Theorem ltexprlemopu 7535

Description: The upper cut of our constructed difference is open. Lemma for ltexpri 7545. (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 7531	. . . 4 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
3	2	simprbi 273	. . 3 |- ( r e. ( 2nd ` C ) -> E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) )
4		19.42v 1893	. . . . . . . 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 1893	. . . . . . . . 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 453	. . . . . . . 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 183	. . . . . . 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 7437	. . . . . . . . . . . . . . 15 |- <P C_ ( P. X. P. )
9	8	brel 4650	. . . . . . . . . . . . . 14 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
10	9	simprd 113	. . . . . . . . . . . . 13 |- ( A <P B -> B e. P. )
11		prop 7407	. . . . . . . . . . . . 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 7421	. . . . . . . . . . . 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 281	. . . . . . . . . . 11 |- ( ( A <P B /\ ( y +Q r ) e. ( 2nd ` B ) ) -> E. s e. ( 2nd ` B ) s <Q ( y +Q r ) )
15	14	adantrl 470	. . . . . . . . . 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 470	. . . . . . . . 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 7438	. . . . . . . . . . . . . . 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 143	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 111	. . . . . . . . . . . . . . . 16 |- ( A <P B -> A e. P. )
22	21	ad2antrr 480	. . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 526	. . . . . . . . . . . . . . . 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 111	. . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . 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 7407	. . . . . . . . . . . . . . 15 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
29		prltlu 7419	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) /\ t e. ( 2nd ` A ) ) -> y <Q t )
30	28, 29	syl3an1 1260	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ y e. ( 1st ` A ) /\ t e. ( 2nd ` A ) ) -> y <Q t )
31	23, 26, 27, 30	syl3anc 1227	. . . . . . . . . . . . 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 523	. . . . . . . . . . . . . 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 530	. . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . 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 7419	. . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ t e. ( 1st ` B ) /\ s e. ( 2nd ` B ) ) -> t <Q s )
36	12, 35	syl3an1 1260	. . . . . . . . . . . . . 14 |- ( ( A <P B /\ t e. ( 1st ` B ) /\ s e. ( 2nd ` B ) ) -> t <Q s )
37	32, 33, 34, 36	syl3anc 1227	. . . . . . . . . . . . 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 7330	. . . . . . . . . . . . . 14 |- <Q Or Q.
39		ltrelnq 7297	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
40	38, 39	sotri 4993	. . . . . . . . . . . . 13 |- ( ( y <Q t /\ t <Q s ) -> y <Q s )
41	31, 37, 40	syl2anc 409	. . . . . . . . . . . 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 2586	. . . . . . . . . . 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 7413	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
44	28, 43	sylan 281	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
45	22, 25, 44	syl2anc 409	. . . . . . . . . . . 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 7414	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ s e. ( 2nd ` B ) ) -> s e. Q. )
47	12, 46	sylan 281	. . . . . . . . . . . . 13 |- ( ( A <P B /\ s e. ( 2nd ` B ) ) -> s e. Q. )
48	47	ad2ant2r 501	. . . . . . . . . . . 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 7340	. . . . . . . . . . . 12 |- ( ( y e. Q. /\ s e. Q. ) -> ( y <Q s <-> E. q e. Q. ( y +Q q ) = s ) )
50	45, 48, 49	syl2anc 409	. . . . . . . . . . 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 146	. . . . . . . . . 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 522	. . . . . . . . . . . . . . 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 526	. . . . . . . . . . . . . . 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 3998	. . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . 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 7332	. . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . 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 166	. . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . . 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 2241	. . . . . . . . . . . . . 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 304	. . . . . . . . . . . . 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 308	. . . . . . . . . . . 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 373	. . . . . . . . . . 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 2566	. . . . . . . . . 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 2586	. . . . . . . 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 1587	. . . . . . 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 134	. . . . . 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 2744	. . . . . 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 133	. . . . 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 1893	. . . . . . 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 1893	. . . . . . . 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 453	. . . . . . 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 183	. . . . . 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 2471	. . . . 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 121	. . . 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 7531	. . . . . 6 |- ( q e. ( 2nd ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) )
82	81	anbi2i 453	. . . . 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 2471	. . . 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 133	. . 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 403	. 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 1188	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 103   <-> wb 104   /\ w3a 967   = wceq 1342   E.wex 1479   e. wcel 2135   E.wrex 2443   {crab 2446   <.cop 3573   class class class wbr 3976   ` cfv 5182  (class class class)co 5836   1stc1st 6098   2ndc2nd 6099   Q.cnq 7212   +Q cplq 7214   <Q cltq 7217   P.cnp 7223   <P cltp 7227

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-eprel 4261  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-1o 6375  df-oadd 6379  df-omul 6380  df-er 6492  df-ec 6494  df-qs 6498  df-ni 7236  df-pli 7237  df-mi 7238  df-lti 7239  df-plpq 7276  df-mpq 7277  df-enq 7279  df-nqqs 7280  df-plqqs 7281  df-mqqs 7282  df-1nqqs 7283  df-ltnqqs 7285  df-inp 7398  df-iltp 7402

This theorem is referenced by:  ltexprlemrnd  7537