Theorem ltexprlemfu 7754

Description: Lemma for ltexpri 7756. 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 7648	. . . . . 6 |- <P C_ ( P. X. P. )
2	1	brel 4740	. . . . 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 7751	. . . 4 |- ( A <P B -> C e. P. )
6		df-iplp 7611	. . . . 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 7518	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
8	6, 7	genpelvu 7656	. . . 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 531	. . . . . 6 |- ( ( A <P B /\ ( ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ z = ( w +Q u ) ) ) -> z = ( w +Q u ) )
11	4	ltexprlemelu 7742	. . . . . . . . . . 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 7618	. . . . . . . . . . . . . . 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 7630	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) /\ w e. ( 2nd ` A ) ) -> y <Q w )
19	17, 18	syl3an1 1283	. . . . . . . . . . . . 13 |- ( ( A <P B /\ y e. ( 1st ` A ) /\ w e. ( 2nd ` A ) ) -> y <Q w )
20	19	3com23 1212	. . . . . . . . . . . 12 |- ( ( A <P B /\ w e. ( 2nd ` A ) /\ y e. ( 1st ` A ) ) -> y <Q w )
21	20	3adant2r 1236	. . . . . . . . . . 11 |- ( ( A <P B /\ ( w e. ( 2nd ` A ) /\ u e. ( 2nd ` C ) ) /\ y e. ( 1st ` A ) ) -> y <Q w )
22	21	3adant2r 1236	. . . . . . . . . 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 1238	. . . . . . . . 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 7543	. . . . . . . . . . . 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 7624	. . . . . . . . . . . . . 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 1019	. . . . . . . . . . 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 7625	. . . . . . . . . . . . . . 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 1020	. . . . . . . . . . 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 7618	. . . . . . . . . . . . . . . 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 7625	. . . . . . . . . . . . . . 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 1020	. . . . . . . . . . 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 7524	. . . . . . . . . . . 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 6134	. . . . . . . . . 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 7618	. . . . . . . . . . . . . 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 7628	. . . . . . . . . . . . 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 1019	. . . . . . . . . 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 1206	. . . . . . 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 1923	. . . . . 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 2283	. . . . 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 2632	. . 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 3203	1 |- ( A <P B -> ( 2nd ` ( A +P. C ) ) C_ ( 2nd ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   E.wex 1516   e. wcel 2177   E.wrex 2486   {crab 2489   C_ wss 3170   <.cop 3641   class class class wbr 4054   ` cfv 5285  (class class class)co 5962   1stc1st 6242   2ndc2nd 6243   Q.cnq 7423   +Q cplq 7425   <Q cltq 7428   P.cnp 7434   +P. cpp 7436   <P cltp 7438

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 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

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 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-eprel 4349  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-irdg 6474  df-1o 6520  df-2o 6521  df-oadd 6524  df-omul 6525  df-er 6638  df-ec 6640  df-qs 6644  df-ni 7447  df-pli 7448  df-mi 7449  df-lti 7450  df-plpq 7487  df-mpq 7488  df-enq 7490  df-nqqs 7491  df-plqqs 7492  df-mqqs 7493  df-1nqqs 7494  df-rq 7495  df-ltnqqs 7496  df-enq0 7567  df-nq0 7568  df-0nq0 7569  df-plq0 7570  df-mq0 7571  df-inp 7609  df-iplp 7611  df-iltp 7613

This theorem is referenced by:  ltexpri  7756