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Theorem fvmptnn04if 19223
Description: The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)
Hypotheses
Ref Expression
fvmptnn04if.g  |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )
fvmptnn04if.s  |-  ( ph  ->  S  e.  NN )
fvmptnn04if.n  |-  ( ph  ->  N  e.  NN0 )
fvmptnn04if.y  |-  ( ph  ->  Y  e.  V )
fvmptnn04if.a  |-  ( (
ph  /\  N  = 
0 )  ->  Y  =  [_ N  /  n ]_ A )
fvmptnn04if.b  |-  ( (
ph  /\  0  <  N  /\  N  <  S
)  ->  Y  =  [_ N  /  n ]_ B )
fvmptnn04if.c  |-  ( (
ph  /\  N  =  S )  ->  Y  =  [_ N  /  n ]_ C )
fvmptnn04if.d  |-  ( (
ph  /\  S  <  N )  ->  Y  =  [_ N  /  n ]_ D )
Assertion
Ref Expression
fvmptnn04if  |-  ( ph  ->  ( G `  N
)  =  Y )
Distinct variable groups:    n, N    S, n
Allowed substitution hints:    ph( n)    A( n)    B( n)    C( n)    D( n)    G( n)    V( n)    Y( n)

Proof of Theorem fvmptnn04if
StepHypRef Expression
1 fvmptnn04if.n . . 3  |-  ( ph  ->  N  e.  NN0 )
2 csbif 3976 . . . . 5  |-  [_ N  /  n ]_ if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) )  =  if ( [. N  /  n ]. n  =  0 ,  [_ N  /  n ]_ A ,  [_ N  /  n ]_ if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) )
3 eqsbc3 3353 . . . . . . 7  |-  ( N  e.  NN0  ->  ( [. N  /  n ]. n  =  0  <->  N  = 
0 ) )
41, 3syl 16 . . . . . 6  |-  ( ph  ->  ( [. N  /  n ]. n  =  0  <-> 
N  =  0 ) )
5 csbif 3976 . . . . . . 7  |-  [_ N  /  n ]_ if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) )  =  if ( [. N  /  n ]. n  =  S ,  [_ N  /  n ]_ C ,  [_ N  /  n ]_ if ( S  < 
n ,  D ,  B ) )
6 eqsbc3 3353 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( [. N  /  n ]. n  =  S  <->  N  =  S
) )
71, 6syl 16 . . . . . . . 8  |-  ( ph  ->  ( [. N  /  n ]. n  =  S  <-> 
N  =  S ) )
8 csbif 3976 . . . . . . . . 9  |-  [_ N  /  n ]_ if ( S  <  n ,  D ,  B )  =  if ( [. N  /  n ]. S  <  n ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )
9 sbcbr2g 4493 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( [. N  /  n ]. S  <  n  <->  S  <  [_ N  /  n ]_ n ) )
101, 9syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( [. N  /  n ]. S  <  n  <->  S  <  [_ N  /  n ]_ n ) )
11 csbvarg 3834 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  [_ N  /  n ]_ n  =  N )
121, 11syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  [_ N  /  n ]_ n  =  N
)
1312breq2d 4449 . . . . . . . . . . 11  |-  ( ph  ->  ( S  <  [_ N  /  n ]_ n  <->  S  <  N ) )
1410, 13bitrd 253 . . . . . . . . . 10  |-  ( ph  ->  ( [. N  /  n ]. S  <  n  <->  S  <  N ) )
1514ifbid 3948 . . . . . . . . 9  |-  ( ph  ->  if ( [. N  /  n ]. S  < 
n ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )  =  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )
168, 15syl5eq 2496 . . . . . . . 8  |-  ( ph  ->  [_ N  /  n ]_ if ( S  < 
n ,  D ,  B )  =  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )
177, 16ifbieq2d 3951 . . . . . . 7  |-  ( ph  ->  if ( [. N  /  n ]. n  =  S ,  [_ N  /  n ]_ C ,  [_ N  /  n ]_ if ( S  < 
n ,  D ,  B ) )  =  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) ) )
185, 17syl5eq 2496 . . . . . 6  |-  ( ph  ->  [_ N  /  n ]_ if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) )  =  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) ) )
194, 18ifbieq2d 3951 . . . . 5  |-  ( ph  ->  if ( [. N  /  n ]. n  =  0 ,  [_ N  /  n ]_ A ,  [_ N  /  n ]_ if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) )  =  if ( N  =  0 ,  [_ N  /  n ]_ A ,  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) ) ) )
202, 19syl5eq 2496 . . . 4  |-  ( ph  ->  [_ N  /  n ]_ if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) )  =  if ( N  =  0 ,  [_ N  /  n ]_ A ,  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) ) ) )
21 fvmptnn04if.a . . . . . 6  |-  ( (
ph  /\  N  = 
0 )  ->  Y  =  [_ N  /  n ]_ A )
22 fvmptnn04if.y . . . . . . 7  |-  ( ph  ->  Y  e.  V )
2322adantr 465 . . . . . 6  |-  ( (
ph  /\  N  = 
0 )  ->  Y  e.  V )
2421, 23eqeltrrd 2532 . . . . 5  |-  ( (
ph  /\  N  = 
0 )  ->  [_ N  /  n ]_ A  e.  V )
25 fvmptnn04if.c . . . . . . . . 9  |-  ( (
ph  /\  N  =  S )  ->  Y  =  [_ N  /  n ]_ C )
2625eqcomd 2451 . . . . . . . 8  |-  ( (
ph  /\  N  =  S )  ->  [_ N  /  n ]_ C  =  Y )
