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Theorem monotuz 35702
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 3336 . . 3  |-  ( a  =  b  ->  [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C )
2 csbeq1 3336 . . 3  |-  ( a  =  A  ->  [_ a  /  x ]_ C  = 
[_ A  /  x ]_ C )
3 csbeq1 3336 . . 3  |-  ( a  =  B  ->  [_ a  /  x ]_ C  = 
[_ B  /  x ]_ C )
4 monotuz.3 . . . 4  |-  H  =  ( ZZ>= `  I )
5 uzssz 11124 . . . . 5  |-  ( ZZ>= `  I )  C_  ZZ
6 zssre 10890 . . . . 5  |-  ZZ  C_  RR
75, 6sstri 3411 . . . 4  |-  ( ZZ>= `  I )  C_  RR
84, 7eqsstri 3432 . . 3  |-  H  C_  RR
9 nfv 1755 . . . . 5  |-  F/ x
( ph  /\  a  e.  H )
10 nfcsb1v 3349 . . . . . 6  |-  F/_ x [_ a  /  x ]_ C
1110nfel1 2578 . . . . 5  |-  F/ x [_ a  /  x ]_ C  e.  RR
129, 11nfim 1980 . . . 4  |-  F/ x
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  e.  RR )
13 eleq1 2489 . . . . . 6  |-  ( x  =  a  ->  (
x  e.  H  <->  a  e.  H ) )
1413anbi2d 708 . . . . 5  |-  ( x  =  a  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  a  e.  H ) ) )
15 csbeq1a 3342 . . . . . 6  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
1615eleq1d 2485 . . . . 5  |-  ( x  =  a  ->  ( C  e.  RR  <->  [_ a  /  x ]_ C  e.  RR ) )
1714, 16imbi12d 321 . . . 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 2073 . . 3  |-  ( (
ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  e.  RR )
20 simpl 458 . . . . . 6  |-  ( ( ( ph  /\  a  e.  H )  /\  a  <  b )  ->  ( ph  /\  a  e.  H
) )
2120adantlrr 725 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  ( ph  /\  a  e.  H ) )
224, 5eqsstri 3432 . . . . . . 7  |-  H  C_  ZZ
23 simplrl 768 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  e.  H )
2422, 23sseldi 3400 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  e.  ZZ )
25 simplrr 769 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  b  e.  H )
2622, 25sseldi 3400 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  b  e.  ZZ )
27 simpr 462 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  a  <  b )
28 csbeq1 3336 . . . . . . . . 9  |-  ( c  =  ( a  +  1 )  ->  [_ c  /  x ]_ C  = 
[_ ( a  +  1 )  /  x ]_ C )
2928breq2d 4373 . . . . . . . 8  |-  ( c  =  ( a  +  1 )  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ (
a  +  1 )  /  x ]_ C
) )
3029imbi2d 317 . . . . . . 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 3336 . . . . . . . . 9  |-  ( c  =  d  ->  [_ c  /  x ]_ C  = 
[_ d  /  x ]_ C )
3231breq2d 4373 . . . . . . . 8  |-  ( c  =  d  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C ) )
3332imbi2d 317 . . . . . . 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 3336 . . . . . . . . 9  |-  ( c  =  ( d  +  1 )  ->  [_ c  /  x ]_ C  = 
[_ ( d  +  1 )  /  x ]_ C )
3534breq2d 4373 . . . . . . . 8  |-  ( c  =  ( d  +  1 )  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) )
3635imbi2d 317 . . . . . . 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 3336 . . . . . . . . 9  |-  ( c  =  b  ->  [_ c  /  x ]_ C  = 
[_ b  /  x ]_ C )
3837breq2d 4373 . . . . . . . 8  |-  ( c  =  b  ->  ( [_ a  /  x ]_ C  <  [_ c  /  x ]_ C  <->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
3938imbi2d 317 . . . . . . 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 2489 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  e.  H  <->  a  e.  H ) )
4140anbi2d 708 . . . . . . . . 9  |-  ( y  =  a  ->  (
( ph  /\  y  e.  H )  <->  ( ph  /\  a  e.  H ) ) )
42 vex 3020 . . . . . . . . . . . 12  |-  y  e. 
