Theorem ltexprlemdisj 7596

Description: Our constructed difference is disjoint. Lemma for ltexpri 7603. (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 7388	. . . . . 6 |- <Q Or Q.
2		ltrelnq 7355	. . . . . 6 |- <Q C_ ( Q. X. Q. )
3	1, 2	son2lpi 5021	. . . . 5 |- -. ( y <Q z /\ z <Q y )
4		ltrelpr 7495	. . . . . . . . . . . . . . . 16 |- <P C_ ( P. X. P. )
5	4	brel 4675	. . . . . . . . . . . . . . 15 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
6	5	simprd 114	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
7		prop 7465	. . . . . . . . . . . . . 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 7477	. . . . . . . . . . . . 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 1271	. . . . . . . . . . . 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 1204	. . . . . . . . . . 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 7390	. . . . . . . . . 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 7465	. . . . . . . . . . . . 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 7472	. . . . . . . . . . . 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 7471	. . . . . . . . . . . 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 7371	. . . . . . . . . 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 6038	. . . . . . . 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 7477	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> z <Q y )
34	19, 33	syl3an1 1271	. . . . . . . . . . . 12 |- ( ( A <P B /\ z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) -> z <Q y )
35	34	3com23 1209	. . . . . . . . . . 11 |- ( ( A <P B /\ y e. ( 2nd ` A ) /\ z e. ( 1st ` A ) ) -> z <Q y )
36	35	3expb 1204	. . . . . . . . . 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 664	. . . 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 1875	. . 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 7588	. . . . . . . . . . 11 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
46	44	ltexprlemelu 7589	. . . . . . . . . . 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 919	. . . . . . . 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 2240	. . . . . . . . . . 11 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
52		oveq1 5876	. . . . . . . . . . . 12 |- ( y = z -> ( y +Q q ) = ( z +Q q ) )
53	52	eleq1d 2246	. . . . . . . . . . 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 1918	. . . . . . . . 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 1932	. . . . . . 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 667	. . . . 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 1499	. . . . . . 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 1470	. . . . . 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 1499	. . . . . 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 2550	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 978   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   A.wral 2455   {crab 2459   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270   +Q cplq 7272   <Q cltq 7275   P.cnp 7281   <P cltp 7285

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-ltnqqs 7343  df-inp 7456  df-iltp 7460

This theorem is referenced by:  ltexprlempr  7598