Theorem ltexprlemopl 7932

Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7944. (Contributed by Jim Kingdon, 21-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
ltexprlemopl	|- ( ( A <P B /\ q e. Q. /\ q e. ( 1st ` C ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) )

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

Proof of Theorem ltexprlemopl

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexprlem.1	. . . . 5 |- 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 ) ) } >.
2	1	ltexprlemell 7929	. . . 4 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
3	2	simprbi 275	. . 3 |- ( q e. ( 1st ` C ) -> E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) )
4		19.42v 1958	. . . . . . . 8 |- ( E. y ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) <-> ( A <P B /\ E. y ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
5		19.42v 1958	. . . . . . . . 9 |- ( E. y ( q e. Q. /\ ( 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 ) ) ) )
6	5	anbi2i 457	. . . . . . . 8 |- ( ( A <P B /\ E. y ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) <-> ( A <P B /\ ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
7	4, 6	bitri 184	. . . . . . 7 |- ( E. y ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) <-> ( A <P B /\ ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
8		ltrelpr 7836	. . . . . . . . . . . . . 14 |- <P C_ ( P. X. P. )
9	8	brel 4807	. . . . . . . . . . . . 13 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
10	9	simprd 114	. . . . . . . . . . . 12 |- ( A <P B -> B e. P. )
11		prop 7806	. . . . . . . . . . . . 13 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnmaxl 7819	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q q ) e. ( 1st ` B ) ) -> E. s e. ( 1st ` B ) ( y +Q q ) <Q s )
13	11, 12	sylan 283	. . . . . . . . . . . 12 |- ( ( B e. P. /\ ( y +Q q ) e. ( 1st ` B ) ) -> E. s e. ( 1st ` B ) ( y +Q q ) <Q s )
14	10, 13	sylan 283	. . . . . . . . . . 11 |- ( ( A <P B /\ ( y +Q q ) e. ( 1st ` B ) ) -> E. s e. ( 1st ` B ) ( y +Q q ) <Q s )
15	14	adantrl 478	. . . . . . . . . 10 |- ( ( A <P B /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) -> E. s e. ( 1st ` B ) ( y +Q q ) <Q s )
16	15	adantrl 478	. . . . . . . . 9 |- ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) -> E. s e. ( 1st ` B ) ( y +Q q ) <Q s )
17	9	simpld 112	. . . . . . . . . . . . . . 15 |- ( A <P B -> A e. P. )
18	17	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> A e. P. )
19		simplrr 538	. . . . . . . . . . . . . . 15 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) )
20	19	simpld 112	. . . . . . . . . . . . . 14 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> y e. ( 2nd ` A ) )
21		prop 7806	. . . . . . . . . . . . . . 15 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
22		elprnqu 7813	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
23	21, 22	sylan 283	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
24	18, 20, 23	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> y e. Q. )
25		simplrl 537	. . . . . . . . . . . . 13 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> q e. Q. )
26		ltaddnq 7738	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ q e. Q. ) -> y <Q ( y +Q q ) )
27	24, 25, 26	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> y <Q ( y +Q q ) )
28		simprr 533	. . . . . . . . . . . 12 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> ( y +Q q ) <Q s )
29		ltsonq 7729	. . . . . . . . . . . . 13 |- <Q Or Q.
30		ltrelnq 7696	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
31	29, 30	sotri 5163	. . . . . . . . . . . 12 |- ( ( y <Q ( y +Q q ) /\ ( y +Q q ) <Q s ) -> y <Q s )
32	27, 28, 31	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> y <Q s )
33	10	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> B e. P. )
34		simprl 531	. . . . . . . . . . . . 13 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> s e. ( 1st ` B ) )
35		elprnql 7812	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ s e. ( 1st ` B ) ) -> s e. Q. )
36	11, 35	sylan 283	. . . . . . . . . . . . 13 |- ( ( B e. P. /\ s e. ( 1st ` B ) ) -> s e. Q. )
37	33, 34, 36	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> s e. Q. )
38		ltexnqq 7739	. . . . . . . . . . . 12 |- ( ( y e. Q. /\ s e. Q. ) -> ( y <Q s <-> E. r e. Q. ( y +Q r ) = s ) )
39	24, 37, 38	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> ( y <Q s <-> E. r e. Q. ( y +Q r ) = s ) )
40	32, 39	mpbid 147	. . . . . . . . . 10 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> E. r e. Q. ( y +Q r ) = s )
41		simplrr 538	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> ( y +Q q ) <Q s )
42		simprr 533	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> ( y +Q r ) = s )
