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Theorem monotuz 27004
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 3254 . . 3  |-  ( a  =  b  ->  [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C )
2 csbeq1 3254 . . 3  |-  ( a  =  A  ->  [_ a  /  x ]_ C  = 
[_ A  /  x ]_ C )
3 csbeq1 3254 . . 3  |-  ( a  =  B  ->  [_ a  /  x ]_ C  = 
[_ B  /  x ]_ C )
4 monotuz.3 . . . 4  |-  H  =  ( ZZ>= `  I )
5 uzssz 10505 . . . . 5  |-  ( ZZ>= `  I )  C_  ZZ
6 zssre 10289 . . . . 5  |-  ZZ  C_  RR
75, 6sstri 3357 . . . 4  |-  ( ZZ>= `  I )  C_  RR
84, 7eqsstri 3378 . . 3  |-  H  C_  RR
9 nfv 1629 . . . . 5  |-  F/ x
( ph  /\  a  e.  H )
10 nfcsb1v 3283 . . . . . 6  |-  F/_ x [_ a  /  x ]_ C
1110nfel1 2582 . . . . 5  |-  F/ x [_ a  /  x ]_ C  e.  RR
129, 11nfim 1832 . . . 4  |-  F/ x
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  e.  RR )
13 eleq1 2496 . . . . . 6  |-  ( x  =  a  ->  (
x  e.  H  <->  a  e.  H ) )
1413anbi2d 685 . . . . 5  |-  ( x  =  a  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  a  e.  H ) ) )
15 csbeq1a 3259 . . . . . 6  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
1615eleq1d 2502 . . . . 5  |-  ( x  =  a  ->  ( C  e.  RR  <->  [_ a  /  x ]_ C  e.  RR ) )
1714, 16imbi12d 312 . . . 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 1968 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  e.  RR )
20 simpl 444 . . . . . 6  |-  ( ( ( ph  /\  a  e.  H )  /\  a  <  b )  ->  ( ph  /\  a  e.  H
) )
2120adantlrr 702 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  ( ph  /\  a  e.  H ) )
224, 5eqsstri 3378 . . . . . . 7  |-  H  C_  ZZ
23 simplrl 737 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  e.  H )
2422, 23sseldi 3346 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  e.  ZZ )
25 simplrr 738 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  b  e.  H )
2622, 25sseldi 3346 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  b  e.  ZZ )
27 simpr 448 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  <  b )
28 csbeq1 3254 . . . . . . . . 9  |-  ( c  =  ( a  +  1 )  ->  [_ c  /  x ]_ C  = 
[_ ( a  +  1 )  /  x ]_ C )
2928breq2d 4224 . . . . . . . 8  |-  ( c  =  ( a  +  1 )  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ (
a  +  1 )  /  x ]_ C
) )
3029imbi2d 308 . . . . . . 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 3254 . . . . . . . . 9  |-  ( c  =  d  ->  [_ c  /  x ]_ C  = 
[_ d  /  x ]_ C )
3231breq2d 4224 . . . . . . . 8  |-  ( c  =  d  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C ) )
3332imbi2d 308 . . . . . . 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 3254 . . . . . . . . 9  |-  ( c  =  ( d  +  1 )  ->  [_ c  /  x ]_ C  = 
[_ ( d  +  1 )  /  x ]_ C )
3534breq2d 4224 . . . . . . . 8  |-  ( c  =  ( d  +  1 )  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) )
3635imbi2d 308 . . . . . . 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 3254 . . . . . . . . 9  |-  ( c  =  b  ->  [_ c  /  x ]_ C  = 
[_ b  /  x ]_ C )
3837breq2d 4224 . . . . . . . 8  |-  ( c  =  b  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
3938imbi2d 308 . . . . . . 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 2496 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  e.  H  <->  a  e.  H ) )
4140anbi2d 685 . . . . . . . . 9  |-  ( y  =  a  ->  (
( ph  /\  y  e.  H )  <->  ( ph  /\  a  e.  H ) ) )
42 vex 2959 . . . . . . . . . . . 12  |-  y  e. 
_V
43 nfcv 2572 . . . . . . . . . . . 12  |-  F/_ x F
44 monotuz.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  C  =  F )
4542, 43, 44csbief 3292 . . . . . . . . . . 11  |-  [_ y  /  x ]_ C  =  F
46 csbeq1 3254 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ y  /  x ]_ C  = 
[_ a  /  x ]_ C )
4745, 46syl5eqr 2482 . . . . . . . . . 10  |-  ( y  =  a  ->  F  =  [_ a  /  x ]_ C )
48 ovex 6106 . . . . . . . . . . . 12  |-  ( y  +  1 )  e. 
