Theorem ltexprlemdisj 6704

Description: Our constructed difference is disjoint. Lemma for ltexpri 6711. (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
ltexprlemdisj	|- ( A <P B -> A. q e. Q. -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) )

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

Proof of Theorem ltexprlemdisj

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

Step	Hyp	Ref	Expression
1		ltsonq 6496	. . . . . 6 |- <Q Or Q.
2		ltrelnq 6463	. . . . . 6 |- <Q C_ ( Q. X. Q. )
3	1, 2	son2lpi 4721	. . . . 5 |- -. ( y <Q z /\ z <Q y )
4		ltrelpr 6603	. . . . . . . . . . . . . . . 16 |- <P C_ ( P. X. P. )
5	4	brel 4392	. . . . . . . . . . . . . . 15 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
6	5	simprd 107	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
7		prop 6573	. . . . . . . . . . . . . 14 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
8	6, 7	syl 14	. . . . . . . . . . . . 13 |- ( A <P B -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
9		prltlu 6585	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q q ) e. ( 1st ` B ) /\ ( z +Q q ) e. ( 2nd ` B ) ) -> ( y +Q q ) <Q ( z +Q q ) )
10	8, 9	syl3an1 1168	. . . . . . . . . . . 12 |- ( ( A <P B /\ ( y +Q q ) e. ( 1st ` B ) /\ ( z +Q q ) e. ( 2nd ` B ) ) -> ( y +Q q ) <Q ( z +Q q ) )
11	10	3expb 1105	. . . . . . . . . . 11 |- ( ( A <P B /\ ( ( y +Q q ) e. ( 1st ` B ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) -> ( y +Q q ) <Q ( z +Q q ) )
12	11	adantlr 446	. . . . . . . . . 10 |- ( ( ( A <P B /\ q e. Q. ) /\ ( ( y +Q q ) e. ( 1st ` B ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) -> ( y +Q q ) <Q ( z +Q q ) )
13	12	adantrll 453	. . . . . . . . 9 |- ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) -> ( y +Q q ) <Q ( z +Q q ) )
14	13	adantrrl 455	. . . . . . . 8 |- ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) -> ( y +Q q ) <Q ( z +Q q ) )
15		ltanqg 6498	. . . . . . . . . 10 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
16	15	adantl 262	. . . . . . . . 9 |- ( ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
17	5	simpld 105	. . . . . . . . . . . . 13 |- ( A <P B -> A e. P. )
18		prop 6573	. . . . . . . . . . . . 13 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
19	17, 18	syl 14	. . . . . . . . . . . 12 |- ( A <P B -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
20		elprnqu 6580	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
21	19, 20	sylan 267	. . . . . . . . . . 11 |- ( ( A <P B /\ y e. ( 2nd ` A ) ) -> y e. Q. )
22	21	ad2ant2r 478	. . . . . . . . . 10 |- ( ( ( A <P B /\ q e. Q. ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) -> y e. Q. )
23	22	adantrr 448	. . . . . . . . 9 |- ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) -> y e. Q. )
24		elprnql 6579	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
25	19, 24	sylan 267	. . . . . . . . . . 11 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
26	25	ad2ant2r 478	. . . . . . . . . 10 |- ( ( ( A <P B /\ q e. Q. ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) -> z e. Q. )
27	26	adantrl 447	. . . . . . . . 9 |- ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) -> z e. Q. )
28		simplr 482	. . . . . . . . 9 |- ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) -> q e. Q. )
29		addcomnqg 6479	. . . . . . . . . 10 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
30	29	adantl 262	. . . . . . . . 9 |- ( ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
31	16, 23, 27, 28, 30	caovord2d 5670	. . . . . . . 8 |- ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) -> ( y <Q z <-> ( y +Q q ) <Q ( z +Q q ) ) )
32	14, 31	mpbird 156	. . . . . . 7 |- ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) -> y <Q z )
33		prltlu 6585	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> z <Q y )
34	19, 33	syl3an1 1168	. . . . . . . . . . . 12 |- ( ( A <P B /\ z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> z <Q y )
35	34	3com23 1110	. . . . . . . . . . 11 |- ( ( A <P B /\ y e. ( 2nd ` A ) /\ z e. ( 1st ` A ) ) -> z <Q y )
36	35	3expb 1105	. . . . . . . . . 10 |- ( ( A <P B /\ ( y e. ( 2nd ` A ) /\ z e. ( 1st ` A ) ) ) -> z <Q y )
37	36	adantlr 446	. . . . . . . . 9 |- ( ( ( A <P B /\ q e. Q. ) /\ ( y e. ( 2nd ` A ) /\ z e. ( 1st ` A ) ) ) -> z <Q y )
38	37	adantrlr 454	. . . . . . . 8 |- ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ z e. ( 1st ` A ) ) ) -> z <Q y )
