Theorem ltexprlemloc 6797

Description: Our constructed difference is located. Lemma for ltexpri 6803. (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
ltexprlemloc	|- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )

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

Proof of Theorem ltexprlemloc

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

Step	Hyp	Ref	Expression
1		ltexnqi 6599	. . . . . 6 |- ( q <Q r -> E. w e. Q. ( q +Q w ) = r )
2	1	adantl 271	. . . . 5 |- ( ( A <P B /\ q <Q r ) -> E. w e. Q. ( q +Q w ) = r )
3		ltrelpr 6695	. . . . . . . . . 10 |- <P C_ ( P. X. P. )
4	3	brel 4410	. . . . . . . . 9 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
5	4	simpld 110	. . . . . . . 8 |- ( A <P B -> A e. P. )
6		prop 6665	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
7		prarloc 6693	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
8	6, 7	sylan 277	. . . . . . . 8 |- ( ( A e. P. /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
9	5, 8	sylan 277	. . . . . . 7 |- ( ( A <P B /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
10	9	ad2ant2r 492	. . . . . 6 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
11	4	simprd 112	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
12	11	ad2antrr 471	. . . . . . . . . . . . 13 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> B e. P. )
13	12	ad2antrr 471	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> B e. P. )
14		ltanqg 6590	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
15	14	adantl 271	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
16		elprnqu 6672	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
17	6, 16	sylan 277	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
18	5, 17	sylan 277	. . . . . . . . . . . . . . . . 17 |- ( ( A <P B /\ y e. ( 2nd ` A ) ) -> y e. Q. )
19	18	adantlr 460	. . . . . . . . . . . . . . . 16 |- ( ( ( A <P B /\ q <Q r ) /\ y e. ( 2nd ` A ) ) -> y e. Q. )
20	19	ad2ant2rl 494	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> y e. Q. )
21		elprnql 6671	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
22	6, 21	sylan 277	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
23	5, 22	sylan 277	. . . . . . . . . . . . . . . . . 18 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
24	23	adantlr 460	. . . . . . . . . . . . . . . . 17 |- ( ( ( A <P B /\ q <Q r ) /\ z e. ( 1st ` A ) ) -> z e. Q. )
25	24	ad2ant2r 492	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> z e. Q. )
26		simplrl 501	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> w e. Q. )
27		addclnq 6565	. . . . . . . . . . . . . . . 16 |- ( ( z e. Q. /\ w e. Q. ) -> ( z +Q w ) e. Q. )
28	25, 26, 27	syl2anc 403	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q w ) e. Q. )
29		ltrelnq 6555	. . . . . . . . . . . . . . . . . . 19 |- <Q C_ ( Q. X. Q. )
30	29	brel 4410	. . . . . . . . . . . . . . . . . 18 |- ( q <Q r -> ( q e. Q. /\ r e. Q. ) )
31	30	simpld 110	. . . . . . . . . . . . . . . . 17 |- ( q <Q r -> q e. Q. )
32	31	adantl 271	. . . . . . . . . . . . . . . 16 |- ( ( A <P B /\ q <Q r ) -> q e. Q. )
33	32	ad2antrr 471	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> q e. Q. )
34		addcomnqg 6571	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
35	34	adantl 271	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
36	15, 20, 28, 33, 35	caovord2d 5690	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) <-> ( y +Q q ) <Q ( ( z +Q w ) +Q q ) ) )
37		addassnqg 6572	. . . . . . . . . . . . . . . . 17 |- ( ( z e. Q. /\ w e. Q. /\ q e. Q. ) -> ( ( z +Q w ) +Q q ) = ( z +Q ( w +Q q ) ) )
38	25, 26, 33, 37	syl3anc 1169	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( z +Q w ) +Q q ) = ( z +Q ( w +Q q ) ) )
39		addcomnqg 6571	. . . . . . . . . . . . . . . . . 18 |- ( ( w e. Q. /\ q e. Q. ) -> ( w +Q q ) = ( q +Q w ) )
40	26, 33, 39	syl2anc 403	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( w +Q q ) = ( q +Q w ) )
41	40	oveq2d 5548	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q ( w +Q q ) ) = ( z +Q ( q +Q w ) ) )
42		simplrr 502	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( q +Q w ) = r )
43	42	oveq2d 5548	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q ( q +Q w ) ) = ( z +Q r ) )
44	38, 41, 43	3eqtrd 2117	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( z +Q w ) +Q q ) = ( z +Q r ) )
45	44	breq2d 3797	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( y +Q q ) <Q ( ( z +Q w ) +Q q ) <-> ( y +Q q ) <Q ( z +Q r ) ) )
46	36, 45	bitrd 186	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) <-> ( y +Q q ) <Q ( z +Q r ) ) )
47	46	biimpa 290	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> ( y +Q q ) <Q ( z +Q r ) )
48		prop 6665	. . . . . . . . . . . . 13 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
49		prloc 6681	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q q ) <Q ( z +Q r ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
50	48, 49	sylan 277	. . . . . . . . . . . 12 |- ( ( B e. P. /\ ( y +Q q ) <Q ( z +Q r ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
51	13, 47, 50	syl2anc 403	. . . . . . . . . . 11 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
52	51	ex 113	. . . . . . . . . 10 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
53	52	anassrs 392	. . . . . . . . 9 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ z e. ( 1st ` A ) ) /\ y e. ( 2nd ` A ) ) -> ( y <Q ( z +Q w ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
54	53	reximdva 2463	. . . . . . . 8 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ z e. ( 1st ` A ) ) -> ( E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
