Theorem ltexprlemopl 7542

Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7554. (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 7539	. . . 4 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
3	2	simprbi 273	. . 3 |- ( q e. ( 1st ` C ) -> E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) )
4		19.42v 1894	. . . . . . . 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 1894	. . . . . . . . 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 453	. . . . . . . 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 183	. . . . . . 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 7446	. . . . . . . . . . . . . 14 |- <P C_ ( P. X. P. )
9	8	brel 4656	. . . . . . . . . . . . 13 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
10	9	simprd 113	. . . . . . . . . . . 12 |- ( A <P B -> B e. P. )
11		prop 7416	. . . . . . . . . . . . 13 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnmaxl 7429	. . . . . . . . . . . . 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 281	. . . . . . . . . . . 12 |- ( ( B e. P. /\ ( y +Q q ) e. ( 1st ` B ) ) -> E. s e. ( 1st ` B ) ( y +Q q ) <Q s )
14	10, 13	sylan 281	. . . . . . . . . . 11 |- ( ( A <P B /\ ( y +Q q ) e. ( 1st ` B ) ) -> E. s e. ( 1st ` B ) ( y +Q q ) <Q s )
15	14	adantrl 470	. . . . . . . . . 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 470	. . . . . . . . 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 111	. . . . . . . . . . . . . . 15 |- ( A <P B -> A e. P. )
18	17	ad2antrr 480	. . . . . . . . . . . . . 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 526	. . . . . . . . . . . . . . 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 111	. . . . . . . . . . . . . 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 7416	. . . . . . . . . . . . . . 15 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
22		elprnqu 7423	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
23	21, 22	sylan 281	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
24	18, 20, 23	syl2anc 409	. . . . . . . . . . . . 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 525	. . . . . . . . . . . . 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 7348	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ q e. Q. ) -> y <Q ( y +Q q ) )
27	24, 25, 26	syl2anc 409	. . . . . . . . . . . 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 522	. . . . . . . . . . . 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 7339	. . . . . . . . . . . . 13 |- <Q Or Q.
30		ltrelnq 7306	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
31	29, 30	sotri 4999	. . . . . . . . . . . 12 |- ( ( y <Q ( y +Q q ) /\ ( y +Q q ) <Q s ) -> y <Q s )
32	27, 28, 31	syl2anc 409	. . . . . . . . . . 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 480	. . . . . . . . . . . . 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 521	. . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ s e. ( 1st ` B ) ) -> s e. Q. )
36	11, 35	sylan 281	. . . . . . . . . . . . 13 |- ( ( B e. P. /\ s e. ( 1st ` B ) ) -> s e. Q. )
37	33, 34, 36	syl2anc 409	. . . . . . . . . . . 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 7349	. . . . . . . . . . . 12 |- ( ( y e. Q. /\ s e. Q. ) -> ( y <Q s <-> E. r e. Q. ( y +Q r ) = s ) )
39	24, 37, 38	syl2anc 409	. . . . . . . . . . 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 146	. . . . . . . . . 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 526	. . . . . . . . . . . . . . 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 522	. . . . . . . . . . . . . . 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 4010	. . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . 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 7341	. . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . 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 166	. . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . . 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 2243	. . . . . . . . . . . . . 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 304	. . . . . . . . . . . . 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 308	. . . . . . . . . . . 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 373	. . . . . . . . . . 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 2568	. . . . . . . . . 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 2588	. . . . . . . 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 1588	. . . . . . 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 134	. . . . . 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 2749	. . . . . 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 133	. . . . 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 1894	. . . . . . 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 1894	. . . . . . . 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 453	. . . . . . 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 183	. . . . . 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 2473	. . . . 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 121	. . . 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 7539	. . . . . 6 |- ( r e. ( 1st ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) )
70	69	anbi2i 453	. . . . 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 2473	. . . 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 133	. . 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 403	. 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 1189	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 103   <-> wb 104   /\ w3a 968   = wceq 1343   E.wex 1480   e. wcel 2136   E.wrex 2445   {crab 2448   <.cop 3579   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226   P.cnp 7232   <P cltp 7236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-ltnqqs 7294  df-inp 7407  df-iltp 7411

This theorem is referenced by:  ltexprlemrnd  7546