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Theorem monotuz 27129
Description: A function defined on a set of upper 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 3097 . . 3  |-  ( a  =  b  ->  [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C )
2 csbeq1 3097 . . 3  |-  ( a  =  A  ->  [_ a  /  x ]_ C  = 
[_ A  /  x ]_ C )
3 csbeq1 3097 . . 3  |-  ( a  =  B  ->  [_ a  /  x ]_ C  = 
[_ B  /  x ]_ C )
4 monotuz.3 . . . 4  |-  H  =  ( ZZ>= `  I )
5 uzssz 10263 . . . . 5  |-  ( ZZ>= `  I )  C_  ZZ
6 zssre 10047 . . . . 5  |-  ZZ  C_  RR
75, 6sstri 3201 . . . 4  |-  ( ZZ>= `  I )  C_  RR
84, 7eqsstri 3221 . . 3  |-  H  C_  RR
9 nfv 1609 . . . . 5  |-  F/ x
( ph  /\  a  e.  H )
10 nfcsb1v 3126 . . . . . 6  |-  F/_ x [_ a  /  x ]_ C
1110nfel1 2442 . . . . 5  |-  F/ x [_ a  /  x ]_ C  e.  RR
129, 11nfim 1781 . . . 4  |-  F/ x
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  e.  RR )
13 eleq1 2356 . . . . . 6  |-  ( x  =  a  ->  (
x  e.  H  <->  a  e.  H ) )
1413anbi2d 684 . . . . 5  |-  ( x  =  a  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  a  e.  H ) ) )
15 csbeq1a 3102 . . . . . 6  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
1615eleq1d 2362 . . . . 5  |-  ( x  =  a  ->  ( C  e.  RR  <->  [_ a  /  x ]_ C  e.  RR ) )
1714, 16imbi12d 311 . . . 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 1939 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  e.  RR )
20 simpl 443 . . . . . 6  |-  ( ( ( ph  /\  a  e.  H )  /\  a  <  b )  ->  ( ph  /\  a  e.  H
) )
2120adantlrr 701 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  ( ph  /\  a  e.  H ) )
224, 5eqsstri 3221 . . . . . . 7  |-  H  C_  ZZ
23 simplrl 736 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  e.  H )
2422, 23sseldi 3191 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  e.  ZZ )
25 simplrr 737 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  b  e.  H )
2622, 25sseldi 3191 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  b  e.  ZZ )
27 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  <  b )
28 csbeq1 3097 . . . . . . . . 9  |-  ( c  =  ( a  +  1 )  ->  [_ c  /  x ]_ C  = 
[_ ( a  +  1 )  /  x ]_ C )
2928breq2d 4051 . . . . . . . 8  |-  ( c  =  ( a  +  1 )  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ (
a  +  1 )  /  x ]_ C
) )
3029imbi2d 307 . . . . . . 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 3097 . . . . . . . . 9  |-  ( c  =  d  ->  [_ c  /  x ]_ C  = 
[_ d  /  x ]_ C )
3231breq2d 4051 . . . . . . . 8  |-  ( c  =  d  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C ) )
3332imbi2d 307 . . . . . . 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 3097 . . . . . . . . 9  |-  ( c  =  ( d  +  1 )  ->  [_ c  /  x ]_ C  = 
[_ ( d  +  1 )  /  x ]_ C )
3534breq2d 4051 . . . . . . . 8  |-  ( c  =  ( d  +  1 )  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) )
3635imbi2d 307 . . . . . . 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 3097 . . . . . . . . 9  |-  ( c  =  b  ->  [_ c  /  x ]_ C  = 
[_ b  /  x ]_ C )
3837breq2d 4051 . . . . . . . 8  |-  ( c  =  b  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
3938imbi2d 307 . . . . . . 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 2356 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  e.  H  <->  a  e.  H ) )
4140anbi2d 684 . . . . . . . . 9  |-  ( y  =  a  ->  (
( ph  /\  y  e.  H )  <->  ( ph  /\  a  e.  H ) ) )
42 vex 2804 . . . . . . . . . . . 12  |-  y  e. 
_V
43 nfcv 2432 . . . . . . . . . . . 12  |-  F/_ x F
44 monotuz.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  C  =  F )
4542, 43, 44csbief 3135 . . . . . . . . . . 11  |-  [_ y  /  x ]_ C  =  F
46 csbeq1 3097 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ y  /  x ]_ C  = 
[_ a  /  x ]_ C )
4745, 46syl5eqr 2342 . . . . . . . . . 10  |-  ( y  =  a  ->  F  =  [_ a  /  x ]_ C )
48 ovex 5899 . . . . . . . . . . . 12  |-  ( y  +  1 )  e. 
