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Theorem fvmptnn04if 19872
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 3961 . . . . 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 3339 . . . . . . 7  |-  ( N  e.  NN0  ->  ( [. N  /  n ]. n  =  0  <->  N  = 
0 ) )
41, 3syl 17 . . . . . 6  |-  ( ph  ->  ( [. N  /  n ]. n  =  0  <-> 
N  =  0 ) )
5 csbif 3961 . . . . . . 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 3339 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( [. N  /  n ]. n  =  S  <->  N  =  S
) )
71, 6syl 17 . . . . . . . 8  |-  ( ph  ->  ( [. N  /  n ]. n  =  S  <-> 
N  =  S ) )
8 csbif 3961 . . . . . . . . 9  |-  [_ N  /  n ]_ if ( S  <  n ,  D ,  B )  =  if ( [. N  /  n ]. S  <  n ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )
9 sbcbr2g 4479 . . . . . . . . . . . 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 3822 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  [_ N  /  n ]_ n  =  N )
121, 11syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  [_ N  /  n ]_ n  =  N
)
1312breq2d 4435 . . . . . . . . . . 11  |-  ( ph  ->  ( S  <  [_ N  /  n ]_ n  <->  S  <  N ) )
1410, 13bitrd 256 . . . . . . . . . 10  |-  ( ph  ->  ( [. N  /  n ]. S  <  n  <->  S  <  N ) )
1514ifbid 3933 . . . . . . . . 9  |-  ( ph  ->  if ( [. N  /  n ]. S  < 
n ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )  =  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )
168, 15syl5eq 2475 . . . . . . . 8  |-  ( ph  ->  [_ N  /  n ]_ if ( S  < 
n ,  D ,  B )  =  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )
177, 16ifbieq2d 3936 . . . . . . 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 2475 . . . . . 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 3936 . . . . 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 2475 . . . 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 466 . . . . . 6  |-  ( (
ph  /\  N  = 
0 )  ->  Y  e.  V )
2421, 23eqeltrrd 2508 . . . . 5  |-  ( (
ph  /\  N  = 
0 )  ->  [_ N  /  n ]_ A  e.  V )
25 fvmptnn04if.c . . . . . . . . 9  |-  ( (
ph  /\  N  =  S )  ->  Y  =  [_ N  /  n ]_ C )
2625eqcomd 2430 . . . . . . . 8  |-  ( (
ph  /\  N  =  S )  ->  [_ N  /  n ]_ C  =  Y )
2726adantlr 719 . . . . . . 7  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  N  =  S )  ->  [_ N  /  n ]_ C  =  Y )
2822ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  N  =  S )  ->  Y  e.  V )
2927, 28eqeltrd 2507 . . . . . 6  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  N  =  S )  ->  [_ N  /  n ]_ C  e.  V
)
30 fvmptnn04if.d . . . . . . . . . . . 12  |-  ( (
ph  /\  S  <  N )  ->  Y  =  [_ N  /  n ]_ D )
3130eqcomd 2430 . . . . . . . . . . 11  |-  ( (
ph  /\  S  <  N )  ->  [_ N  /  n ]_ D  =  Y )
3231ex 435 . . . . . . . . . 10  |-  ( ph  ->  ( S  <  N  ->  [_ N  /  n ]_ D  =  Y
) )
3332ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  -.  N  =  S )  ->  ( S  <  N  ->  [_ N  /  n ]_ D  =  Y ) )
3433imp 430 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  S  <  N )  ->  [_ N  /  n ]_ D  =  Y )
3522ad3antrrr 734 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  S  <  N )  ->  Y  e.  V )
3634, 35eqeltrd 2507 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  S  <  N )  ->  [_ N  /  n ]_ D  e.  V )
37 simplll 766 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  ph )
38 ancom 451 . . . . . . . . . . . . 13  |-  ( ( -.  S  <  N  /\  ph )  <->  ( ph  /\ 
-.  S  <  N
) )
3938anbi2i 698 . . . . . . . . . . . 12  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( -.  S  <  N  /\  ph ) )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( ph  /\ 
-.  S  <  N
) ) )
40 ancom 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N
) )  /\  ph ) )
41 anass 653 . . . . . . . . . . . . . . 15  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N )  <->  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )
4241bicomi 205 . . . . . . . . . . . . . 14  |-  ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N ) )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N ) )
4342anbi1i 699 . . . . . . . . . . . . 13  |-  ( ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N ) )  /\  ph )  <->  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N
)  /\  ph ) )
44 anass 653 . . . . . . . . . . . . 13  |-  ( ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  -.  S  <  N
)  /\  ph )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( -.  S  <  N  /\  ph ) ) )
4540, 43, 443bitri 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( -.  S  < 
N  /\  ph ) ) )
46 anass 653 . . . . . . . . . . . 12  |-  ( ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph )  /\  -.  S  <  N )  <->  ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ( ph  /\ 
-.  S  <  N
) ) )
4739, 45, 463bitr4i 280 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( ( -.  N  =  0  /\ 
-.  N  =  S )  /\  ph )  /\  -.  S  <  N
) )
48 an32 805 . . . . . . . . . . . . 13  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph ) 
<->  ( ( -.  N  =  0  /\  ph )  /\  -.  N  =  S ) )
49 ancom 451 . . . . . . . . . . . . . 14  |-  ( ( -.  N  =  0  /\  ph )  <->  ( ph  /\ 
-.  N  =  0 ) )
5049anbi1i 699 . . . . . . . . . . . . 13  |-  ( ( ( -.  N  =  0  /\  ph )  /\  -.  N  =  S )  <->  ( ( ph  /\ 