_V
43 monotuz.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  C  =  F )
4442, 43csbie 3359 . . . . . . . . . . 11  |-  [_ y  /  x ]_ C  =  F
45 csbeq1 3336 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ y  /  x ]_ C  = 
[_ a  /  x ]_ C )
4644, 45syl5eqr 2471 . . . . . . . . . 10  |-  ( y  =  a  ->  F  =  [_ a  /  x ]_ C )
47 ovex 6272 . . . . . . . . . . . 12  |-  ( y  +  1 )  e. 
_V
48 monotuz.4 . . . . . . . . . . . 12  |-  ( x  =  ( y  +  1 )  ->  C  =  G )
4947, 48csbie 3359 . . . . . . . . . . 11  |-  [_ (
y  +  1 )  /  x ]_ C  =  G
50 oveq1 6251 . . . . . . . . . . . 12  |-  ( y  =  a  ->  (
y  +  1 )  =  ( a  +  1 ) )
5150csbeq1d 3340 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ (
y  +  1 )  /  x ]_ C  =  [_ ( a  +  1 )  /  x ]_ C )
5249, 51syl5eqr 2471 . . . . . . . . . 10  |-  ( y  =  a  ->  G  =  [_ ( a  +  1 )  /  x ]_ C )
5346, 52breq12d 4374 . . . . . . . . 9  |-  ( y  =  a  ->  ( F  <  G  <->  [_ a  /  x ]_ C  <  [_ (
a  +  1 )  /  x ]_ C
) )
5441, 53imbi12d 321 . . . . . . . 8  |-  ( y  =  a  ->  (
( ( ph  /\  y  e.  H )  ->  F  <  G )  <-> 
( ( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ (
a  +  1 )  /  x ]_ C
) ) )
55 monotuz.1 . . . . . . . 8  |-  ( (
ph  /\  y  e.  H )  ->  F  <  G )
5654, 55vtoclg 3077 . . . . . . 7  |-  ( a  e.  ZZ  ->  (
( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ ( a  +  1 )  /  x ]_ C ) )
57193ad2ant2 1027 . . . . . . . . . 10  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  [_ a  /  x ]_ C  e.  RR )
58 simp2l 1031 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  ph )
59 zre 10887 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  ZZ  ->  a  e.  RR )
60593ad2ant1 1026 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  e.  RR )
61 zre 10887 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ZZ  ->  d  e.  RR )
62613ad2ant2 1027 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  d  e.  RR )
63 simp3 1007 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  <  d )
6460, 62, 63ltled 9729 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  ->  a  <_  d )
65643ad2ant1 1026 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  <_  d )
66 simp11 1035 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  e.  ZZ )
67 simp12 1036 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  ZZ )
68 eluz 11118 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  ZZ  /\  d  e.  ZZ )  ->  ( d  e.  (
ZZ>= `  a )  <->  a  <_  d ) )
6966, 67, 68syl2anc 665 . . . . . . . . . . . . . 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 ) )
7065, 69mpbird 235 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  ( ZZ>= `  a )
)
71 simp2r 1032 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  e.  H )
7271, 4syl6eleq 2511 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  a  e.  ( ZZ>= `  I )
)
73 uztrn 11121 . . . . . . . . . . . . 13  |-  ( ( d  e.  ( ZZ>= `  a )  /\  a  e.  ( ZZ>= `  I )
)  ->  d  e.  ( ZZ>= `  I )
)
7470, 72, 73syl2anc 665 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  ( ZZ>= `  I )
)
7574, 4syl6eleqr 2512 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  d  e.  H )
76 nfv 1755 . . . . . . . . . . . . 13  |-  F/ x
( ph  /\  d  e.  H )
77 nfcsb1v 3349 . . . . . . . . . . . . . 14  |-  F/_ x [_ d  /  x ]_ C
