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Theorem caucvgprlemloc 8048
Description: Lemma for caucvgpr 8055. 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.
StepHypRef Expression
1 ltexnqi 7782 . . . . 5 |- ( s <Q r -> E. y e. Q. ( s +Q y ) = r )
21adantl 277 . . . 4 |- ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) -> E. y e. Q. ( s +Q y ) = r )
3 subhalfnqq 7787 . . . . . 6 |- ( y e. Q. -> E. x e. Q. ( x +Q x ) <Q y )
43ad2antrl 494 . . . . 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 8036 . . . . . . 7 |- ( x e. Q. -> E. m e. N. ( *Q ` [ <. m , 1o >. ] ~Q ) <Q x )
65ad2antrl 494 . . . . . 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 537 . . . . . . . . . . . 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 7795 . . . . . . . . . . . . . . 15 |- ( m e. N. -> [ <. m , 1o >. ] ~Q e. Q. )
9 recclnq 7765 . . . . . . . . . . . . . . 15 |- ( [ <. m , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
108, 9syl 14 . . . . . . . . . . . . . 14 |- ( m e. N. -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
1110ad2antrl 494 . . . . . . . . . . . . 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 541 . . . . . . . . . . . . 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 7777 . . . . . . . . . . . . 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 ) ) )
1411, 12, 11, 12, 13syl22anc 1279 . . . . . . . . . . . 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 ) ) )
157, 7, 14mp2and 437 . . . . . . . . . . 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 542 . . . . . . . . . . 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 7771 . . . . . . . . . . . 12 |- <Q Or Q.
18 ltrelnq 7738 . . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
1917, 18sotri 5186 . . . . . . . . . . 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 )
2015, 16, 19syl2anc 415 . . . . . . . . . 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 541 . . . . . . . . . . 11 |- ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) -> s e. Q. )
2221ad3antrrr 496 . . . . . . . . . 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 7775 . . . . . . . . . 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 ) )
2420, 22, 23syl2anc 415 . . . . . . . . 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 537 . . . . . . . . . 10 |- ( ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) /\ ( y e. Q. /\ ( s +Q y ) = r ) ) -> ( s +Q y ) = r )
2625ad2antrr 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 ) ) -> ( s +Q y ) = r )
2724, 26breqtrd 4157 . . . . . . . 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 7748 . . . . . . . . . . 11 |- ( ( ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. ) -> ( ( *Q ` [ <. m , 1o >. ] ~Q ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
2911, 11, 28syl2anc 415 . . . . . . . . . 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 7748 . . . . . . . . . 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. )
3122, 29, 30syl2anc 415 . . . . . . . . 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 542 . . . . . . . . . 10 |- ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) -> r e. Q. )
3332ad3antrrr 496 . . . . . . . . 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. )
3534ad5antr 500 . . . . . . . . . . 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 535 . . . . . . . . . . 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. )
3735, 36ffvelcdmd 5847 . . . . . . . . . 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 7748 . . . . . . . . . 10 |- ( ( ( F ` m ) e. Q. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
3937, 11, 38syl2anc 415 . . . . . . . . 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 4467 . . . . . . . . . 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 ) ) )
4117, 40mpan 428 . . . . . . . . 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 ) ) )
4231, 33, 39, 41syl3anc 1278 . . . . . . . 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 ) ) )
4327, 42mpd 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 ) )
4422adantr 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 541 . . . . . . . . . . 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 ) ) )
4711adantr 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 7755 . . . . . . . . . . . . . . 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 ) ) ) )
4944, 47, 47, 48syl3anc 1278 . . . . . . . . . . . . . 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 ) ) ) )
5049breq1d 4141 . . . . . . . . . . . . 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 ) ) ) )
5146, 50mpbird 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 7773 . . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
5352adantl 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 7748 . . . . . . . . . . . . . 14 |- ( ( s e. Q. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
