Theorem ltexprlemopl 7823

Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7835. (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 7820	. . . 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 1954	. . . . . . . 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 1954	. . . . . . . . 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 7727	. . . . . . . . . . . . . 14 |- <P C_ ( P. X. P. )
9	8	brel 4777	. . . . . . . . . . . . 13 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
10	9	simprd 114	. . . . . . . . . . . 12 |- ( A <P B -> B e. P. )
11		prop 7697	. . . . . . . . . . . . 13 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnmaxl 7710	. . . . . . . . . . . . 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 7697	. . . . . . . . . . . . . . 15 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
22		elprnqu 7704	. . . . . . . . . . . . . . 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 7629	. . . . . . . . . . . . 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 7620	. . . . . . . . . . . . 13 |- <Q Or Q.
30		ltrelnq 7587	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
31	29, 30	sotri 5131	. . . . . . . . . . . 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 7703	. . . . . . . . . . . . . 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 7630	. . . . . . . . . . . 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 4115	. . . . . . . . . . . . . 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 7622	. . . . . . . . . . . . . . 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 1273	. . . . . . . . . . . . . 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 2307	. . . . . . . . . . . . . 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 2633	. . . . . . . . . 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 2654	. . . . . . . 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 1648	. . . . . . 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 2825	. . . . . 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 1954	. . . . . . 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 1954	. . . . . . . 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 2538	. . . . 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 7820	. . . . . 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 2538	. . . 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 1225	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 1004   = wceq 1397   E.wex 1540   e. wcel 2201   E.wrex 2510   {crab 2513   <.cop 3671   class class class wbr 4087   ` cfv 5325  (class class class)co 6020   1stc1st 6303   2ndc2nd 6304   Q.cnq 7502   +Q cplq 7504   <Q cltq 7507   P.cnp 7513   <P cltp 7517

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-iinf 4685

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-tr 4187  df-eprel 4385  df-id 4389  df-po 4392  df-iso 4393  df-iord 4462  df-on 4464  df-suc 4467  df-iom 4688  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-ov 6023  df-oprab 6024  df-mpo 6025  df-1st 6305  df-2nd 6306  df-recs 6473  df-irdg 6538  df-1o 6584  df-oadd 6588  df-omul 6589  df-er 6704  df-ec 6706  df-qs 6710  df-ni 7526  df-pli 7527  df-mi 7528  df-lti 7529  df-plpq 7566  df-mpq 7567  df-enq 7569  df-nqqs 7570  df-plqqs 7571  df-mqqs 7572  df-1nqqs 7573  df-ltnqqs 7575  df-inp 7688  df-iltp 7692

This theorem is referenced by:  ltexprlemrnd  7827