Theorem ltexprlemopu 7960

Description: The upper cut of our constructed difference is open. Lemma for ltexpri 7970. (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
ltexprlemopu	|- ( ( A <P B /\ r e. Q. /\ r e. ( 2nd ` C ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) )

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

Proof of Theorem ltexprlemopu

Dummy variables s t are mutually distinct and 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	ltexprlemelu 7956	. . . 4 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
3	2	simprbi 275	. . 3 |- ( r e. ( 2nd ` C ) -> E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) )
4		19.42v 1962	. . . . . . . 8 |- ( E. y ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) <-> ( A <P B /\ E. y ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) )
5		19.42v 1962	. . . . . . . . 9 |- ( E. y ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
6	5	anbi2i 461	. . . . . . . 8 |- ( ( A <P B /\ E. y ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) <-> ( A <P B /\ ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) )
7	4, 6	bitri 184	. . . . . . 7 |- ( E. y ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) <-> ( A <P B /\ ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) )
8		ltrelpr 7862	. . . . . . . . . . . . . . 15 |- <P C_ ( P. X. P. )
9	8	brel 4822	. . . . . . . . . . . . . 14 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
10	9	simprd 114	. . . . . . . . . . . . 13 |- ( A <P B -> B e. P. )
11		prop 7832	. . . . . . . . . . . . 13 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12	10, 11	syl 14	. . . . . . . . . . . 12 |- ( A <P B -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
13		prnminu 7846	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q r ) e. ( 2nd ` B ) ) -> E. s e. ( 2nd ` B ) s <Q ( y +Q r ) )
14	12, 13	sylan 283	. . . . . . . . . . 11 |- ( ( A <P B /\ ( y +Q r ) e. ( 2nd ` B ) ) -> E. s e. ( 2nd ` B ) s <Q ( y +Q r ) )
15	14	adantrl 482	. . . . . . . . . 10 |- ( ( A <P B /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) -> E. s e. ( 2nd ` B ) s <Q ( y +Q r ) )
16	15	adantrl 482	. . . . . . . . 9 |- ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) -> E. s e. ( 2nd ` B ) s <Q ( y +Q r ) )
17		ltdfpr 7863	. . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> E. t e. Q. ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) ) )
18	17	biimpd 144	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B -> E. t e. Q. ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) ) )
19	9, 18	mpcom 36	. . . . . . . . . . . . 13 |- ( A <P B -> E. t e. Q. ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) )
20	19	ad2antrr 492	. . . . . . . . . . . 12 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> E. t e. Q. ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) )
21	9	simpld 112	. . . . . . . . . . . . . . . 16 |- ( A <P B -> A e. P. )
22	21	ad2antrr 492	. . . . . . . . . . . . . . 15 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> A e. P. )
23	22	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( t e. Q. /\ ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) ) ) -> A e. P. )
24		simplrr 542	. . . . . . . . . . . . . . . 16 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) )
25	24	simpld 112	. . . . . . . . . . . . . . 15 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> y e. ( 1st ` A ) )
26	25	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( t e. Q. /\ ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) ) ) -> y e. ( 1st ` A ) )
27		simprrl 545	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( t e. Q. /\ ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) ) ) -> t e. ( 2nd ` A ) )
28		prop 7832	. . . . . . . . . . . . . . 15 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
29		prltlu 7844	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) /\ t e. ( 2nd ` A ) ) -> y <Q t )
30	28, 29	syl3an1 1311	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ y e. ( 1st ` A ) /\ t e. ( 2nd ` A ) ) -> y <Q t )
31	23, 26, 27, 30	syl3anc 1278	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( t e. Q. /\ ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) ) ) -> y <Q t )
32		simplll 539	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( t e. Q. /\ ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) ) ) -> A <P B )
33		simprrr 546	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( t e. Q. /\ ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) ) ) -> t e. ( 1st ` B ) )
34		simplrl 541	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( t e. Q. /\ ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) ) ) -> s e. ( 2nd ` B ) )
35		prltlu 7844	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ t e. ( 1st ` B ) /\ s e. ( 2nd ` B ) ) -> t <Q s )
36	12, 35	syl3an1 1311	. . . . . . . . . . . . . 14 |- ( ( A <P B /\ t e. ( 1st ` B ) /\ s e. ( 2nd ` B ) ) -> t <Q s )
37	32, 33, 34, 36	syl3anc 1278	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( t e. Q. /\ ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) ) ) -> t <Q s )
38		ltsonq 7755	. . . . . . . . . . . . . 14 |- <Q Or Q.
39		ltrelnq 7722	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
40	38, 39	sotri 5178	. . . . . . . . . . . . 13 |- ( ( y <Q t /\ t <Q s ) -> y <Q s )
41	31, 37, 40	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( t e. Q. /\ ( t e. ( 2nd ` A ) /\ t e. ( 1st ` B ) ) ) ) -> y <Q s )
42	20, 41	rexlimddv 2673	. . . . . . . . . . 11 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> y <Q s )
43		elprnql 7838	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
44	28, 43	sylan 283	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