39	38	adantrrr 456	. . . . . . 7 |- ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) -> z <Q y )
40	32, 39	jca 290	. . . . . 6 |- ( ( ( A <P B /\ q e. Q. ) /\ ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) -> ( y <Q z /\ z <Q y ) )
41	40	ex 108	. . . . 5 |- ( ( A <P B /\ q e. Q. ) -> ( ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) -> ( y <Q z /\ z <Q y ) ) )
42	3, 41	mtoi 590	. . . 4 |- ( ( A <P B /\ q e. Q. ) -> -. ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) )
43	42	alrimivv 1755	. . 3 |- ( ( A <P B /\ q e. Q. ) -> A. y A. z -. ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) )
44		ltexprlem.1	. . . . . . . . . . . 12 |- 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 ) ) } >.
45	44	ltexprlemell 6696	. . . . . . . . . . 11 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
46	44	ltexprlemelu 6697	. . . . . . . . . . 11 |- ( q e. ( 2nd ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) )
47	45, 46	anbi12i 433	. . . . . . . . . 10 |- ( ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) /\ ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
48		anandi 524	. . . . . . . . . 10 |- ( ( q e. Q. /\ ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) /\ ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
49	47, 48	bitr4i 176	. . . . . . . . 9 |- ( ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) <-> ( q e. Q. /\ ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
50	49	baib 828	. . . . . . . 8 |- ( q e. Q. -> ( ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
51		eleq1 2100	. . . . . . . . . . 11 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
52		oveq1 5519	. . . . . . . . . . . 12 |- ( y = z -> ( y +Q q ) = ( z +Q q ) )
53	52	eleq1d 2106	. . . . . . . . . . 11 |- ( y = z -> ( ( y +Q q ) e. ( 2nd ` B ) <-> ( z +Q q ) e. ( 2nd ` B ) ) )
54	51, 53	anbi12d 442	. . . . . . . . . 10 |- ( y = z -> ( ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) <-> ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) )
55	54	cbvexv 1795	. . . . . . . . 9 |- ( E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) <-> E. z ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) )
56	55	anbi2i 430	. . . . . . . 8 |- ( ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) )
57	50, 56	syl6bb 185	. . . . . . 7 |- ( q e. Q. -> ( ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) )
58		eeanv 1807	. . . . . . 7 |- ( E. y E. z ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) )
59	57, 58	syl6bbr 187	. . . . . 6 |- ( q e. Q. -> ( ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) <-> E. y E. z ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) )
60	59	notbid 592	. . . . 5 |- ( q e. Q. -> ( -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) <-> -. E. y E. z ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) )
61		alnex 1388	. . . . . . 7 |- ( A. z -. ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) <-> -. E. z ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) )
62	61	albii 1359	. . . . . 6 |- ( A. y A. z -. ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) <-> A. y -. E. z ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) )
63		alnex 1388	. . . . . 6 |- ( A. y -. E. z ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) <-> -. E. y E. z ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) )
64	62, 63	bitri 173	. . . . 5 |- ( A. y A. z -. ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) <-> -. E. y E. z ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) )
65	60, 64	syl6bbr 187	. . . 4 |- ( q e. Q. -> ( -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) <-> A. y A. z -. ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) )
66	65	adantl 262	. . 3 |- ( ( A <P B /\ q e. Q. ) -> ( -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) <-> A. y A. z -. ( ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) ) )
67	43, 66	mpbird 156	. 2 |- ( ( A <P B /\ q e. Q. ) -> -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) )
68	67	ralrimiva 2392	1 |- ( A <P B -> A. q e. Q. -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   A.wral 2306   {crab 2310   <.cop 3378   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   +Q cplq 6380   <Q cltq 6383   P.cnp 6389   <P cltp 6393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-ltnqqs 6451  df-inp 6564  df-iltp 6568

This theorem is referenced by:  ltexprlempr  6706