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Theorem monotuz 29282
Description: A function defined on an upper set of integers which increases at every adjacent pair is globally strictly monotonic by induction. (Contributed by Stefan O'Rear, 24-Sep-2014.)
Hypotheses
Ref Expression
monotuz.1  |-  ( (
ph  /\  y  e.  H )  ->  F  <  G )
monotuz.2  |-  ( (
ph  /\  x  e.  H )  ->  C  e.  RR )
monotuz.3  |-  H  =  ( ZZ>= `  I )
monotuz.4  |-  ( x  =  ( y  +  1 )  ->  C  =  G )
monotuz.5  |-  ( x  =  y  ->  C  =  F )
monotuz.6  |-  ( x  =  A  ->  C  =  D )
monotuz.7  |-  ( x  =  B  ->  C  =  E )
Assertion
Ref Expression
monotuz  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( A  <  B  <->  D  <  E ) )
Distinct variable groups:    x, A, y    x, B, y    y, C    x, D, y    x, E, y    x, F    x, G    x, H, y    ph, x, y
Allowed substitution hints:    C( x)    F( y)    G( y)    I( x, y)

Proof of Theorem monotuz
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbeq1 3291 . . 3  |-  ( a  =  b  ->  [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C )
2 csbeq1 3291 . . 3  |-  ( a  =  A  ->  [_ a  /  x ]_ C  = 
[_ A  /  x ]_ C )
3 csbeq1 3291 . . 3  |-  ( a  =  B  ->  [_ a  /  x ]_ C  = 
[_ B  /  x ]_ C )
4 monotuz.3 . . . 4  |-  H  =  ( ZZ>= `  I )
5 uzssz 10880 . . . . 5  |-  ( ZZ>= `  I )  C_  ZZ
6 zssre 10653 . . . . 5  |-  ZZ  C_  RR
75, 6sstri 3365 . . . 4  |-  ( ZZ>= `  I )  C_  RR
84, 7eqsstri 3386 . . 3  |-  H  C_  RR
9 nfv 1673 . . . . 5  |-  F/ x
( ph  /\  a  e.  H )
10 nfcsb1v 3304 . . . . . 6  |-  F/_ x [_ a  /  x ]_ C
1110nfel1 2589 . . . . 5  |-  F/ x [_ a  /  x ]_ C  e.  RR
129, 11nfim 1853 . . . 4  |-  F/ x
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  e.  RR )
13 eleq1 2503 . . . . . 6  |-  ( x  =  a  ->  (
x  e.  H  <->  a  e.  H ) )
1413anbi2d 703 . . . . 5  |-  ( x  =  a  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  a  e.  H ) ) )
15 csbeq1a 3297 . . . . . 6  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
1615eleq1d 2509 . . . . 5  |-  ( x  =  a  ->  ( C  e.  RR  <->  [_ a  /  x ]_ C  e.  RR ) )
1714, 16imbi12d 320 . . . 4  |-  ( x  =  a  ->  (
( ( ph  /\  x  e.  H )  ->  C  e.  RR )  <-> 
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  e.  RR ) ) )
18 monotuz.2 . . . 4  |-  ( (
ph  /\  x  e.  H )  ->  C  e.  RR )
1912, 17, 18chvar 1957 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  e.  RR )
20 simpl 457 . . . . . 6  |-  ( ( ( ph  /\  a  e.  H )  /\  a  <  b )  ->  ( ph  /\  a  e.  H
) )
2120adantlrr 720 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  ( ph  /\  a  e.  H ) )
224, 5eqsstri 3386 . . . . . . 7  |-  H  C_  ZZ
23 simplrl 759 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  e.  H )
2422, 23sseldi 3354 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  e.  ZZ )
25 simplrr 760 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  b  e.  H )
2622, 25sseldi 3354 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  b  e.  ZZ )
27 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  <  b )
28 csbeq1 3291 . . . . . . . . 9  |-  ( c  =  ( a  +  1 )  ->  [_ c  /  x ]_ C  = 
[_ ( a  +  1 )  /  x ]_ C )
2928breq2d 4304 . . . . . . . 8  |-  ( c  =  ( a  +  1 )  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ (
a  +  1 )  /  x ]_ C
) )
3029imbi2d 316 . . . . . . 7  |-  ( c  =  ( a  +  1 )  ->  (