55	54	reximdva 2463	. . . . . . 7 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
56		prml 6667	. . . . . . . . . . . 12 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. z e. Q. z e. ( 1st ` A ) )
57		rexex 2410	. . . . . . . . . . . 12 |- ( E. z e. Q. z e. ( 1st ` A ) -> E. z z e. ( 1st ` A ) )
58	6, 56, 57	3syl 17	. . . . . . . . . . 11 |- ( A e. P. -> E. z z e. ( 1st ` A ) )
59		r19.45mv 3335	. . . . . . . . . . 11 |- ( E. z z e. ( 1st ` A ) -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
60	5, 58, 59	3syl 17	. . . . . . . . . 10 |- ( A <P B -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
61	60	adantr 270	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
62		prmu 6668	. . . . . . . . . . . . 13 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 2nd ` A ) )
63		rexex 2410	. . . . . . . . . . . . 13 |- ( E. x e. Q. x e. ( 2nd ` A ) -> E. x x e. ( 2nd ` A ) )
64	6, 62, 63	3syl 17	. . . . . . . . . . . 12 |- ( A e. P. -> E. x x e. ( 2nd ` A ) )
65		r19.9rmv 3333	. . . . . . . . . . . . . 14 |- ( E. x x e. ( 2nd ` A ) -> ( ( z +Q r ) e. ( 2nd ` B ) <-> E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) )
66	65	orbi2d 736	. . . . . . . . . . . . 13 |- ( E. x x e. ( 2nd ` A ) -> ( ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
67		r19.43 2512	. . . . . . . . . . . . 13 |- ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) )
68	66, 67	syl6rbbr 197	. . . . . . . . . . . 12 |- ( E. x x e. ( 2nd ` A ) -> ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
69	5, 64, 68	3syl 17	. . . . . . . . . . 11 |- ( A <P B -> ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
70	69	rexbidv 2369	. . . . . . . . . 10 |- ( A <P B -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
71	70	adantr 270	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
72		ibar 295	. . . . . . . . . . . . . . 15 |- ( q e. Q. -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
73	72	adantr 270	. . . . . . . . . . . . . 14 |- ( ( q e. Q. /\ r e. Q. ) -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
74		ibar 295	. . . . . . . . . . . . . . 15 |- ( r e. Q. -> ( E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
75	74	adantl 271	. . . . . . . . . . . . . 14 |- ( ( q e. Q. /\ r e. Q. ) -> ( E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
76	73, 75	orbi12d 739	. . . . . . . . . . . . 13 |- ( ( q e. Q. /\ r e. Q. ) -> ( ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) ) )
77	30, 76	syl 14	. . . . . . . . . . . 12 |- ( q <Q r -> ( ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) ) )
78		ltexprlem.1	. . . . . . . . . . . . . 14 |- 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 ) ) } >.
79	78	ltexprlemell 6788	. . . . . . . . . . . . 13 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
80	78	ltexprlemelu 6789	. . . . . . . . . . . . . 14 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
81		eleq1 2141	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
82		oveq1 5539	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( y +Q r ) = ( z +Q r ) )
83	82	eleq1d 2147	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( ( y +Q r ) e. ( 2nd ` B ) <-> ( z +Q r ) e. ( 2nd ` B ) ) )
84	81, 83	anbi12d 456	. . . . . . . . . . . . . . . 16 |- ( y = z -> ( ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) <-> ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
85	84	cbvexv 1836	. . . . . . . . . . . . . . 15 |- ( E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) <-> E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) )
86	85	anbi2i 444	. . . . . . . . . . . . . 14 |- ( ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
87	80, 86	bitri 182	. . . . . . . . . . . . 13 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
88	79, 87	orbi12i 713	. . . . . . . . . . . 12 |- ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
89	77, 88	syl6rbbr 197	. . . . . . . . . . 11 |- ( q <Q r -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
90		df-rex 2354	. . . . . . . . . . . 12 |- ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) <-> E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) )
91		df-rex 2354	. . . . . . . . . . . 12 |- ( E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) <-> E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) )
92	90, 91	orbi12i 713	. . . . . . . . . . 11 |- ( ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
93	89, 92	syl6bbr 196	. . . . . . . . . 10 |- ( q <Q r -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
94	93	adantl 271	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
95	61, 71, 94	3bitr4rd 219	. . . . . . . 8 |- ( ( A <P B /\ q <Q r ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
96	95	adantr 270	. . . . . . 7 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
97	55, 96	sylibrd 167	. . . . . 6 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
98	10, 97	mpd 13	. . . . 5 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) )
99	2, 98	rexlimddv 2481	. . . 4 |- ( ( A <P B /\ q <Q r ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) )
100	99	ex 113	. . 3 |- ( A <P B -> ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
101	100	ralrimivw 2435	. 2 |- ( A <P B -> A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
102	101	ralrimivw 2435	1 |- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   /\ w3a 919   = wceq 1284   E.wex 1421   e. wcel 1433   A.wral 2348   E.wrex 2349   {crab 2352   <.cop 3401   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   1stc1st 5785   2ndc2nd 5786   Q.cnq 6470   +Q cplq 6472   <Q cltq 6475   P.cnp 6481   <P cltp 6485

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-iltp 6660

This theorem is referenced by:  ltexprlempr  6798