_V
49 nfcv 2572 . . . . . . . . . . . 12  |-  F/_ x G
50 monotuz.4 . . . . . . . . . . . 12  |-  ( x  =  ( y  +  1 )  ->  C  =  G )
5148, 49, 50csbief 3292 . . . . . . . . . . 11  |-  [_ (
y  +  1 )  /  x ]_ C  =  G
52 oveq1 6088 . . . . . . . . . . . 12  |-  ( y  =  a  ->  (
y  +  1 )  =  ( a  +  1 ) )
5352csbeq1d 3257 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ (
y  +  1 )  /  x ]_ C  =  [_ ( a  +  1 )  /  x ]_ C )
5451, 53syl5eqr 2482 . . . . . . . . . 10  |-  ( y  =  a  ->  G  =  [_ ( a  +  1 )  /  x ]_ C )
5547, 54breq12d 4225 . . . . . . . . 9  |-  ( y  =  a  ->  ( F  <  G  <->  [_ a  /  x ]_ C  <  [_ (
a  +  1 )  /  x ]_ C
) )
5641, 55imbi12d 312 . . . . . . . 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 3011 . . . . . . 7  |-  ( a  e.  ZZ  ->  (
( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ ( a  +  1 )  /  x ]_ C ) )
59193ad2ant2 979 . . . . . . . . . 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 983 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  ph )
61 zre 10286 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  ZZ  ->  a  e.  RR )
62613ad2ant1 978 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  e.  RR )
63 zre 10286 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ZZ  ->  d  e.  RR )
64633ad2ant2 979 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  d  e.  RR )
65 simp3 959 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  <  d )
6662, 64, 65ltled 9221 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  <_  d )
67663ad2ant1 978 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  <_  d )
68 simp11 987 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  e.  ZZ )
69 simp12 988 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  ZZ )
70 eluz 10499 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ )  ->  ( d  e.  (
ZZ>= `  a )  <->  a  <_  d ) )
7168, 69, 70syl2anc 643 . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . 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 984 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  e.  H )
7473, 4syl6eleq 2526 . . . . . . . . . . . . 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 10502 . . . . . . . . . . . . 13  |-  ( ( d  e.  ( ZZ>= `  a )  /\  a  e.  ( ZZ>= `  I )
)  ->  d  e.  ( ZZ>= `  I )
)
7672, 74, 75syl2anc 643 . . . . . . . . . . . 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 2527 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  H )
78 nfv 1629 . . . . . . . . . . . . 13  |-  F/ x
( ph  /\  d  e.  H )
79 nfcsb1v 3283 . . . . . . . . . . . . . 14  |-  F/_ x [_ d  /  x ]_ C
8079nfel1 2582 . . . . . . . . . . . . 13  |-  F/ x [_ d  /  x ]_ C  e.  RR
8178, 80nfim 1832 . . . . . . . . . . . 12  |-  F/ x
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR )
82 eleq1 2496 . . . . . . . . . . . . . 14  |-  ( x  =  d  ->  (
x  e.  H  <->  d  e.  H ) )
8382anbi2d 685 . . . . . . . . . . . . 13  |-  ( x  =  d  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  d  e.  H ) ) )
84 csbeq1a 3259 . . . . . . . . . . . . . 14  |-  ( x  =  d  ->  C  =  [_ d  /  x ]_ C )
8584eleq1d 2502 . . . . . . . . . . . . 13  |-  ( x  =  d  ->  ( C  e.  RR  <->  [_ d  /  x ]_ C  e.  RR ) )
8683, 85imbi12d 312 . . . . . . . . . . . 12  |-  ( x  =  d  ->  (
( ( ph  /\  x  e.  H )  ->  C  e.  RR )  <-> 
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR ) ) )
8781, 86, 18chvar 1968 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR )
8860, 77, 87syl2anc 643 . . . . . . . . . 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 10530 . . . . . . . . . . . . 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 2527 . . . . . . . . . . 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 1629 . . . . . . . . . . . . 13  |-  F/ x
( ph  /\  (
d  +  1 )  e.  H )
93 nfcsb1v 3283 . . . . . . . . . . . . . 14  |-  F/_ x [_ ( d  +  1 )  /  x ]_ C
9493nfel1 2582 . . . . . . . . . . . . 13  |-  F/ x [_ ( d  +  1 )  /  x ]_ C  e.  RR
9592, 94nfim 1832 . . . . . . . . . . . 12  |-  F/ x
( ( ph  /\  ( d  +  1 )  e.  H )  ->  [_ ( d  +  1 )  /  x ]_ C  e.  RR )
96 ovex 6106 . . . . . . . . . . . 12  |-  ( d  +  1 )  e. 
