Theorem caucvgprlemloc 8006

Description: Lemma for caucvgpr 8013. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	|- ( ph -> F : N. --> Q. )
caucvgpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
caucvgpr.bnd	|- ( ph -> A. j e. N. A <Q ( F ` j ) )
caucvgpr.lim	|- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.

Assertion

Ref	Expression
caucvgprlemloc	|- ( ph -> A. s e. Q. A. r e. Q. ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) )

Distinct variable groups:   A, j   j, F, l   u, F   ph, j, r, s   s, l   u, j, r

Allowed substitution hints:   ph( u, k, n, l)   A( u, k, n, s, r, l)   F( k, n, s, r)   L( u, j, k, n, s, r, l)

Proof of Theorem caucvgprlemloc

Dummy variables f g h m x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexnqi 7740	. . . . 5 |- ( s <Q r -> E. y e. Q. ( s +Q y ) = r )
2	1	adantl 277	. . . 4 |- ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) -> E. y e. Q. ( s +Q y ) = r )
3		subhalfnqq 7745	. . . . . 6 |- ( y e. Q. -> E. x e. Q. ( x +Q x ) <Q y )
4	3	ad2antrl 490	. . . . 5 |- ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) -> E. x e. Q. ( x +Q x ) <Q y )
5		archrecnq 7994	. . . . . . 7 |- ( x e. Q. -> E. m e. N. ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x )
6	5	ad2antrl 490	. . . . . 6 |- ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> E. m e. N. ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x )
7		simprr 533	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x )
8		nnnq 7753	. . . . . . . . . . . . . . 15 |- ( m e. N. -> [ <. m , 1o >. ] ~Q e. Q. )
9		recclnq 7723	. . . . . . . . . . . . . . 15 |- ( [ <. m , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
10	8, 9	syl 14	. . . . . . . . . . . . . 14 |- ( m e. N. -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
11	10	ad2antrl 490	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
12		simplrl 537	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> x e. Q. )
13		lt2addnq 7735	. . . . . . . . . . . . 13 |- ( ( ( ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. /\ x e. Q. ) /\ ( ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. /\ x e. Q. ) ) -> ( ( ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) -> ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( x +Q x ) ) )
14	11, 12, 11, 12, 13	syl22anc 1275	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( ( ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) -> ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( x +Q x ) ) )
15	7, 7, 14	mp2and 433	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( x +Q x ) )
16		simplrr 538	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( x +Q x ) <Q y )
17		ltsonq 7729	. . . . . . . . . . . 12 |- <Q Or Q.
18		ltrelnq 7696	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
19	17, 18	sotri 5163	. . . . . . . . . . 11 |- ( ( ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( x +Q x ) /\ ( x +Q x ) <Q y ) -> ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q y )
20	15, 16, 19	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q y )
21		simplrl 537	. . . . . . . . . . 11 |- ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) -> s e. Q. )
22	21	ad3antrrr 492	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> s e. Q. )
23		ltanqi 7733	. . . . . . . . . 10 |- ( ( ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q y /\ s e. Q. ) -> ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( s +Q y ) )
24	20, 22, 23	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( s +Q y ) )
25		simprr 533	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) -> ( s +Q y ) = r )
26	25	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( s +Q y ) = r )
27	24, 26	breqtrd 4140	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q r )
28		addclnq 7706	. . . . . . . . . . 11 |- ( ( ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. ) -> ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
29	11, 11, 28	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
30		addclnq 7706	. . . . . . . . . 10 |- ( ( s e. Q. /\ ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. ) -> ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) e. Q. )
31	22, 29, 30	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) e. Q. )
32		simplrr 538	. . . . . . . . . 10 |- ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) -> r e. Q. )
33	32	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> r e. Q. )
34		caucvgpr.f	. . . . . . . . . . . 12 |- ( ph -> F : N. --> Q. )
35	34	ad5antr 496	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> F : N. --> Q. )
36		simprl 531	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> m e. N. )
37	35, 36	ffvelcdmd 5818	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( F ` m ) e. Q. )
38		addclnq 7706	. . . . . . . . . 10 |- ( ( ( F ` m ) e. Q. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
39	37, 11, 38	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
40		sowlin 4446	. . . . . . . . . 10 |- ( ( <Q Or Q. /\ ( ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) e. Q. /\ r e. Q. /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. ) ) -> ( ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q r -> ( ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) \/ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r ) ) )
41	17, 40	mpan 424	. . . . . . . . 9 |- ( ( ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) e. Q. /\ r e. Q. /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. ) -> ( ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q r -> ( ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) \/ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r ) ) )
42	31, 33, 39, 41	syl3anc 1274	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q r -> ( ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) \/ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r ) ) )
43	27, 42	mpd 13	. . . . . . 7 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) \/ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r ) )
44	22	adantr 276	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> s e. Q. )
45		simplrl 537	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> m e. N. )
46		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) )
47	11	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
48		addassnqg 7713	. . . . . . . . . . . . . . 15 |- ( ( s e. Q. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. ) -> ( ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) = ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) )