_V
49 nfcv 2432 . . . . . . . . . . . 12  |-  F/_ x G
50 monotuz.4 . . . . . . . . . . . 12  |-  ( x  =  ( y  +  1 )  ->  C  =  G )
5148, 49, 50csbief 3135 . . . . . . . . . . 11  |-  [_ (
y  +  1 )  /  x ]_ C  =  G
52 oveq1 5881 . . . . . . . . . . . 12  |-  ( y  =  a  ->  (
y  +  1 )  =  ( a  +  1 ) )
5352csbeq1d 3100 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ (
y  +  1 )  /  x ]_ C  =  [_ ( a  +  1 )  /  x ]_ C )
5451, 53syl5eqr 2342 . . . . . . . . . 10  |-  ( y  =  a  ->  G  =  [_ ( a  +  1 )  /  x ]_ C )
5547, 54breq12d 4052 . . . . . . . . 9  |-  ( y  =  a  ->  ( F  <  G  <->  [_ a  /  x ]_ C  <  [_ (
a  +  1 )  /  x ]_ C
) )
5641, 55imbi12d 311 . . . . . . . 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 2856 . . . . . . 7  |-  ( a  e.  ZZ  ->  (
( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ ( a  +  1 )  /  x ]_ C ) )
59193ad2ant2 977 . . . . . . . . . 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 981 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  ph )
61 zre 10044 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  ZZ  ->  a  e.  RR )
62613ad2ant1 976 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  e.  RR )
63 zre 10044 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ZZ  ->  d  e.  RR )
64633ad2ant2 977 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  d  e.  RR )
65 simp3 957 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  <  d )
6662, 64, 65ltled 8983 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  <_  d )
67663ad2ant1 976 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  <_  d )
68 simp11 985 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  e.  ZZ )
69 simp12 986 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  ZZ )
70 eluz 10257 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ )  ->  ( d  e.  (
ZZ>= `  a )  <->  a  <_  d ) )
7168, 69, 70syl2anc 642 . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 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 982 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  e.  H )
7473, 4syl6eleq 2386 . . . . . . . . . . . . 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 10260 . . . . . . . . . . . . 13  |-  ( ( d  e.  ( ZZ>= `  a )  /\  a  e.  ( ZZ>= `  I )
)  ->  d  e.  ( ZZ>= `  I )
)
7672, 74, 75syl2anc 642 . . . . . . . . . . . 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 2387 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  H )
78 nfv 1609 . . . . . . . . . . . . 13  |-  F/ x
( ph  /\  d  e.  H )
79 nfcsb1v 3126 . . . . . . . . . . . . . 14  |-  F/_ x [_ d  /  x ]_ C
8079nfel1 2442 . . . . . . . . . . . . 13  |-  F/ x [_ d  /  x ]_ C  e.  RR
8178, 80nfim 1781 . . . . . . . . . . . 12  |-  F/ x
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR )
82 eleq1 2356 . . . . . . . . . . . . . 14  |-  ( x  =  d  ->  (
x  e.  H  <->  d  e.  H ) )
8382anbi2d 684 . . . . . . . . . . . . 13  |-  ( x  =  d  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  d  e.  H ) ) )
84 csbeq1a 3102 . . . . . . . . . . . . . 14  |-  ( x  =  d  ->  C  =  [_ d  /  x ]_ C )
8584eleq1d 2362 . . . . . . . . . . . . 13  |-  ( x  =  d  ->  ( C  e.  RR  <->  [_ d  /  x ]_ C  e.  RR ) )
8683, 85imbi12d 311 . . . . . . . . . . . 12  |-  ( x  =  d  ->  (
( ( ph  /\  x  e.  H )  ->  C  e.  RR )  <-> 
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR ) ) )
8781, 86, 18chvar 1939 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR )
8860, 77, 87syl2anc 642 . . . . . . . . . 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 10288 . . . . . . . . . . . . 13  |-  ( d  e.  ( ZZ>= `  I
)  ->  ( d  +  1 )  e.  ( ZZ>= `  I )
)
9076, 89syl 15 . . . . . . . . . . . 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 2387 . . . . . . . . . . 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 1609 . . . . . . . . . . . . 13  |-  F/ x
( ph  /\  (
d  +  1 )  e.  H )
93 nfcsb1v 3126 . . . . . . . . . . . . . 14  |-  F/_ x [_ ( d  +  1 )  /  x ]_ C
9493nfel1 2442 . . . . . . . . . . . . 13  |-  F/ x [_ ( d  +  1 )  /  x ]_ C  e.  RR
9592, 94nfim 1781 . . . . . . . . . . . 12  |-  F/ x
( ( ph  /\  ( d  +  1 )  e.  H )  ->  [_ ( d  +  1 )  /  x ]_ C  e.  RR )
96 ovex 5899 . . . . . . . . . . . 12  |-  ( d  +  1 )  e. 