2726adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  N  =  S )  ->  [_ N  /  n ]_ C  =  Y )
2822ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  N  =  S )  ->  Y  e.  V )
2927, 28eqeltrd 2531 . . . . . 6  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  N  =  S )  ->  [_ N  /  n ]_ C  e.  V
)
30 fvmptnn04if.d . . . . . . . . . . . 12  |-  ( (
ph  /\  S  <  N )  ->  Y  =  [_ N  /  n ]_ D )
3130eqcomd 2451 . . . . . . . . . . 11  |-  ( (
ph  /\  S  <  N )  ->  [_ N  /  n ]_ D  =  Y )
3231ex 434 . . . . . . . . . 10  |-  ( ph  ->  ( S  <  N  ->  [_ N  /  n ]_ D  =  Y
) )
3332ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  -.  N  =  S )  ->  ( S  <  N  ->  [_ N  /  n ]_ D  =  Y ) )
3433imp 429 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  S  <  N )  ->  [_ N  /  n ]_ D  =  Y )
3522ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  S  <  N )  ->  Y  e.  V )
3634, 35eqeltrd 2531 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  S  <  N )  ->  [_ N  /  n ]_ D  e.  V )
37 simplll 759 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  ph )
38 ancom 450 . . . . . . . . . . . . 13  |-  ( ( -.  S  <  N  /\  ph )  <->  ( ph  /\ 
-.  S  <  N
) )
3938anbi2i 694 . . . . . . . . . . . 12  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( -.  S  <  N  /\  ph ) )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( ph  /\ 
-.  S  <  N
) ) )
40 ancom 450 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N
) )  /\  ph ) )
41 anass 649 . . . . . . . . . . . . . . 15  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N )  <->  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )
4241bicomi 202 . . . . . . . . . . . . . 14  |-  ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N ) )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N ) )
4342anbi1i 695 . . . . . . . . . . . . 13  |-  ( ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N ) )  /\  ph )  <->  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N
)  /\  ph ) )
44 anass 649 . . . . . . . . . . . . 13  |-  ( ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N
)  /\  ph )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( -.  S  <  N  /\  ph ) ) )
4540, 43, 443bitri 271 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( -.  S  < 
N  /\  ph ) ) )
46 anass 649 . . . . . . . . . . . 12  |-  ( ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph )  /\  -.  S  <  N )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( ph  /\ 
-.  S  <  N
) ) )
4739, 45, 463bitr4i 277 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( ( -.  N  =  0  /\ 
-.  N  =  S )  /\  ph )  /\  -.  S  <  N
) )
48 an32 798 . . . . . . . . . . . . 13  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph ) 
<->  ( ( -.  N  =  0  /\  ph )  /\  -.  N  =  S ) )
49 ancom 450 . . . . . . . . . . . . . 14  |-  ( ( -.  N  =  0  /\  ph )  <->  ( ph  /\ 
-.  N  =  0 ) )
5049anbi1i 695 . . . . . . . . . . . . 13  |-  ( ( ( -.  N  =  0  /\  ph )  /\  -.  N  =  S )  <->  ( ( ph  /\ 
-.  N  =  0 )  /\  -.  N  =  S ) )
5148, 50bitri 249 . . . . . . . . . . . 12  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph ) 
<->  ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S ) )
5251anbi1i 695 . . . . . . . . . . 11  |-  ( ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph )  /\  -.  S  <  N )  <->  ( (
( ph  /\  -.  N  =  0 )  /\  -.  N  =  S
)  /\  -.  S  <  N ) )
5347, 52bitri 249 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( ( ph  /\ 
-.  N  =  0 )  /\  -.  N  =  S )  /\  -.  S  <  N ) )
54 df-ne 2640 . . . . . . . . . . . . 13  |-  ( N  =/=  0  <->  -.  N  =  0 )
55 elnnne0 10815 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
56 nngt0 10571 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  0  <  N )
5755, 56sylbir 213 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  N  =/=  0 )  -> 
0  <  N )
5857expcom 435 . . . . . . . . . . . . 13  |-  ( N  =/=  0  ->  ( N  e.  NN0  ->  0  <  N ) )
5954, 58sylbir 213 . . . . . . . . . . . 12  |-  ( -.  N  =  0  -> 
( N  e.  NN0  ->  0  <  N ) )
6059adantr 465 . . . . . . . . . . 11  |-  ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N ) )  -> 
( N  e.  NN0  ->  0  <  N ) )
