Theorem ltexprlemopl 7591

Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7603. (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 7588	. . . 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 1906	. . . . . . . 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 1906	. . . . . . . . 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 7495	. . . . . . . . . . . . . 14 |- <P C_ ( P. X. P. )
9	8	brel 4675	. . . . . . . . . . . . 13 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
10	9	simprd 114	. . . . . . . . . . . 12 |- ( A <P B -> B e. P. )
11		prop 7465	. . . . . . . . . . . . 13 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnmaxl 7478	. . . . . . . . . . . . 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 536	. . . . . . . . . . . . . . 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 7465	. . . . . . . . . . . . . . 15 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
22		elprnqu 7472	. . . . . . . . . . . . . . 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 535	. . . . . . . . . . . . 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 7397	. . . . . . . . . . . . 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 531	. . . . . . . . . . . 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 7388	. . . . . . . . . . . . 13 |- <Q Or Q.
30		ltrelnq 7355	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
31	29, 30	sotri 5020	. . . . . . . . . . . 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 529	. . . . . . . . . . . . 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 7471	. . . . . . . . . . . . . 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 7398	. . . . . . . . . . . 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 536	. . . . . . . . . . . . . . 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 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 ) ) -> ( y +Q r ) = s )
43	41, 42	breqtrrd 4028	. . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . 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 1238	. . . . . . . . . . . . . 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 535	. . . . . . . . . . . . . . 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 2254	. . . . . . . . . . . . . 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 2579	. . . . . . . . . 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 2599	. . . . . . . 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 1600	. . . . . . 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 2760	. . . . . 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 1906	. . . . . . 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 1906	. . . . . . . 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 2484	. . . . 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 7588	. . . . . 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 2484	. . . 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 1199	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 978   = wceq 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   {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-1o 6411  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-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-ltnqqs 7343  df-inp 7456  df-iltp 7460

This theorem is referenced by:  ltexprlemrnd  7595