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ c  /  x ]_ C )  <-> 
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ (
a  +  1 )  /  x ]_ C
) ) )
31 csbeq1 3291 . . . . . . . . 9  |-  ( c  =  d  ->  [_ c  /  x ]_ C  = 
[_ d  /  x ]_ C )
3231breq2d 4304 . . . . . . . 8  |-  ( c  =  d  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C ) )
3332imbi2d 316 . . . . . . 7  |-  ( c  =  d  ->  (
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ c  /  x ]_ C )  <-> 
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C ) ) )
34 csbeq1 3291 . . . . . . . . 9  |-  ( c  =  ( d  +  1 )  ->  [_ c  /  x ]_ C  = 
[_ ( d  +  1 )  /  x ]_ C )
3534breq2d 4304 . . . . . . . 8  |-  ( c  =  ( d  +  1 )  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) )
3635imbi2d 316 . . . . . . 7  |-  ( c  =  ( d  +  1 )  ->  (
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ c  /  x ]_ C )  <-> 
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) ) )
37 csbeq1 3291 . . . . . . . . 9  |-  ( c  =  b  ->  [_ c  /  x ]_ C  = 
[_ b  /  x ]_ C )
3837breq2d 4304 . . . . . . . 8  |-  ( c  =  b  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
3938imbi2d 316 . . . . . . 7  |-  ( c  =  b  ->  (
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ c  /  x ]_ C )  <-> 
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) ) )
40 eleq1 2503 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  e.  H  <->  a  e.  H ) )
4140anbi2d 703 . . . . . . . . 9  |-  ( y  =  a  ->  (
( ph  /\  y  e.  H )  <->  ( ph  /\  a  e.  H ) ) )
42 vex 2975 . . . . . . . . . . . 12  |-  y  e. 
_V
43 nfcv 2579 . . . . . . . . . . . 12  |-  F/_ x F
44 monotuz.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  C  =  F )
4542, 43, 44csbief 3313 . . . . . . . . . . 11  |-  [_ y  /  x ]_ C  =  F
46 csbeq1 3291 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ y  /  x ]_ C  = 
[_ a  /  x ]_ C )
4745, 46syl5eqr 2489 . . . . . . . . . 10  |-  ( y  =  a  ->  F  =  [_ a  /  x ]_ C )
48 ovex 6116 . . . . . . . . . . . 12  |-  ( y  +  1 )  e. 
_V
49 nfcv 2579 . . . . . . . . . . . 12  |-  F/_ x G
50 monotuz.4 . . . . . . . . . . . 12  |-  ( x  =  ( y  +  1 )  ->  C  =  G )
5148, 49, 50csbief 3313 . . . . . . . . . . 11  |-  [_ (
y  +  1 )  /  x ]_ C  =  G
52 oveq1 6098 . . . . . . . . . . . 12  |-  ( y  =  a  ->  (
y  +  1 )  =  ( a  +  1 ) )
5352csbeq1d 3295 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ (
y  +  1 )  /  x ]_ C  =  [_ ( a  +  1 )  /  x ]_ C )
5451, 53syl5eqr 2489 . . . . . . . . . 10  |-  ( y  =  a  ->  G  =  [_ ( a  +  1 )  /  x ]_ C )
5547, 54breq12d 4305 . . . . . . . . 9  |-  ( y  =  a  ->  ( F  <  G  <->  [_ a  /  x ]_ C  <  [_ (
a  +  1 )  /  x ]_ C
) )
5641, 55imbi12d 320 . . . . . . . 8  |-  ( y  =  a  ->  (
( ( ph  /\  y  e.  H )  ->  F  <  G )  <-> 
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ (
a  +  1 )  /  x ]_ C
) ) )
57 monotuz.1 . . . . . . . 8  |-  ( (
ph  /\  y  e.  H )  ->  F  <  G )
5856, 57vtoclg 3030 . . . . . . 7  |-  ( a  e.  ZZ  ->  (
( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ ( a  +  1 )  /  x ]_ C ) )
59193ad2ant2 1010 . . . . . . . . . 10  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  [_ a  /  x ]_ C  e.  RR )
60 simp2l 1014 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  ph )
61 zre 10650 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  ZZ  ->  a  e.  RR )
