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Theorem fvmptnn04if 19921
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 3942 . . . . 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 3318 . . . . . . 7  |-  ( N  e.  NN0  ->  ( [. N  /  n ]. n  =  0  <->  N  = 
0 ) )
41, 3syl 17 . . . . . 6  |-  ( ph  ->  ( [. N  /  n ]. n  =  0  <-> 
N  =  0 ) )
5 csbif 3942 . . . . . . 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 3318 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( [. N  /  n ]. n  =  S  <->  N  =  S
) )
71, 6syl 17 . . . . . . . 8  |-  ( ph  ->  ( [. N  /  n ]. n  =  S  <-> 
N  =  S ) )
8 csbif 3942 . . . . . . . . 9  |-  [_ N  /  n ]_ if ( S  <  n ,  D ,  B )  =  if ( [. N  /  n ]. S  <  n ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )
9 sbcbr2g 4471 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( [. N  /  n ]. S  <  n  <->  S  <  [_ N  /  n ]_ n ) )
101, 9syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( [. N  /  n ]. S  <  n  <->  S  <  [_ N  /  n ]_ n ) )
11 csbvarg 3803 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  [_ N  /  n ]_ n  =  N )
121, 11syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  [_ N  /  n ]_ n  =  N
)
1312breq2d 4427 . . . . . . . . . . 11  |-  ( ph  ->  ( S  <  [_ N  /  n ]_ n  <->  S  <  N ) )
1410, 13bitrd 261 . . . . . . . . . 10  |-  ( ph  ->  ( [. N  /  n ]. S  <  n  <->  S  <  N ) )
1514ifbid 3914 . . . . . . . . 9  |-  ( ph  ->  if ( [. N  /  n ]. S  < 
n ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )  =  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )
168, 15syl5eq 2507 . . . . . . . 8  |-  ( ph  ->  [_ N  /  n ]_ if ( S  < 
n ,  D ,  B )  =  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )
177, 16ifbieq2d 3917 . . . . . . 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 2507 . . . . . 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 3917 . . . . 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 2507 . . . 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 471 . . . . . 6  |-  ( (
ph  /\  N  = 
0 )  ->  Y  e.  V )
2421, 23eqeltrrd 2540 . . . . 5  |-  ( (
ph  /\  N  = 
0 )  ->  [_ N  /  n ]_ A  e.  V )
25 fvmptnn04if.c . . . . . . . . 9  |-  ( (
ph  /\  N  =  S )  ->  Y  =  [_ N  /  n ]_ C )
2625eqcomd 2467 . . . . . . . 8  |-  ( (
ph  /\  N  =  S )  ->  [_ N  /  n ]_ C  =  Y )
2726adantlr 726 . . . . . . 7  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  N  =  S )  ->  [_ N  /  n ]_ C  =  Y )
2822ad2antrr 737 . . . . . . 7  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  N  =  S )  ->  Y  e.  V )
2927, 28eqeltrd 2539 . . . . . 6  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  N  =  S )  ->  [_ N  /  n ]_ C  e.  V
)
30 fvmptnn04if.d . . . . . . . . . . . 12  |-  ( (
ph  /\  S  <  N )  ->  Y  =  [_ N  /  n ]_ D )
3130eqcomd 2467 . . . . . . . . . . 11  |-  ( (
ph  /\  S  <  N )  ->  [_ N  /  n ]_ D  =  Y )
3231ex 440 . . . . . . . . . 10  |-  ( ph  ->  ( S  <  N  ->  [_ N  /  n ]_ D  =  Y
) )
3332ad2antrr 737 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  -.  N  =  S )  ->  ( S  <  N  ->  [_ N  /  n ]_ D  =  Y ) )
3433imp 435 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  S  <  N )  ->  [_ N  /  n ]_ D  =  Y )
3522ad3antrrr 741 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  S  <  N )  ->  Y  e.  V )
3634, 35eqeltrd 2539 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  S  <  N )  ->  [_ N  /  n ]_ D  e.  V )
37 simplll 773 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  ph )
38 ancom 456 . . . . . . . . . . . . 13  |-  ( ( -.  S  <  N  /\  ph )  <->  ( ph  /\ 
-.  S  <  N
) )
3938anbi2i 705 . . . . . . . . . . . 12  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( -.  S  <  N  /\  ph ) )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( ph  /\ 
-.  S  <  N
) ) )
40 ancom 456 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N
) )  /\  ph ) )
41 anass 659 . . . . . . . . . . . . . . 15  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N )  <->  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )
4241bicomi 207 . . . . . . . . . . . . . 14  |-  ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N ) )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N ) )
4342anbi1i 706 . . . . . . . . . . . . 13  |-  ( ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N ) )  /\  ph )  <->  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N
)  /\  ph ) )
44 anass 659 . . . . . . . . . . . . 13  |-  ( ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N
)  /\  ph )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( -.  S  <  N  /\  ph ) ) )
4540, 43, 443bitri 279 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( -.  S  < 
N  /\  ph ) ) )
46 anass 659 . . . . . . . . . . . 12  |-  ( ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph )  /\  -.  S  <  N )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( ph  /\ 
-.  S  <  N
) ) )
4739, 45, 463bitr4i 285 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( ( -.  N  =  0  /\ 
-.  N  =  S )  /\  ph )  /\  -.  S  <  N
) )
48 an32 812 . . . . . . . . . . . . 13  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph ) 
