- ( ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
26		elprnql 7796	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
27	17, 26	sylan 283	. . . . . . . . . . . . 13 |- ( ( A <P B /\ y e. ( 1st ` A ) ) -> y e. Q. )
28	27	adantrr 479	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> y e. Q. )
29	28	3adant2 1043	. . . . . . . . . . 11 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> y e. Q. )
30		elprnqu 7797	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. ( 2nd ` A ) ) -> w e. Q. )
31	17, 30	sylan 283	. . . . . . . . . . . . . 14 |- ( ( A <P B /\ w e. ( 2nd ` A ) ) -> w e. Q. )
32	31	adantrr 479	. . . . . . . . . . . . 13 |- ( ( A <P B /\ ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) ) -> w e. Q. )
33	32	adantrr 479	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> w e. Q. )
34	33	3adant3 1044	. . . . . . . . . . 11 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> w e. Q. )
35		prop 7790	. . . . . . . . . . . . . . . 16 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
36	5, 35	syl 14	. . . . . . . . . . . . . . 15 |- ( A <P B -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
37		elprnqu 7797	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ u e. ( 2nd ` C ) ) -> u e. Q. )
38	36, 37	sylan 283	. . . . . . . . . . . . . 14 |- ( ( A <P B /\ u e. ( 2nd ` C ) ) -> u e. Q. )
39	38	adantrl 478	. . . . . . . . . . . . 13 |- ( ( A <P B /\ ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) ) -> u e. Q. )
40	39	adantrr 479	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> u e. Q. )
41	40	3adant3 1044	. . . . . . . . . . 11 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> u e. Q. )
42		addcomnqg 7696	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
43	42	adantl 277	. . . . . . . . . . 11 |- ( ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
44	25, 29, 34, 41, 43	caovord2d 6224	. . . . . . . . . 10 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( y <Q w <-> ( y +Q u ) <Q ( w +Q u ) ) )
45	2	simprd 114	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
46		prop 7790	. . . . . . . . . . . . . 14 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
47	45, 46	syl 14	. . . . . . . . . . . . 13 |- ( A <P B -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
48		prcunqu 7800	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q u ) e. ( 2nd ` B ) ) -> ( ( y +Q u ) <Q ( w +Q u ) -> ( w +Q u ) e. ( 2nd ` B ) ) )
49	47, 48	sylan 283	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( y +Q u ) e. ( 2nd ` B ) ) -> ( ( y +Q u ) <Q ( w +Q u ) -> ( w +Q u ) e. ( 2nd ` B ) ) )
50	49	adantrl 478	. . . . . . . . . . 11 |- ( ( A <P B /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( ( y +Q u ) <Q ( w +Q u ) -> ( w +Q u ) e. ( 2nd ` B ) ) )
51	50	3adant2 1043	. . . . . . . . . 10 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( ( y +Q u ) <Q ( w +Q u ) -> ( w +Q u ) e. ( 2nd ` B ) ) )
52	44, 51	sylbid 150	. . . . . . . . 9 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( y <Q w -> ( w +Q u ) e. ( 2nd ` B ) ) )
53	23, 52	mpd 13	. . . . . . . 8 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( w +Q u ) e. ( 2nd ` B ) )
54	53	3expa 1230	. . . . . . 7 |- ( ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) /\ ( y e. ( 1st ` A ) /\ ( y +Q u ) e. ( 2nd ` B ) ) ) -> ( w +Q u ) e. ( 2nd ` B ) )
55	15, 54	exlimddv 1948	. . . . . 6 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> ( w +Q u ) e. ( 2nd ` B ) )
56	10, 55	eqeltrd 2309	. . . . 5 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> z e. ( 2nd ` B ) )
57	56	expr 375	. . . 4 |- ( ( A <P B /\ ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) ) -> ( z = ( w +Q u ) -> z e. ( 2nd ` B ) ) )
58	57	rexlimdvva 2668	. . 3 |- ( A <P B -> ( E. w e. ( 2nd ` A ) E. u e. ( 2nd ` C ) z = ( w +Q u ) -> z e. ( 2nd ` B ) ) )
59	9, 58	sylbid 150	. 2 |- ( A <P B -> ( z e. ( 2nd ` ( A +P. C ) ) -> z e. ( 2nd ` B ) ) )
60	59	ssrdv 3244	1 |- ( A <P B -> ( 2nd ` ( A +P. C ) ) C_ ( 2nd ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2203   E.wrex 2521   {crab 2524   C_ wss 3211   <.cop 3692   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   1stc1st 6332   2ndc2nd 6333   Q.cnq 7595   +Q cplq 7597   <Q cltq 7600   P.cnp 7606   +P. cpp 7608   <P cltp 7610

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iplp 7783  df-iltp 7785

This theorem is referenced by:  ltexpri  7928