5544, 47, 54syl2anc 415 . . . . . . . . . . . . 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. )
5637adantr 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 7754 . . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
5857adantl 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 ) )
5953, 55, 56, 47, 58caovord2d 6262 . . . . . . . . . . . 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 ) ) ) )
6051, 59mpbird 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 3904 . . . . . . . . . . . . . . . 16 |- ( j = m -> <. j , 1o >. = <. m , 1o >. )
6261eceq1d 6847 . . . . . . . . . . . . . . 15 |- ( j = m -> [ <. j , 1o >. ] ~Q = [ <. m , 1o >. ] ~Q )
6362fveq2d 5702 . . . . . . . . . . . . . 14 |- ( j = m -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. m , 1o >. ] ~Q ) )
6463oveq2d 6104 . . . . . . . . . . . . 13 |- ( j = m -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) )
65 fveq2 5698 . . . . . . . . . . . . 13 |- ( j = m -> ( F ` j ) = ( F ` m ) )
6664, 65breq12d 4144 . . . . . . . . . . . 12 |- ( j = m -> ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q ( F ` m ) ) )
6766rspcev 2929 . . . . . . . . . . 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 ) )
6845, 60, 67syl2anc 415 . . . . . . . . . 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 6095 . . . . . . . . . . . . 13 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
7069breq1d 4141 . . . . . . . . . . . 12 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
7170rexbidv 2551 . . . . . . . . . . 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 } >.
7372fveq2i 5701 . . . . . . . . . . . 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 7736 . . . . . . . . . . . . . 14 |- Q. e. _V
7574rabex 4282 . . . . . . . . . . . . 13 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
7674rabex 4282 . . . . . . . . . . . . 13 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
7775, 76op1st 6384 . . . . . . . . . . . 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 ) }
7873, 77eqtri 2259 . . . . . . . . . . 11 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
7971, 78elrab2 2985 . . . . . . . . . 10 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
8044, 68, 79sylanbrc 421 . . . . . . . . 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 ) )
8180ex 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 ) ) )
8233adantr 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. )
8365, 63oveq12d 6106 . . . . . . . . . . . . 13 |- ( j = m -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) )
8483breq1d 4141 . . . . . . . . . . . 12 |- ( j = m -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r <-> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q r ) )
8584rspcev 2929 . . . . . . . . . . 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 )
8636, 85sylan 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 4135 . . . . . . . . . . . 12 |- ( u = r -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) )
8887rexbidv 2551 . . . . . . . . . . 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 ) )
8972fveq2i 5701 . . . . . . . . . . . 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 } >. )
9075, 76op2nd 6385 . . . . . . . . . . . 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 }
9189, 90eqtri 2259 . . . . . . . . . . 11 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
9288, 91elrab2 2985 . . . . . . . . . 10 |- ( r e. ( 2nd ` L ) <-> ( r e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q r ) )
9382, 86, 92sylanbrc 421 . . . . . . . . 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 ) )
9493ex 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 ) ) )
9581, 94orim12d 798 . . . . . . 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 ) ) ) )
9643, 95mpd 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 ) ) )
976, 96rexlimddv 2673 . . . . 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 ) ) )
984, 97rexlimddv 2673 . . . 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 ) ) )
992, 98rexlimddv 2673 . . 3 |- ( ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) /\ s <Q r ) -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) )
10099ex 115 . 2 |- ( ( ph /\ ( s e. Q. /\ r e. Q. ) ) -> ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) )
101100ralrimivva 2632 1 |- ( ph -> A. s e. Q. A. r e. Q. ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   <.cop 3712   class class class wbr 4131   Or wor 4442   -->wf 5376   ` cfv 5380  (class class class)co 6088   1stc1st 6376   2ndc2nd 6377   1oc1o 6684   [cec 6809   N.cnpi 7645   <N clti 7648   ~Q ceq 7652   Q.cnq 7653   +Q cplq 7655   *Qcrq 7657   <Q cltq 7658
This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737
This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726
This theorem is used by:  caucvgprlemcl  8049
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