<->  ( ( -.  N  =  0  /\  ph )  /\  -.  N  =  S ) )
49 ancom 456 . . . . . . . . . . . . . 14  |-  ( ( -.  N  =  0  /\  ph )  <->  ( ph  /\ 
-.  N  =  0 ) )
5049anbi1i 706 . . . . . . . . . . . . 13  |-  ( ( ( -.  N  =  0  /\  ph )  /\  -.  N  =  S )  <->  ( ( ph  /\ 
-.  N  =  0 )  /\  -.  N  =  S ) )
5148, 50bitri 257 . . . . . . . . . . . 12  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph ) 
<->  ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S ) )
5251anbi1i 706 . . . . . . . . . . 11  |-  ( ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph )  /\  -.  S  <  N )  <->  ( (
( ph  /\  -.  N  =  0 )  /\  -.  N  =  S
)  /\  -.  S  <  N ) )
5347, 52bitri 257 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( ( ph  /\ 
-.  N  =  0 )  /\  -.  N  =  S )  /\  -.  S  <  N ) )
54 df-ne 2634 . . . . . . . . . . . . 13  |-  ( N  =/=  0  <->  -.  N  =  0 )
55 elnnne0 10911 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
56 nngt0 10665 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  0  <  N )
5755, 56sylbir 218 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  N  =/=  0 )  -> 
0  <  N )
5857expcom 441 . . . . . . . . . . . . 13  |-  ( N  =/=  0  ->  ( N  e.  NN0  ->  0  <  N ) )
5954, 58sylbir 218 . . . . . . . . . . . 12  |-  ( -.  N  =  0  -> 
( N  e.  NN0  ->  0  <  N ) )
6059adantr 471 . . . . . . . . . . 11  |-  ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N ) )  -> 
( N  e.  NN0  ->  0  <  N ) )
611, 60mpan9 476 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  0  <  N
)
6253, 61sylbir 218 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  -> 
0  <  N )
631nn0red 10954 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  RR )
64 fvmptnn04if.s . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  S  e.  NN )
6564nnred 10651 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  e.  RR )
6663, 65lenltd 9806 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N  <_  S  <->  -.  S  <  N ) )
6766biimprd 231 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( -.  S  < 
N  ->  N  <_  S ) )
6867adantld 473 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( -.  N  =  S  /\  -.  S  <  N )  ->  N  <_  S ) )
6968adantld 473 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N
) )  ->  N  <_  S ) )
7069imp 435 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  N  <_  S
)
71 nesym 2691 . . . . . . . . . . . . . 14  |-  ( S  =/=  N  <->  -.  N  =  S )
7271biimpri 211 . . . . . . . . . . . . 13  |-  ( -.  N  =  S  ->  S  =/=  N )
7372adantr 471 . . . . . . . . . . . 12  |-  ( ( -.  N  =  S  /\  -.  S  < 
N )  ->  S  =/=  N )
7473ad2antll 740 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  S  =/=  N
)
7563adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  N  e.  RR )
7665adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  S  e.  RR )
7775, 76ltlend 9805 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  ( N  < 
S  <->  ( N  <_  S  /\  S  =/=  N
) ) )
7870, 74, 77mpbir2and 938 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  N  <  S
)
7953, 78sylbir 218 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  N  <  S )
80 fvmptnn04if.b . . . . . . . . . 10  |-  ( (
ph  /\  0  <  N  /\  N  <  S
)  ->  Y  =  [_ N  /  n ]_ B )
8180eqcomd 2467 . . . . . . . . 9  |-  ( (
ph  /\  0  <  N  /\  N  <  S
)  ->  [_ N  /  n ]_ B  =  Y )
8237, 62, 79, 81syl3anc 1276 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  [_ N  /  n ]_ B  =  Y
)
8322ad3antrrr 741 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  Y  e.  V )
8482, 83eqeltrd 2539 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  [_ N  /  n ]_ B  e.  V
)
8536, 84ifclda 3924 . . . . . 6  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  -.  N  =  S )  ->  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )  e.  V )
8629, 85ifclda 3924 . . . . 5  |-  ( (
ph  /\  -.  N  =  0 )  ->  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )  e.  V )
8724, 86ifclda 3924 . . . 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 2539 . . 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 5973 . . 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 671 . 2  |-  ( ph  ->  ( G `  N
)  =  [_ N  /  n ]_ if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )
9221eqcomd 2467 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  [_ N  /  n ]_ A  =  Y )
9334, 82ifeqda 3925 . . . 4  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  -.  N  =  S )  ->  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )  =  Y )
9427, 93ifeqda 3925 . . 3  |-  ( (
ph  /\  -.  N  =  0 )  ->  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )  =  Y )
9592, 94ifeqda 3925 . 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 2499 1  |-  ( ph  ->  ( G `  N
)  =  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   [.wsbc 3278   [_csb 3374   ifcif 3892   class class class wbr 4415    |-> cmpt 4474   ` cfv 5600   RRcr 9563   0cc0 9564    < clt 9700    <_ cle 9701   NNcn 10636   NN0cn0 10897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-n0 10898
This theorem is referenced by:  fvmptnn04ifa  19922  fvmptnn04ifb  19923  fvmptnn04ifc  19924  fvmptnn04ifd  19925
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