7877nfel1 2578 . . . . . . . . . . . . 13  |-  F/ x [_ d  /  x ]_ C  e.  RR
7976, 78nfim 1980 . . . . . . . . . . . 12  |-  F/ x
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR )
80 eleq1 2489 . . . . . . . . . . . . . 14  |-  ( x  =  d  ->  (
x  e.  H  <->  d  e.  H ) )
8180anbi2d 708 . . . . . . . . . . . . 13  |-  ( x  =  d  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  d  e.  H ) ) )
82 csbeq1a 3342 . . . . . . . . . . . . . 14  |-  ( x  =  d  ->  C  =  [_ d  /  x ]_ C )
8382eleq1d 2485 . . . . . . . . . . . . 13  |-  ( x  =  d  ->  ( C  e.  RR  <->  [_ d  /  x ]_ C  e.  RR ) )
8481, 83imbi12d 321 . . . . . . . . . . . 12  |-  ( x  =  d  ->  (
( ( ph  /\  x  e.  H )  ->  C  e.  RR )  <-> 
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR ) ) )
8579, 84, 18chvar 2073 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  e.  RR )
8658, 75, 85syl2anc 665 . . . . . . . . . 10  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  [_ d  /  x ]_ C  e.  RR )
87 peano2uz 11158 . . . . . . . . . . . . 13  |-  ( d  e.  ( ZZ>= `  I
)  ->  ( d  +  1 )  e.  ( ZZ>= `  I )
)
8874, 87syl 17 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  (
d  +  1 )  e.  ( ZZ>= `  I
) )
8988, 4syl6eleqr 2512 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ  /\  d  e.  ZZ  /\  a  <  d )  /\  ( ph  /\  a  e.  H )  /\  [_ a  /  x ]_ C  <  [_ d  /  x ]_ C )  ->  (
d  +  1 )  e.  H )
90 nfv 1755 . . . . . . . . . . . . 13  |-  F/ x
( ph  /\  (
d  +  1 )  e.  H )
91 nfcsb1v 3349 . . . . . . . . . . . . . 14  |-  F/_ x [_ ( d  +  1 )  /  x ]_ C
9291nfel1 2578 . . . . . . . . . . . . 13  |-  F/ x [_ ( d  +  1 )  /  x ]_ C  e.  RR
9390, 92nfim 1980 . . . . . . . . . . . 12  |-  F/ x
( ( ph  /\  ( d  +  1 )  e.  H )  ->  [_ ( d  +  1 )  /  x ]_ C  e.  RR )
94 ovex 6272 . . . . . . . . . . . 12  |-  ( d  +  1 )  e. 
_V
95 eleq1 2489 . . . . . . . . . . . . . 14  |-  ( x  =  ( d  +  1 )  ->  (
x  e.  H  <->  ( d  +  1 )  e.  H ) )
9695anbi2d 708 . . . . . . . . . . . . 13  |-  ( x  =  ( d  +  1 )  ->  (
( ph  /\  x  e.  H )  <->  ( ph  /\  ( d  +  1 )  e.  H ) ) )
97 csbeq1a 3342 . . . . . . . . . . . . . 14  |-  ( x  =  ( d  +  1 )  ->  C  =  [_ ( d  +  1 )  /  x ]_ C )
9897eleq1d 2485 . . . . . . . . . . . . 13  |-  ( x  =  ( d  +  1 )  ->  ( C  e.  RR  <->  [_ ( d  +  1 )  /  x ]_ C  e.  RR ) )
9996, 98imbi12d 321 . . . . . . . . . . . 12  |-  ( x  =  ( d  +  1 )  ->  (
( ( ph  /\  x  e.  H )  ->  C  e.  RR )  <-> 
( ( ph  /\  ( d  +  1 )  e.  H )  ->  [_ ( d  +  1 )  /  x ]_ C  e.  RR ) ) )
10093, 94, 99, 18vtoclf 3069 . . . . . . . . . . 11  |-  ( (
ph  /\  ( d  +  1 )  e.  H )  ->  [_ (
d  +  1 )  /  x ]_ C  e.  RR )
10158, 89, 100syl2anc 665 . . . . . . . . . 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 )
102 simp3 1007 . . . . . . . . . 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 )
103 nfv 1755 . . . . . . . . . . . 12  |-  F/ y ( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
)
104 eleq1 2489 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  (
y  e.  H  <->  d  e.  H ) )
105104anbi2d 708 . . . . . . . . . . . . 13  |-  ( y  =  d  ->  (
( ph  /\  y  e.  H )  <->  ( ph  /\  d  e.  H ) ) )
106 csbeq1 3336 . . . . . . . . . . . . . . 15  |-  ( y  =  d  ->  [_ y  /  x ]_ C  = 
[_ d  /  x ]_ C )