-.  N  =  0 )  /\  -.  N  =  S ) )
5148, 50bitri 252 . . . . . . . . . . . 12  |-  ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph ) 
<->  ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S ) )
5251anbi1i 699 . . . . . . . . . . 11  |-  ( ( ( ( -.  N  =  0  /\  -.  N  =  S )  /\  ph )  /\  -.  S  <  N )  <->  ( (
( ph  /\  -.  N  =  0 )  /\  -.  N  =  S
)  /\  -.  S  <  N ) )
5347, 52bitri 252 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  <-> 
( ( ( ph  /\ 
-.  N  =  0 )  /\  -.  N  =  S )  /\  -.  S  <  N ) )
54 df-ne 2616 . . . . . . . . . . . . 13  |-  ( N  =/=  0  <->  -.  N  =  0 )
55 elnnne0 10891 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
56 nngt0 10646 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  0  <  N )
5755, 56sylbir 216 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  N  =/=  0 )  -> 
0  <  N )
5857expcom 436 . . . . . . . . . . . . 13  |-  ( N  =/=  0  ->  ( N  e.  NN0  ->  0  <  N ) )
5954, 58sylbir 216 . . . . . . . . . . . 12  |-  ( -.  N  =  0  -> 
( N  e.  NN0  ->  0  <  N ) )
6059adantr 466 . . . . . . . . . . 11  |-  ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N ) )  -> 
( N  e.  NN0  ->  0  <  N ) )
611, 60mpan9 471 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  0  <  N
)
6253, 61sylbir 216 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  -> 
0  <  N )
631nn0red 10934 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  RR )
64 fvmptnn04if.s . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  S  e.  NN )
6564nnred 10632 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  e.  RR )
6663, 65lenltd 9789 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N  <_  S  <->  -.  S  <  N ) )
6766biimprd 226 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( -.  S  < 
N  ->  N  <_  S ) )
6867adantld 468 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( -.  N  =  S  /\  -.  S  <  N )  ->  N  <_  S ) )
6968adantld 468 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  <  N
) )  ->  N  <_  S ) )
7069imp 430 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  N  <_  S
)
71 nesym 2692 . . . . . . . . . . . . . 14  |-  ( S  =/=  N  <->  -.  N  =  S )
7271biimpri 209 . . . . . . . . . . . . 13  |-  ( -.  N  =  S  ->  S  =/=  N )
7372adantr 466 . . . . . . . . . . . 12  |-  ( ( -.  N  =  S  /\  -.  S  < 
N )  ->  S  =/=  N )
7473ad2antll 733 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  S  =/=  N
)
7563adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  N  e.  RR )
7665adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  S  e.  RR )
7775, 76ltlend 9788 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  ( N  < 
S  <->  ( N  <_  S  /\  S  =/=  N
) ) )
7870, 74, 77mpbir2and 930 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  N  =  0  /\  ( -.  N  =  S  /\  -.  S  < 
N ) ) )  ->  N  <  S
)
7953, 78sylbir 216 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  N  <  S )
80 fvmptnn04if.b . . . . . . . . . 10  |-  ( (
ph  /\  0  <  N  /\  N  <  S
)  ->  Y  =  [_ N  /  n ]_ B )
8180eqcomd 2430 . . . . . . . . 9  |-  ( (
ph  /\  0  <  N  /\  N  <  S
)  ->  [_ N  /  n ]_ B  =  Y )
8237, 62, 79, 81syl3anc 1264 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  [_ N  /  n ]_ B  =  Y
)
8322ad3antrrr 734 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  Y  e.  V )
8482, 83eqeltrd 2507 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  N  =  0
)  /\  -.  N  =  S )  /\  -.  S  <  N )  ->  [_ N  /  n ]_ B  e.  V
)
8536, 84ifclda 3943 . . . . . 6  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  -.  N  =  S )  ->  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )  e.  V )
8629, 85ifclda 3943 . . . . 5  |-  ( (
ph  /\  -.  N  =  0 )  ->  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )  e.  V )
8724, 86ifclda 3943 . . . 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 2507 . . 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 5968 . . 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 665 . 2  |-  ( ph  ->  ( G `  N
)  =  [_ N  /  n ]_ if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )
9221eqcomd 2430 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  [_ N  /  n ]_ A  =  Y )
9334, 82ifeqda 3944 . . . 4  |-  ( ( ( ph  /\  -.  N  =  0 )  /\  -.  N  =  S )  ->  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B )  =  Y )
9427, 93ifeqda 3944 . . 3  |-  ( (
ph  /\  -.  N  =  0 )  ->  if ( N  =  S ,  [_ N  /  n ]_ C ,  if ( S  <  N ,  [_ N  /  n ]_ D ,  [_ N  /  n ]_ B ) )  =  Y )
9592, 94ifeqda 3944 . 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 2467 1  |-  ( ph  ->  ( G `  N
)  =  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   [.wsbc 3299   [_csb 3395   ifcif 3911   class class class wbr 4423    |-> cmpt 4482   ` cfv 5601   RRcr 9546   0cc0 9547    < clt 9683    <_ cle 9684   NNcn 10617   NN0cn0 10877
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 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-n0 10878
This theorem is referenced by:  fvmptnn04ifa  19873  fvmptnn04ifb  19874  fvmptnn04ifc  19875  fvmptnn04ifd  19876
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