49	44, 47, 47, 48	syl3anc 1274	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> ( ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) = ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) )
50	49	breq1d 4124	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> ( ( ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <-> ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) )
51	46, 50	mpbird 167	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> ( ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) )
52		ltanqg 7731	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
53	52	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
54		addclnq 7706	. . . . . . . . . . . . . 14 |- ( ( s e. Q. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
55	44, 47, 54	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
56	37	adantr 276	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> ( F ` m ) e. Q. )
57		addcomnqg 7712	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
58	57	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
59	53, 55, 56, 47, 58	caovord2d 6232	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> ( ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( F ` m ) <-> ( ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) )
60	51, 59	mpbird 167	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( F ` m ) )
61		opeq1 3888	. . . . . . . . . . . . . . . 16 |- ( j = m -> <. j , 1o >. = <. m , 1o >. )
62	61	eceq1d 6816	. . . . . . . . . . . . . . 15 |- ( j = m -> [ <. j , 1o >. ] ~Q = [ <. m , 1o >. ] ~Q )
63	62	fveq2d 5679	. . . . . . . . . . . . . 14 |- ( j = m -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. m , 1o >. ] ~Q ) )
64	63	oveq2d 6074	. . . . . . . . . . . . 13 |- ( j = m -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) )
65		fveq2 5675	. . . . . . . . . . . . 13 |- ( j = m -> ( F ` j ) = ( F ` m ) )
66	64, 65	breq12d 4127	. . . . . . . . . . . 12 |- ( j = m -> ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( F ` m ) ) )
67	66	rspcev 2923	. . . . . . . . . . 11 |- ( ( m e. N. /\ ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( F ` m ) ) -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
68	45, 60, 67	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
69		oveq1 6065	. . . . . . . . . . . . 13 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
70	69	breq1d 4124	. . . . . . . . . . . 12 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
71	70	rexbidv 2545	. . . . . . . . . . 11 |- ( l = s -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
72		caucvgpr.lim	. . . . . . . . . . . . 13 |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
73	72	fveq2i 5678	. . . . . . . . . . . 12 |- ( 1st ` L ) = ( 1st ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. )
74		nqex 7694	. . . . . . . . . . . . . 14 |- Q. e. _V
75	74	rabex 4261	. . . . . . . . . . . . 13 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
76	74	rabex 4261	. . . . . . . . . . . . 13 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
77	75, 76	op1st 6353	. . . . . . . . . . . 12 |- ( 1st ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
78	73, 77	eqtri 2255	. . . . . . . . . . 11 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
79	71, 78	elrab2 2979	. . . . . . . . . 10 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
80	44, 68, 79	sylanbrc 417	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) -> s e. ( 1st ` L ) )
81	80	ex 115	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) -> s e. ( 1st ` L ) ) )
82	33	adantr 276	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r ) -> r e. Q. )
83	65, 63	oveq12d 6076	. . . . . . . . . . . . 13 |- ( j = m -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) )
84	83	breq1d 4124	. . . . . . . . . . . 12 |- ( j = m -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r <-> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r ) )
85	84	rspcev 2923	. . . . . . . . . . 11 |- ( ( m e. N. /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r )
86	36, 85	sylan 283	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r )
87		breq2 4118	. . . . . . . . . . . 12 |- ( u = r -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) )
88	87	rexbidv 2545	. . . . . . . . . . 11 |- ( u = r -> ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) )
89	72	fveq2i 5678	. . . . . . . . . . . 12 |- ( 2nd ` L ) = ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. )
90	75, 76	op2nd 6354	. . . . . . . . . . . 12 |- ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
91	89, 90	eqtri 2255	. . . . . . . . . . 11 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
92	88, 91	elrab2 2979	. . . . . . . . . 10 |- ( r e. ( 2nd ` L ) <-> ( r e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) )
93	82, 86, 92	sylanbrc 417	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r ) -> r e. ( 2nd ` L ) )
94	93	ex 115	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r -> r e. ( 2nd ` L ) ) )
95	81, 94	orim12d 794	. . . . . . 7 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( ( ( s +Q ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) ) <Q ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) \/ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r ) -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) )
96	43, 95	mpd 13	. . . . . 6 |- ( ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) /\ ( m e. N. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x ) ) -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) )
97	6, 96	rexlimddv 2667	. . . . 5 |- ( ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) /\ ( x e. Q. /\ ( x +Q x ) <Q y ) ) -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) )
98	4, 97	rexlimddv 2667	. . . 4 |- ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) )
99	2, 98	rexlimddv 2667	. . 3 |- ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) )
100	99	ex 115	. 2 |- ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) -> ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) )
101	100	ralrimivva 2626	1 |- ( ph -> A. s e. Q. A. r e. Q. ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   {crab 2526   <.cop 3697   class class class wbr 4114   Or wor 4421   -->wf 5353   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   1oc1o 6653   [cec 6778   N.cnpi 7603   <N clti 7606   ~Q ceq 7610   Q.cnq 7611   +Q cplq 7613   *Qcrq 7615   <Q cltq 7616

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684

This theorem is referenced by:  caucvgprlemcl  8007