611, 60mpan9 469 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  0  <  N
)
6253, 61sylbir 213 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  -> 
0  <  N )
631nn0red 10859 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  RR )
64 fvmptnn04if.s . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  S  e.  NN )
6564nnred 10557 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  e.  RR )
6663, 65lenltd 9734 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N  <_  S  <->  -.  S  <  N ) )
6766biimprd 223 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( -.  S  < 
N  ->  N  <_  S ) )
6867adantld 467 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( -.  N  =  S  /\  -.  S  <  N )  ->  N  <_  S ) )
6968adantld 467 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N
) )  ->  N  <_  S ) )
7069imp 429 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  N  <_  S
)
71 nesym 2715 . . . . . . . . . . . . . 14  |-  ( S  =/=  N  <->  -.  N  =  S )
7271biimpri 206 . . . . . . . . . . . . 13  |-  ( -.  N  =  S  ->  S  =/=  N )
7372adantr 465 . . . . . . . . . . . 12  |-  ( ( -.  N  =  S  /\  -.  S  < 
N )  ->  S  =/=  N )
7473ad2antll 728 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  S  =/=  N
)
7563adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  N  e.  RR )
7665adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  S  e.  RR )
7775, 76ltlend 9733 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  ( N  < 
S  <->  ( N  <_  S  /\  S  =/=  N
) ) )
7870, 74, 77mpbir2and 922 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  N  <  S
)
7953, 78sylbir 213 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  N  <  S )
80 fvmptnn04if.b . . . . . . . . . 10  |-  ( (
ph  /\  0  <  N  /\  N  <  S
)  ->  Y  =  [_ N  /  n ]_ B )
8180eqcomd 2451 . . . . . . . . 9  |-  ( (
ph  /\  0  <  N  /\  N  <  S
)  ->  [_ N  /  n ]_ B  =  Y )
8237, 62, 79, 81syl3anc 1229 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  [_ N  /  n ]_ B  =  Y
)
8322ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  Y  e.  V )
8482, 83eqeltrd 2531 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  [_ N  /  n ]_ B  e.  V
)
8536, 84ifclda 3958 . . . . . 6  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  -.  N  =  S )  ->  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )  e.  V )
8629, 85ifclda 3958 . . . . 5  |-  ( (
ph  /\  -.  N  =  0 )  ->  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )  e.  V )
8724, 86ifclda 3958 . . . 4  |-  ( ph  ->  if ( N  =  0 ,  [_ N  /  n ]_ A ,  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) ) )  e.  V
)
8820, 87eqeltrd 2531 . . 3  |-  ( ph  ->  [_ N  /  n ]_ if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) )  e.  V
)
89 fvmptnn04if.g . . . 4  |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )
9089fvmpts 5943 . . 3  |-  ( ( N  e.  NN0  /\  [_ N  /  n ]_ if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) )  e.  V
)  ->  ( G `  N )  =  [_ N  /  n ]_ if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )
911, 88, 90syl2anc 661 . 2  |-  ( ph  ->  ( G `  N
)  =  [_ N  /  n ]_ if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )
9221eqcomd 2451 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  [_ N  /  n ]_ A  =  Y )
9334, 82ifeqda 3959 . . . 4  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  -.  N  =  S )  ->  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )  =  Y )
9427, 93ifeqda 3959 . . 3  |-  ( (
ph  /\  -.  N  =  0 )  ->  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )  =  Y )
9592, 94ifeqda 3959 . 2  |-  ( ph  ->  if ( N  =  0 ,  [_ N  /  n ]_ A ,  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) ) )  =  Y )
9691, 20, 953eqtrd 2488 1  |-  ( ph  ->  ( G `  N
)  =  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   [.wsbc 3313   [_csb 3420   ifcif 3926   class class class wbr 4437    |-> cmpt 4495   ` cfv 5578   RRcr 9494   0cc0 9495    < clt 9631    <_ cle 9632   NNcn 10542   NN0cn0 10801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-n0 10802
This theorem is referenced by:  fvmptnn04ifa  19224  fvmptnn04ifb  19225  fvmptnn04ifc  19226  fvmptnn04ifd  19227
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