Theorem ltexprlemdisj 7690

Description: Our constructed difference is disjoint. Lemma for ltexpri 7697. (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 7482	. . . . . 6 |- <Q Or Q.
2		ltrelnq 7449	. . . . . 6 |- <Q C_ ( Q. X. Q. )
3	1, 2	son2lpi 5067	. . . . 5 |- -. ( y <Q z /\ z <Q y )
4		ltrelpr 7589	. . . . . . . . . . . . . . . 16 |- <P C_ ( P. X. P. )
5	4	brel 4716	. . . . . . . . . . . . . . 15 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
6	5	simprd 114	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
7		prop 7559	. . . . . . . . . . . . . 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 7571	. . . . . . . . . . . . 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 1282	. . . . . . . . . . . 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 1206	. . . . . . . . . . 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 7484	. . . . . . . . . 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 7559	. . . . . . . . . . . . 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 7566	. . . . . . . . . . . 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 7565	. . . . . . . . . . . 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 528	. . . . . . . . 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 7465	. . . . . . . . . 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 6097	. . . . . . . 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 7571	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> z <Q y )
34	19, 33	syl3an1 1282	. . . . . . . . . . . 12 |- ( ( A <P B /\ z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> z <Q y )
35	34	3com23 1211	. . . . . . . . . . 11 |- ( ( A <P B /\ y e. ( 2nd ` A ) /\ z e. ( 1st ` A ) ) -> z <Q y )
36	35	3expb 1206	. . . . . . . . . 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 665	. . . 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 1889	. . 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 7682	. . . . . . . . . . 11 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
46	44	ltexprlemelu 7683	. . . . . . . . . . 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 590	. . . . . . . . . 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 920	. . . . . . . 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 2259	. . . . . . . . . . 11 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
52		oveq1 5932	. . . . . . . . . . . 12 |- ( y = z -> ( y +Q q ) = ( z +Q q ) )
53	52	eleq1d 2265	. . . . . . . . . . 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 1933	. . . . . . . . 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 1951	. . . . . . 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 668	. . . . 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 1513	. . . . . . 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 1484	. . . . . 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 1513	. . . . . 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 2570	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 980   A.wal 1362   = wceq 1364   E.wex 1506   e. wcel 2167   A.wral 2475   {crab 2479   <.cop 3626   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   1stc1st 6205   2ndc2nd 6206   Q.cnq 7364   +Q cplq 7366   <Q cltq 7369   P.cnp 7375   <P cltp 7379

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-ltnqqs 7437  df-inp 7550  df-iltp 7554

This theorem is referenced by:  ltexprlempr  7692