Theorem ltexprlemdisj 7414

Description: Our constructed difference is disjoint. Lemma for ltexpri 7421. (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 7206	. . . . . 6 |- <Q Or Q.
2		ltrelnq 7173	. . . . . 6 |- <Q C_ ( Q. X. Q. )
3	1, 2	son2lpi 4935	. . . . 5 |- -. ( y <Q z /\ z <Q y )
4		ltrelpr 7313	. . . . . . . . . . . . . . . 16 |- <P C_ ( P. X. P. )
5	4	brel 4591	. . . . . . . . . . . . . . 15 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
6	5	simprd 113	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
7		prop 7283	. . . . . . . . . . . . . 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 7295	. . . . . . . . . . . . 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 1249	. . . . . . . . . . . 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 1182	. . . . . . . . . . 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 468	. . . . . . . . . 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 475	. . . . . . . . 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 477	. . . . . . . 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 7208	. . . . . . . . . 10 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
16	15	adantl 275	. . . . . . . . 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 111	. . . . . . . . . . . . 13 |- ( A <P B -> A e. P. )
18		prop 7283	. . . . . . . . . . . . 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 7290	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
21	19, 20	sylan 281	. . . . . . . . . . 11 |- ( ( A <P B /\ y e. ( 2nd ` A ) ) -> y e. Q. )
22	21	ad2ant2r 500	. . . . . . . . . 10 |- ( ( ( A <P B /\ q e. Q. ) /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) -> y e. Q. )
23	22	adantrr 470	. . . . . . . . 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 7289	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
25	19, 24	sylan 281	. . . . . . . . . . 11 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
26	25	ad2ant2r 500	. . . . . . . . . 10 |- ( ( ( A <P B /\ q e. Q. ) /\ ( z e. ( 1st ` A ) /\ ( z +Q q ) e. ( 2nd ` B ) ) ) -> z e. Q. )
27	26	adantrl 469	. . . . . . . . 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 519	. . . . . . . . 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 7189	. . . . . . . . . 10 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
30	29	adantl 275	. . . . . . . . 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 5940	. . . . . . . 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 166	. . . . . . 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 7295	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> z <Q y )
34	19, 33	syl3an1 1249	. . . . . . . . . . . 12 |- ( ( A <P B /\ z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> z <Q y )
35	34	3com23 1187	. . . . . . . . . . 11 |- ( ( A <P B /\ y e. ( 2nd ` A ) /\ z e. ( 1st ` A ) ) -> z <Q y )
36	35	3expb 1182	. . . . . . . . . 10 |- ( ( A <P B /\ ( y e. ( 2nd ` A ) /\ z e. ( 1st ` A ) ) ) -> z <Q y )
37	36	adantlr 468	. . . . . . . . 9 |- ( ( ( A <P B /\ q e. Q. ) /\ ( y e. ( 2nd ` A ) /\ z e. ( 1st ` A ) ) ) -> z <Q y )
38	37	adantrlr 476	. . . . . . . 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 478	. . . . . . 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 304	. . . . . 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 114	. . . . 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 653	. . . 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 1847	. . 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 7406	. . . . . . . . . . 11 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
46	44	ltexprlemelu 7407	. . . . . . . . . . 11 |- ( q e. ( 2nd ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) )
47	45, 46	anbi12i 455	. . . . . . . . . 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 579	. . . . . . . . . 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 186	. . . . . . . . 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 904	. . . . . . . 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 2202	. . . . . . . . . . 11 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
52		oveq1 5781	. . . . . . . . . . . 12 |- ( y = z -> ( y +Q q ) = ( z +Q q ) )
53	52	eleq1d 2208	. . . . . . . . . . 11 |- ( y = z -> ( ( y +Q q ) e. ( 2nd ` B ) <-> ( z +Q q ) e. ( 2nd ` B ) ) )
54	51, 53	anbi12d 464	. . . . . . . . . 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 1890	. . . . . . . . 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 452	. . . . . . . 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 195	. . . . . . 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 1904	. . . . . . 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 197	. . . . . 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 656	. . . . 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 1475	. . . . . . 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 1446	. . . . . 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 1475	. . . . . 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 183	. . . . 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 197	. . . 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 275	. . 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 166	. 2 |- ( ( A <P B /\ q e. Q. ) -> -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) )
68	67	ralrimiva 2505	1 |- ( A <P B -> A. q e. Q. -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   A.wal 1329   = wceq 1331   E.wex 1468   e. wcel 1480   A.wral 2416   {crab 2420   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7088   +Q cplq 7090   <Q cltq 7093   P.cnp 7099   <P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-ltnqqs 7161  df-inp 7274  df-iltp 7278

This theorem is referenced by:  ltexprlempr  7416