62613ad2ant1 1009 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  e.  RR )
63 zre 10650 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ZZ  ->  d  e.  RR )
64633ad2ant2 1010 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  d  e.  RR )
65 simp3 990 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  <  d )
6662, 64, 65ltled 9522 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  <_  d )
67663ad2ant1 1009 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  <_  d )
68 simp11 1018 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  e.  ZZ )
69 simp12 1019 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  ZZ )
70 eluz 10874 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ )  ->  ( d  e.  (
ZZ>= `  a )  <->  a  <_  d ) )
7168, 69, 70syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  (
d  e.  ( ZZ>= `  a )  <->  a  <_  d ) )
7267, 71mpbird 232 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  ( ZZ>= `  a )
)
73 simp2r 1015 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  e.  H )
7473, 4syl6eleq 2533 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  e.  ( ZZ>= `  I )
)
75 uztrn 10877 . . . . . . . . . . . . 13  |-  ( ( d  e.  ( ZZ>= `  a )  /\  a  e.  ( ZZ>= `  I )
)  ->  d  e.  ( ZZ>= `  I )
)
7672, 74, 75syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  ( ZZ>= `  I )
)
7776, 4syl6eleqr 2534 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  H )
78 nfv 1673 . . . . . . . . . . . . 13  |-  F/ x
( ph  /\  d  e.  H )
79 nfcsb1v 3304 . . . . . . . . . . . . . 14  |-  F/_ x [_ d  /  x ]_ C
8079nfel1 2589 . . . . . . . . . . . . 13  |-  F/ x [_ d  /  x ]_ C  e.  RR
8178, 80nfim 1853 . . . . . . . . . . . 12  |-  F/ x
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR )
82 eleq1 2503 . . . . . . . . . . . . . 14  |-  ( x  =  d  ->  (
x  e.  H  <->  d  e.  H ) )
8382anbi2d 703 . . . . . . . . . . . . 13  |-  ( x  =  d  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  d  e.  H ) ) )
84 csbeq1a 3297 . . . . . . . . . . . . . 14  |-  ( x  =  d  ->  C  =  [_ d  /  x ]_ C )
8584eleq1d 2509 . . . . . . . . . . . . 13  |-  ( x  =  d  ->  ( C  e.  RR  <->  [_ d  /  x ]_ C  e.  RR ) )
8683, 85imbi12d 320 . . . . . . . . . . . 12  |-  ( x  =  d  ->  (
( ( ph  /\  x  e.  H )  ->  C  e.  RR )  <-> 
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR ) ) )
8781, 86, 18chvar 1957 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR )
8860, 77, 87syl2anc 661 . . . . . . . . . 10  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  [_ d  /  x ]_ C  e.  RR )
89 peano2uz 10908 . . . . . . . . . . . . 13  |-  ( d  e.  ( ZZ>= `  I
)  ->  ( d  +  1 )  e.  ( ZZ>= `  I )
)
9076, 89syl 16 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  (
d  +  1 )  e.  ( ZZ>= `  I
) )
9190, 4syl6eleqr 2534 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  (
d  +  1 )  e.  H )
92 nfv 1673 . . . . . . . . . . . . 13  |-  F/ x
( ph  /\  (
d  +  1 )  e.  H )
93 nfcsb1v 3304 . . . . . . . . . . . . . 14  |-  F/_ x [_ ( d  +  1 )  /  x ]_ C
9493nfel1 2589 . . . . . . . . . . . . 13  |-  F/ x [_ ( d  +  1 )  /  x ]_ C  e.  RR
9592, 94nfim 1853 . . . . . . . . . . . 12  |-  F/ x
( ( ph  /\  ( d  +  1 )  e.  H )  ->  [_ ( d  +  1 )  /  x ]_ C  e.  RR )
96 ovex 6116 . . . . . . . . . . . 12  |-  ( d  +  1 )  e. 