_V
97 eleq1 2356 . . . . . . . . . . . . . 14  |-  ( x  =  ( d  +  1 )  ->  (
x  e.  H  <->  ( d  +  1 )  e.  H ) )
9897anbi2d 684 . . . . . . . . . . . . 13  |-  ( x  =  ( d  +  1 )  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  ( d  +  1 )  e.  H ) ) )
99 csbeq1a 3102 . . . . . . . . . . . . . 14  |-  ( x  =  ( d  +  1 )  ->  C  =  [_ ( d  +  1 )  /  x ]_ C )
10099eleq1d 2362 . . . . . . . . . . . . 13  |-  ( x  =  ( d  +  1 )  ->  ( C  e.  RR  <->  [_ ( d  +  1 )  /  x ]_ C  e.  RR ) )
10198, 100imbi12d 311 . . . . . . . . . . . 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 2850 . . . . . . . . . . 11  |-  ( (
ph  /\  ( d  +  1 )  e.  H )  ->  [_ (
d  +  1 )  /  x ]_ C  e.  RR )
10360, 91, 102syl2anc 642 . . . . . . . . . 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 957 . . . . . . . . . 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 1609 . . . . . . . . . . . 12  |-  F/ y ( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
)
106 eleq1 2356 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  (
y  e.  H  <->  d  e.  H ) )
107106anbi2d 684 . . . . . . . . . . . . 13  |-  ( y  =  d  ->  (
( ph  /\  y  e.  H )  <->  ( ph  /\  d  e.  H ) ) )
108 csbeq1 3097 . . . . . . . . . . . . . . 15  |-  ( y  =  d  ->  [_ y  /  x ]_ C  = 
[_ d  /  x ]_ C )
10945, 108syl5eqr 2342 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  F  =  [_ d  /  x ]_ C )
110 oveq1 5881 . . . . . . . . . . . . . . . 16  |-  ( y  =  d  ->  (
y  +  1 )  =  ( d  +  1 ) )
111110csbeq1d 3100 . . . . . . . . . . . . . . 15  |-  ( y  =  d  ->  [_ (
y  +  1 )  /  x ]_ C  =  [_ ( d  +  1 )  /  x ]_ C )
11251, 111syl5eqr 2342 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  G  =  [_ ( d  +  1 )  /  x ]_ C )
113109, 112breq12d 4052 . . . . . . . . . . . . 13  |-  ( y  =  d  ->  ( F  <  G  <->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) )
114107, 113imbi12d 311 . . . . . . . . . . . 12  |-  ( y  =  d  ->  (
( ( ph  /\  y  e.  H )  ->  F  <  G )  <-> 
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) ) )
115105, 114, 57chvar 1939 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ ( d  +  1 )  /  x ]_ C )
11660, 77, 115syl2anc 642 . . . . . . . . . 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 8993 . . . . . . . . 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 1150 . . . . . . . 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 23 . . . . . . 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 10120 . . . . . 6  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  a  <  b )  ->  (
( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
12124, 26, 27, 120syl3anc 1182 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  ( ( ph  /\  a  e.  H
)  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
12221, 121mpd 14 . . . 4  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C )
123122ex 423 . . 3  |-  ( (
ph  /\  ( a  e.  H  /\  b  e.  H ) )  -> 
( a  <  b  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
1241, 2, 3, 8, 19, 123ltord1 9315 . 2  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( A  <  B  <->  [_ A  /  x ]_ C  <  [_ B  /  x ]_ C ) )
125 nfcvd 2433 . . . . 5  |-  ( A  e.  H  ->  F/_ x D )
126 monotuz.6 . . . . 5  |-  ( x  =  A  ->  C  =  D )
127125, 126csbiegf 3134 . . . 4  |-  ( A  e.  H  ->  [_ A  /  x ]_ C  =  D )
128 nfcvd 2433 . . . . 5  |-  ( B  e.  H  ->  F/_ x E )
129 monotuz.7 . . . . 5  |-  ( x  =  B  ->  C  =  E )
130128, 129csbiegf 3134 . . . 4  |-  ( B  e.  H  ->  [_ B  /  x ]_ C  =  E )
131127, 130breqan12d 4054 . . 3  |-  ( ( A  e.  H  /\  B  e.  H )  ->  ( [_ A  /  x ]_ C  <  [_ B  /  x ]_ C  <->  D  <  E ) )
132131adantl 452 . 2  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( [_ A  /  x ]_ C  <  [_ B  /  x ]_ C  <->  D  <  E ) )
133124, 132bitrd 244 1  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( A  <  B  <->  D  <  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   [_csb 3094   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884   ZZcz 10040   ZZ>=cuz 10246
This theorem is referenced by:  ltrmynn0  27138  ltrmxnn0  27139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247
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