_V
97 eleq1 2496 . . . . . . . . . . . . . 14  |-  ( x  =  ( d  +  1 )  ->  (
x  e.  H  <->  ( d  +  1 )  e.  H ) )
9897anbi2d 685 . . . . . . . . . . . . 13  |-  ( x  =  ( d  +  1 )  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  ( d  +  1 )  e.  H ) ) )
99 csbeq1a 3259 . . . . . . . . . . . . . 14  |-  ( x  =  ( d  +  1 )  ->  C  =  [_ ( d  +  1 )  /  x ]_ C )
10099eleq1d 2502 . . . . . . . . . . . . 13  |-  ( x  =  ( d  +  1 )  ->  ( C  e.  RR  <->  [_ ( d  +  1 )  /  x ]_ C  e.  RR ) )
10198, 100imbi12d 312 . . . . . . . . . . . 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 3005 . . . . . . . . . . 11  |-  ( (
ph  /\  ( d  +  1 )  e.  H )  ->  [_ (
d  +  1 )  /  x ]_ C  e.  RR )
10360, 91, 102syl2anc 643 . . . . . . . . . 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 959 . . . . . . . . . 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 1629 . . . . . . . . . . . 12  |-  F/ y ( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
)
106 eleq1 2496 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  (
y  e.  H  <->  d  e.  H ) )
107106anbi2d 685 . . . . . . . . . . . . 13  |-  ( y  =  d  ->  (
( ph  /\  y  e.  H )  <->  ( ph  /\  d  e.  H ) ) )
108 csbeq1 3254 . . . . . . . . . . . . . . 15  |-  ( y  =  d  ->  [_ y  /  x ]_ C  = 
[_ d  /  x ]_ C )
10945, 108syl5eqr 2482 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  F  =  [_ d  /  x ]_ C )
110 oveq1 6088 . . . . . . . . . . . . . . . 16  |-  ( y  =  d  ->  (
y  +  1 )  =  ( d  +  1 ) )
111110csbeq1d 3257 . . . . . . . . . . . . . . 15  |-  ( y  =  d  ->  [_ (
y  +  1 )  /  x ]_ C  =  [_ ( d  +  1 )  /  x ]_ C )
11251, 111syl5eqr 2482 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  G  =  [_ ( d  +  1 )  /  x ]_ C )
113109, 112breq12d 4225 . . . . . . . . . . . . 13  |-  ( y  =  d  ->  ( F  <  G  <->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) )
114107, 113imbi12d 312 . . . . . . . . . . . 12  |-  ( y  =  d  ->  (
( ( ph  /\  y  e.  H )  ->  F  <  G )  <-> 
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) ) )
115105, 114, 57chvar 1968 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ ( d  +  1 )  /  x ]_ C )
11660, 77, 115syl2anc 643 . . . . . . . . . 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 9231 . . . . . . . . 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 1152 . . . . . . . 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 24 . . . . . . 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 10362 . . . . . 6  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  a  <  b )  ->  (
( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
12124, 26, 27, 120syl3anc 1184 . . . . 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 424 . . 3  |-  ( (
ph  /\  ( a  e.  H  /\  b  e.  H ) )  -> 
( a  <  b  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
1241, 2, 3, 8, 19, 123ltord1 9553 . 2  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( A  <  B  <->  [_ A  /  x ]_ C  <  [_ B  /  x ]_ C ) )
125 nfcvd 2573 . . . . 5  |-  ( A  e.  H  ->  F/_ x D )
126 monotuz.6 . . . . 5  |-  ( x  =  A  ->  C  =  D )
127125, 126csbiegf 3291 . . . 4  |-  ( A  e.  H  ->  [_ A  /  x ]_ C  =  D )
128 nfcvd 2573 . . . . 5  |-  ( B  e.  H  ->  F/_ x E )
129 monotuz.7 . . . . 5  |-  ( x  =  B  ->  C  =  E )
130128, 129csbiegf 3291 . . . 4  |-  ( B  e.  H  ->  [_ B  /  x ]_ C  =  E )
131127, 130breqan12d 4227 . . 3  |-  ( ( A  e.  H  /\  B  e.  H )  ->  ( [_ A  /  x ]_ C  <  [_ B  /  x ]_ C  <->  D  <  E ) )
132131adantl 453 . 2  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( [_ A  /  x ]_ C  <  [_ B  /  x ]_ C  <->  D  <  E ) )
133124, 132bitrd 245 1  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( A  <  B  <->  D  <  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   [_csb 3251   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989   1c1 8991    + caddc 8993    < clt 9120    <_ cle 9121   ZZcz 10282   ZZ>=cuz 10488
This theorem is referenced by:  ltrmynn0  27013  ltrmxnn0  27014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489
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