10744, 106syl5eqr 2471 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  F  =  [_ d  /  x ]_ C )
108 oveq1 6251 . . . . . . . . . . . . . . . 16  |-  ( y  =  d  ->  (
y  +  1 )  =  ( d  +  1 ) )
109108csbeq1d 3340 . . . . . . . . . . . . . . 15  |-  ( y  =  d  ->  [_ (
y  +  1 )  /  x ]_ C  =  [_ ( d  +  1 )  /  x ]_ C )
11049, 109syl5eqr 2471 . . . . . . . . . . . . . 14  |-  ( y  =  d  ->  G  =  [_ ( d  +  1 )  /  x ]_ C )
111107, 110breq12d 4374 . . . . . . . . . . . . 13  |-  ( y  =  d  ->  ( F  <  G  <->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) )
112105, 111imbi12d 321 . . . . . . . . . . . 12  |-  ( y  =  d  ->  (
( ( ph  /\  y  e.  H )  ->  F  <  G )  <-> 
( ( ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ (
d  +  1 )  /  x ]_ C
) ) )
113103, 112, 55chvar 2073 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  H )  ->  [_ d  /  x ]_ C  <  [_ ( d  +  1 )  /  x ]_ C )
11458, 75, 113syl2anc 665 . . . . . . . . . 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 )
11557, 86, 101, 102, 114lttrd 9742 . . . . . . . . 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 )
1161153exp 1204 . . . . . . . 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
) ) )
117116a2d 29 . . . . . . 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
) ) )
11830, 33, 36, 39, 56, 117uzind2 10974 . . . . . 6  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  a  <  b )  ->  (
( ph  /\  a  e.  H )  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
11924, 26, 27, 118syl3anc 1264 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  ( ( ph  /\  a  e.  H
)  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
12021, 119mpd 15 . . . 4  |-  ( ( ( ph  /\  (
a  e.  H  /\  b  e.  H )
)  /\  a  <  b )  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C )
121120ex 435 . . 3  |-  ( (
ph  /\  ( a  e.  H  /\  b  e.  H ) )  -> 
( a  <  b  ->  [_ a  /  x ]_ C  <  [_ b  /  x ]_ C ) )
1221, 2, 3, 8, 19, 121ltord1 10086 . 2  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( A  <  B  <->  [_ A  /  x ]_ C  <  [_ B  /  x ]_ C ) )
123 nfcvd 2565 . . . . 5  |-  ( A  e.  H  ->  F/_ x D )
124 monotuz.6 . . . . 5  |-  ( x  =  A  ->  C  =  D )
125123, 124csbiegf 3357 . . . 4  |-  ( A  e.  H  ->  [_ A  /  x ]_ C  =  D )
126 nfcvd 2565 . . . . 5  |-  ( B  e.  H  ->  F/_ x E )
127 monotuz.7 . . . . 5  |-  ( x  =  B  ->  C  =  E )
128126, 127csbiegf 3357 . . . 4  |-  ( B  e.  H  ->  [_ B  /  x ]_ C  =  E )
129125, 128breqan12d 4377 . . 3  |-  ( ( A  e.  H  /\  B  e.  H )  ->  ( [_ A  /  x ]_ C  <  [_ B  /  x ]_ C  <->  D  <  E ) )
130129adantl 467 . 2  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( [_ A  /  x ]_ C  <  [_ B  /  x ]_ C  <->  D  <  E ) )
131122, 130bitrd 256 1  |-  ( (
ph  /\  ( A  e.  H  /\  B  e.  H ) )  -> 
( A  <  B  <->  D  <  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   [_csb 3333   class class class wbr 4361   ` cfv 5539  (class class class)co 6244   RRcr 9484   1c1 9486    + caddc 9488    < clt 9621    <_ cle 9622   ZZcz 10883   ZZ>=cuz 11105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-er 7313  df-en 7520  df-dom 7521  df-sdom 7522  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-nn 10556  df-n0 10816  df-z 10884  df-uz 11106
This theorem is referenced by:  ltrmynn0  35711  ltrmxnn0  35712
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