45	22, 25, 44	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> y e. Q. )
46		elprnqu 7839	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ s e. ( 2nd ` B ) ) -> s e. Q. )
47	12, 46	sylan 283	. . . . . . . . . . . . 13 |- ( ( A <P B /\ s e. ( 2nd ` B ) ) -> s e. Q. )
48	47	ad2ant2r 513	. . . . . . . . . . . 12 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> s e. Q. )
49		ltexnqq 7765	. . . . . . . . . . . 12 |- ( ( y e. Q. /\ s e. Q. ) -> ( y <Q s <-> E. q e. Q. ( y +Q q ) = s ) )
50	45, 48, 49	syl2anc 415	. . . . . . . . . . 11 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> ( y <Q s <-> E. q e. Q. ( y +Q q ) = s ) )
51	42, 50	mpbid 147	. . . . . . . . . 10 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> E. q e. Q. ( y +Q q ) = s )
52		simprr 537	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> ( y +Q q ) = s )
53		simplrr 542	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> s <Q ( y +Q r ) )
54	52, 53	eqbrtrd 4147	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> ( y +Q q ) <Q ( y +Q r ) )
55		simprl 535	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> q e. Q. )
56		simplrl 541	. . . . . . . . . . . . . . . 16 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> r e. Q. )
57	56	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> r e. Q. )
58	45	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> y e. Q. )
59		ltanqg 7757	. . . . . . . . . . . . . . 15 |- ( ( q e. Q. /\ r e. Q. /\ y e. Q. ) -> ( q <Q r <-> ( y +Q q ) <Q ( y +Q r ) ) )
60	55, 57, 58, 59	syl3anc 1278	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> ( q <Q r <-> ( y +Q q ) <Q ( y +Q r ) ) )
61	54, 60	mpbird 167	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> q <Q r )
62	25	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> y e. ( 1st ` A ) )
63		simplrl 541	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> s e. ( 2nd ` B ) )
64	52, 63	eqeltrd 2315	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> ( y +Q q ) e. ( 2nd ` B ) )
65	62, 64	jca 306	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) )
66	61, 55, 65	jca32 310	. . . . . . . . . . . 12 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ ( q e. Q. /\ ( y +Q q ) = s ) ) -> ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
67	66	expr 375	. . . . . . . . . . 11 |- ( ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) /\ q e. Q. ) -> ( ( y +Q q ) = s -> ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) )
68	67	reximdva 2652	. . . . . . . . . 10 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> ( E. q e. Q. ( y +Q q ) = s -> E. q e. Q. ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) )
69	51, 68	mpd 13	. . . . . . . . 9 |- ( ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q ( y +Q r ) ) ) -> E. q e. Q. ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
70	16, 69	rexlimddv 2673	. . . . . . . 8 |- ( ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) -> E. q e. Q. ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
71	70	eximi 1653	. . . . . . 7 |- ( E. y ( A <P B /\ ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) -> E. y E. q e. Q. ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
72	7, 71	sylbir 135	. . . . . 6 |- ( ( A <P B /\ ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) -> E. y E. q e. Q. ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
73		rexcom4 2845	. . . . . 6 |- ( E. q e. Q. E. y ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) <-> E. y E. q e. Q. ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
74	72, 73	sylibr 134	. . . . 5 |- ( ( A <P B /\ ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) -> E. q e. Q. E. y ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
75		19.42v 1962	. . . . . . 7 |- ( E. y ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) <-> ( q <Q r /\ E. y ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
76		19.42v 1962	. . . . . . . 8 |- ( E. y ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) <-> ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) )
77	76	anbi2i 461	. . . . . . 7 |- ( ( q <Q r /\ E. y ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) <-> ( q <Q r /\ ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
78	75, 77	bitri 184	. . . . . 6 |- ( E. y ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) <-> ( q <Q r /\ ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
79	78	rexbii 2557	. . . . 5 |- ( E. q e. Q. E. y ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) <-> E. q e. Q. ( q <Q r /\ ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
80	74, 79	sylib 122	. . . 4 |- ( ( A <P B /\ ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) -> E. q e. Q. ( q <Q r /\ ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
81	1	ltexprlemelu 7956	. . . . . 6 |- ( q e. ( 2nd ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) )
82	81	anbi2i 461	. . . . 5 |- ( ( q <Q r /\ q e. ( 2nd ` C ) ) <-> ( q <Q r /\ ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
83	82	rexbii 2557	. . . 4 |- ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) <-> E. q e. Q. ( q <Q r /\ ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
84	80, 83	sylibr 134	. . 3 |- ( ( A <P B /\ ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) )
85	3, 84	sylanr2 409	. 2 |- ( ( A <P B /\ ( r e. Q. /\ r e. ( 2nd ` C ) ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) )
86	85	3impb 1230	1 |- ( ( A <P B /\ r e. Q. /\ r e. ( 2nd ` C ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   {crab 2532   <.cop 3708   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   +Q cplq 7639   <Q cltq 7642   P.cnp 7648   <P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-ltnqqs 7710  df-inp 7823  df-iltp 7827

This theorem is referenced by:  ltexprlemrnd  7962