43	41, 42	breqtrrd 4142	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> ( y +Q q ) <Q ( y +Q r ) )
44	25	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> q e. Q. )
45		simprl 531	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> r e. Q. )
46	24	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> y e. Q. )
47		ltanqg 7731	. . . . . . . . . . . . . . 15 |- ( ( q e. Q. /\ r e. Q. /\ y e. Q. ) -> ( q <Q r <-> ( y +Q q ) <Q ( y +Q r ) ) )
48	44, 45, 46, 47	syl3anc 1274	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> ( q <Q r <-> ( y +Q q ) <Q ( y +Q r ) ) )
49	43, 48	mpbird 167	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> q <Q r )
50	20	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> y e. ( 2nd ` A ) )
51		simplrl 537	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> s e. ( 1st ` B ) )
52	42, 51	eqeltrd 2311	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> ( y +Q r ) e. ( 1st ` B ) )
53	50, 52	jca 306	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) )
54	49, 45, 53	jca32 310	. . . . . . . . . . . 12 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ ( r e. Q. /\ ( y +Q r ) = s ) ) -> ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
55	54	expr 375	. . . . . . . . . . 11 |- ( ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) /\ r e. Q. ) -> ( ( y +Q r ) = s -> ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) )
56	55	reximdva 2646	. . . . . . . . . 10 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> ( E. r e. Q. ( y +Q r ) = s -> E. r e. Q. ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) )
57	40, 56	mpd 13	. . . . . . . . 9 |- ( ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) /\ ( s e. ( 1st ` B ) /\ ( y +Q q ) <Q s ) ) -> E. r e. Q. ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
58	16, 57	rexlimddv 2667	. . . . . . . 8 |- ( ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) -> E. r e. Q. ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
59	58	eximi 1649	. . . . . . 7 |- ( E. y ( A <P B /\ ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) -> E. y E. r e. Q. ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
60	7, 59	sylbir 135	. . . . . 6 |- ( ( A <P B /\ ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) -> E. y E. r e. Q. ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
61		rexcom4 2839	. . . . . 6 |- ( E. r e. Q. E. y ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) <-> E. y E. r e. Q. ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
62	60, 61	sylibr 134	. . . . 5 |- ( ( A <P B /\ ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) -> E. r e. Q. E. y ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
63		19.42v 1958	. . . . . . 7 |- ( E. y ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) <-> ( q <Q r /\ E. y ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
64		19.42v 1958	. . . . . . . 8 |- ( E. y ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) <-> ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) )
65	64	anbi2i 457	. . . . . . 7 |- ( ( q <Q r /\ E. y ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) <-> ( q <Q r /\ ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
66	63, 65	bitri 184	. . . . . 6 |- ( E. y ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) <-> ( q <Q r /\ ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
67	66	rexbii 2551	. . . . 5 |- ( E. r e. Q. E. y ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) <-> E. r e. Q. ( q <Q r /\ ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
68	62, 67	sylib 122	. . . 4 |- ( ( A <P B /\ ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) -> E. r e. Q. ( q <Q r /\ ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
69	1	ltexprlemell 7929	. . . . . 6 |- ( r e. ( 1st ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) )
70	69	anbi2i 457	. . . . 5 |- ( ( q <Q r /\ r e. ( 1st ` C ) ) <-> ( q <Q r /\ ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
71	70	rexbii 2551	. . . 4 |- ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) <-> E. r e. Q. ( q <Q r /\ ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
72	68, 71	sylibr 134	. . 3 |- ( ( A <P B /\ ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) )
73	3, 72	sylanr2 405	. 2 |- ( ( A <P B /\ ( q e. Q. /\ q e. ( 1st ` C ) ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) )
74	73	3impb 1226	1 |- ( ( A <P B /\ q e. Q. /\ q e. ( 1st ` C ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2205   E.wrex 2523   {crab 2526   <.cop 3697   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   Q.cnq 7611   +Q cplq 7613   <Q cltq 7616   P.cnp 7622   <P cltp 7626

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-ltnqqs 7684  df-inp 7797  df-iltp 7801

This theorem is referenced by:  ltexprlemrnd  7936