_V
97 eleq1 2503 . . . . . . . . . . . . . 14  |-  ( x  =  ( d  +  1 )  ->  (
x  e.  H  <->  ( d  +  1 )  e.  H ) )
9897anbi2d 703 . . . . . . . . . . . . 13  |-  ( x  =  ( d  +  1 )  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  ( d  +  1 )  e.  H ) ) )
99 csbeq1a 3297 . . . . . . . . . . . . . 14  |-  ( x  =  ( d  +  1 )  ->  C  =  [_ ( d  +  1 )  /  x ]_ C )
10099eleq1d 2509 . . . . . . . . . . . . 13  |-  ( x  =  ( d  +  1 )  ->  ( C  e.  RR  <->  [_ ( d  +  1 )  /  x ]_ C  e.  RR ) )
10198, 100imbi12d 320 . . . . . . . . . . . 12  |-  ( x  =  ( d  +  1 )  ->  (
( ( ph  /\  x  e.  H )  ->  C  e.  RR )  <-> 
( ( ph  /\  ( d  +  1 )  e.  H )  ->  [_ ( d  +  1 )  /  x ]_ C  e.  RR ) ) )
10295, 96, 101, 18vtoclf 3023 . . . . . . . . . . 11  |-  ( (
ph  /\  ( d  +  1 )  e.  H )  ->  [_ (
d  +  1 )  /  x ]_ C  e.  RR )
10360, 91, 102syl2anc 661 . . . . . . . . . 10  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  [_ (
d  +  1 )  /  x ]_ C  e.  RR )
104 simp3 990 . . . . . . . . . 10  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )
105 nfv 1673 . . . . . . . . . . . 12  |-  F/ y ( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
)
106 eleq1 2503 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  (
y  e.  H  <->  d  e.  H ) )
107106anbi2d 703 . . . . . . . . . . . . 13  |-  ( y  =  d  ->  (
( ph  /\  y  e.  H )  <->  ( ph  /\  d  e.  H ) ) )
108 csbeq1 3291 . . . . . . . . . . . . . . 15  |-  ( y  =  d  ->  [_ y  /  x ]_ C  = 
[_ d  /  x ]_ C )
10945, 108syl5eqr 2489 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  F  =  [_ d  /  x ]_ C )
110 oveq1 6098 . . . . . . . . . . . . . . . 16  |-  ( y  =  d  ->  (
y  +  1 )  =  ( d  +  1 ) )
111110csbeq1d 3295 . . . . . . . . . . . . . . 15  |-  ( y  =  d  ->  [_ (
y  +  1 )  /  x ]_ C  =  [_ ( d  +  1 )  /  x ]_ C )
11251, 111syl5eqr 2489 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  G  =  [_ ( d  +  1 )  /  x ]_ C )
113109, 112breq12d 4305 . . . . . . . . . . . . 13  |-  ( y  =  d  ->  ( F  <  G  <->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) )
114107, 113imbi12d 320 . . . . . . . . . . . 12  |-  ( y  =  d  ->  (
( ( ph  /\  y  e.  H )  ->  F  <  G )  <-> 
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) ) )
115105, 114, 57chvar 1957 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ ( d  +  1 )  /  x ]_ C )
11660, 77, 115syl2anc 661 . . . . . . . . . 10  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  [_ d  /  x ]_ C  <  [_ ( d  +  1 )  /  x ]_ C )
11759, 88, 103, 104, 116lttrd 9532 . . . . . . . . 9  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  [_ a  /  x ]_ C  <  [_ ( d  +  1 )  /  x ]_ C )
1181173exp 1186 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  (
( ph  /\  a  e.  H )  ->  ( [_ a  /  x ]_ C  <  [_ d  /  x ]_ C  ->  [_ a  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) ) )
119118a2d 26 . . . . . . 7  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  (
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  ( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) ) )
12030, 33, 36, 39, 58, 119uzind2 10734 . . . . . 6  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  a  <  b )  ->  (
( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
12124, 26, 27, 120syl3anc 1218 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  ( ( ph  /\  a  e.  H
)  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
12221, 121mpd 15 . . . 4  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C )
123122ex 434 . . 3  |-  ( (
ph  /\  ( a  e.  H  /\  b  e.  H ) )  -> 
( a  <  b  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
1241, 2, 3, 8, 19, 123ltord1 9866 . 2  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( A  <  B  <->  [_ A  /  x ]_ C  <  [_ B  /  x ]_ C ) )
125 nfcvd 2580 . . . . 5  |-  ( A  e.  H  ->  F/_ x D )
126 monotuz.6 . . . . 5  |-  ( x  =  A  ->  C  =  D )
127125, 126csbiegf 3312 . . . 4  |-  ( A  e.  H  ->  [_ A  /  x ]_ C  =  D )
128 nfcvd 2580 . . . . 5  |-  ( B  e.  H  ->  F/_ x E )
129 monotuz.7 . . . . 5  |-  ( x  =  B  ->  C  =  E )
130128, 129csbiegf 3312 . . . 4  |-  ( B  e.  H  ->  [_ B  /  x ]_ C  =  E )
131127, 130breqan12d 4307 . . 3  |-  ( ( A  e.  H  /\  B  e.  H )  ->  ( [_ A  /  x ]_ C  <  [_ B  /  x ]_ C  <->  D  <  E ) )
132131adantl 466 . 2  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( [_ A  /  x ]_ C  <  [_ B  /  x ]_ C  <->  D  <  E ) )
133124, 132bitrd 253 1  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( A  <  B  <->  D  <  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   [_csb 3288   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   RRcr 9281   1c1 9283    + caddc 9285    < clt 9418    <_ cle 9419   ZZcz 10646   ZZ>=cuz 10861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862
This theorem is referenced by:  ltrmynn0  29291  ltrmxnn0  29292
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