Theorem ltexprlemdisj 7825

Description: Our constructed difference is disjoint. Lemma for ltexpri 7832. (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 7617	. . . . . 6 |- <Q Or Q.
2		ltrelnq 7584	. . . . . 6 |- <Q C_ ( Q. X. Q. )
3	1, 2	son2lpi 5133	. . . . 5 |- -. ( y <Q z /\ z <Q y )
4		ltrelpr 7724	. . . . . . . . . . . . . . . 16 |- <P C_ ( P. X. P. )
5	4	brel 4778	. . . . . . . . . . . . . . 15 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
6	5	simprd 114	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
7		prop 7694	. . . . . . . . . . . . . 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 7706	. . . . . . . . . . . . 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 1306	. . . . . . . . . . . 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 1230	. . . . . . . . . . 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 477	. . . . . . . . . 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 484	. . . . . . . . 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 486	. . . . . . . 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 7619	. . . . . . . . . 10 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
16	15	adantl 277	. . . . . . . . 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 112	. . . . . . . . . . . . 13 |- ( A <P B -> A e. P. )
18		prop 7694	. . . . . . . . . . . . 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 7701	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
21	19, 20	sylan 283	. . . . . . . . . . 11 |- ( ( A <P B /\ y e. ( 2nd ` A ) ) -> y e. Q. )
22	21	ad2ant2r 509	. . . . . . . . . 10 |- ( ( ( A <P B /\ q e. Q. ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) -> y e. Q. )
23	22	adantrr 479	. . . . . . . . 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 7700	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
25	19, 24	sylan 283	. . . . . . . . . . 11 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
26	25	ad2ant2r 509	. . . . . . . . . 10 |- ( ( ( A <P B /\ q e. Q. ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) -> z e. Q. )
27	26	adantrl 478	. . . . . . . . 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 529	. . . . . . . . 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 7600	. . . . . . . . . 10 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
30	29	adantl 277	. . . . . . . . 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 6191	. . . . . . . 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 167	. . . . . . 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 7706	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> z <Q y )
34	19, 33	syl3an1 1306	. . . . . . . . . . . 12 |- ( ( A <P B /\ z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> z <Q y )
35	34	3com23 1235	. . . . . . . . . . 11 |- ( ( A <P B /\ y e. ( 2nd ` A ) /\ z e. ( 1st ` A ) ) -> z <Q y )
36	35	3expb 1230	. . . . . . . . . 10 |- ( ( A <P B /\ ( y e. ( 2nd ` A ) /\ z e. ( 1st ` A ) ) ) -> z <Q y )
37	36	adantlr 477	. . . . . . . . 9 |- ( ( ( A <P B /\ q e. Q. ) /\ ( y e. ( 2nd ` A ) /\ z e. ( 1st ` A ) ) ) -> z <Q y )
38	37	adantrlr 485	. . . . . . . 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 487	. . . . . . 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 306	. . . . . 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 115	. . . . 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 670	. . . 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 1923	. . 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 7817	. . . . . . . . . . 11 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
46	44	ltexprlemelu 7818	. . . . . . . . . . 11 |- ( q e. ( 2nd ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) )
47	45, 46	anbi12i 460	. . . . . . . . . 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 594	. . . . . . . . . 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 187	. . . . . . . . 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 926	. . . . . . . 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 2294	. . . . . . . . . . 11 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
52		oveq1 6024	. . . . . . . . . . . 12 |- ( y = z -> ( y +Q q ) = ( z +Q q ) )
53	52	eleq1d 2300	. . . . . . . . . . 11 |- ( y = z -> ( ( y +Q q ) e. ( 2nd ` B ) <-> ( z +Q q ) e. ( 2nd ` B ) ) )
54	51, 53	anbi12d 473	. . . . . . . . . 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 1967	. . . . . . . . 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 457	. . . . . . . 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	bitrdi 196	. . . . . . 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 1985	. . . . . . 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	bitr4di 198	. . . . . 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 673	. . . . 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 1547	. . . . . . 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 1518	. . . . . 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 1547	. . . . . 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 184	. . . . 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	bitr4di 198	. . . 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 277	. . 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 167	. 2 |- ( ( A <P B /\ q e. Q. ) -> -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) )
68	67	ralrimiva 2605	1 |- ( A <P B -> A. q e. Q. -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   A.wal 1395   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   {crab 2514   <.cop 3672   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   1stc1st 6300   2ndc2nd 6301   Q.cnq 7499   +Q cplq 7501   <Q cltq 7504   P.cnp 7510   <P cltp 7514

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-ltnqqs 7572  df-inp 7685  df-iltp 7689

This theorem is referenced by:  ltexprlempr  7827