Metamath Proof Explorer


Theorem fvmptnn04if

Description: The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019)

Ref Expression
Hypotheses 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 fvmptnn04if
|- ( ph -> ( G ` N ) = Y )

Proof

Step Hyp Ref Expression
1 fvmptnn04if.g
 |-  G = ( n e. NN0 |-> if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
2 fvmptnn04if.s
 |-  ( ph -> S e. NN )
3 fvmptnn04if.n
 |-  ( ph -> N e. NN0 )
4 fvmptnn04if.y
 |-  ( ph -> Y e. V )
5 fvmptnn04if.a
 |-  ( ( ph /\ N = 0 ) -> Y = [_ N / n ]_ A )
6 fvmptnn04if.b
 |-  ( ( ph /\ 0 < N /\ N < S ) -> Y = [_ N / n ]_ B )
7 fvmptnn04if.c
 |-  ( ( ph /\ N = S ) -> Y = [_ N / n ]_ C )
8 fvmptnn04if.d
 |-  ( ( ph /\ S < N ) -> Y = [_ N / n ]_ D )
9 csbif
 |-  [_ 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 ) ) )
10 eqsbc3
 |-  ( N e. NN0 -> ( [. N / n ]. n = 0 <-> N = 0 ) )
11 3 10 syl
 |-  ( ph -> ( [. N / n ]. n = 0 <-> N = 0 ) )
12 csbif
 |-  [_ 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 ) )
13 eqsbc3
 |-  ( N e. NN0 -> ( [. N / n ]. n = S <-> N = S ) )
14 3 13 syl
 |-  ( ph -> ( [. N / n ]. n = S <-> N = S ) )
15 csbif
 |-  [_ N / n ]_ if ( S < n , D , B ) = if ( [. N / n ]. S < n , [_ N / n ]_ D , [_ N / n ]_ B )
16 sbcbr2g
 |-  ( N e. NN0 -> ( [. N / n ]. S < n <-> S < [_ N / n ]_ n ) )
17 3 16 syl
 |-  ( ph -> ( [. N / n ]. S < n <-> S < [_ N / n ]_ n ) )
18 csbvarg
 |-  ( N e. NN0 -> [_ N / n ]_ n = N )
19 3 18 syl
 |-  ( ph -> [_ N / n ]_ n = N )
20 19 breq2d
 |-  ( ph -> ( S < [_ N / n ]_ n <-> S < N ) )
21 17 20 bitrd
 |-  ( ph -> ( [. N / n ]. S < n <-> S < N ) )
22 21 ifbid
 |-  ( ph -> if ( [. N / n ]. S < n , [_ N / n ]_ D , [_ N / n ]_ B ) = if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) )
23 15 22 syl5eq
 |-  ( ph -> [_ N / n ]_ if ( S < n , D , B ) = if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) )
24 14 23 ifbieq2d
 |-  ( 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 ) ) )
25 12 24 syl5eq
 |-  ( 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 ) ) )
26 11 25 ifbieq2d
 |-  ( 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 ) ) ) )
27 9 26 syl5eq
 |-  ( 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 ) ) ) )
28 4 adantr
 |-  ( ( ph /\ N = 0 ) -> Y e. V )
29 5 28 eqeltrrd
 |-  ( ( ph /\ N = 0 ) -> [_ N / n ]_ A e. V )
30 7 eqcomd
 |-  ( ( ph /\ N = S ) -> [_ N / n ]_ C = Y )
31 30 adantlr
 |-  ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> [_ N / n ]_ C = Y )
32 4 ad2antrr
 |-  ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> Y e. V )
33 31 32 eqeltrd
 |-  ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> [_ N / n ]_ C e. V )
34 8 eqcomd
 |-  ( ( ph /\ S < N ) -> [_ N / n ]_ D = Y )
35 34 ad4ant14
 |-  ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> [_ N / n ]_ D = Y )
36 4 ad3antrrr
 |-  ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> Y e. V )
37 35 36 eqeltrd
 |-  ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> [_ N / n ]_ D e. V )
38 simplll
 |-  ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> ph )
39 anass
 |-  ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) <-> ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) )
40 39 bicomi
 |-  ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) )
41 40 bianassc
 |-  ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) )
42 an32
 |-  ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) <-> ( ( -. N = 0 /\ ph ) /\ -. N = S ) )
43 ancom
 |-  ( ( -. N = 0 /\ ph ) <-> ( ph /\ -. N = 0 ) )
44 43 anbi1i
 |-  ( ( ( -. N = 0 /\ ph ) /\ -. N = S ) <-> ( ( ph /\ -. N = 0 ) /\ -. N = S ) )
45 42 44 bitri
 |-  ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) <-> ( ( ph /\ -. N = 0 ) /\ -. N = S ) )
46 45 anbi1i
 |-  ( ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) <-> ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) )
47 41 46 bitri
 |-  ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) )
48 df-ne
 |-  ( N =/= 0 <-> -. N = 0 )
49 elnnne0
 |-  ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
50 nngt0
 |-  ( N e. NN -> 0 < N )
51 49 50 sylbir
 |-  ( ( N e. NN0 /\ N =/= 0 ) -> 0 < N )
52 51 expcom
 |-  ( N =/= 0 -> ( N e. NN0 -> 0 < N ) )
53 48 52 sylbir
 |-  ( -. N = 0 -> ( N e. NN0 -> 0 < N ) )
54 53 adantr
 |-  ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) -> ( N e. NN0 -> 0 < N ) )
55 3 54 mpan9
 |-  ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> 0 < N )
56 47 55 sylbir
 |-  ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> 0 < N )
57 3 nn0red
 |-  ( ph -> N e. RR )
58 57 adantr
 |-  ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N e. RR )
59 2 nnred
 |-  ( ph -> S e. RR )
60 59 adantr
 |-  ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> S e. RR )
61 57 59 lenltd
 |-  ( ph -> ( N <_ S <-> -. S < N ) )
62 61 biimprd
 |-  ( ph -> ( -. S < N -> N <_ S ) )
63 62 adantld
 |-  ( ph -> ( ( -. N = S /\ -. S < N ) -> N <_ S ) )
64 63 adantld
 |-  ( ph -> ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) -> N <_ S ) )
65 64 imp
 |-  ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N <_ S )
66 nesym
 |-  ( S =/= N <-> -. N = S )
67 66 biimpri
 |-  ( -. N = S -> S =/= N )
68 67 adantr
 |-  ( ( -. N = S /\ -. S < N ) -> S =/= N )
69 68 ad2antll
 |-  ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> S =/= N )
70 58 60 65 69 leneltd
 |-  ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N < S )
71 47 70 sylbir
 |-  ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> N < S )
72 6 eqcomd
 |-  ( ( ph /\ 0 < N /\ N < S ) -> [_ N / n ]_ B = Y )
73 38 56 71 72 syl3anc
 |-  ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> [_ N / n ]_ B = Y )
74 4 ad3antrrr
 |-  ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> Y e. V )
75 73 74 eqeltrd
 |-  ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> [_ N / n ]_ B e. V )
76 37 75 ifclda
 |-  ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) e. V )
77 33 76 ifclda
 |-  ( ( ph /\ -. N = 0 ) -> if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) e. V )
78 29 77 ifclda
 |-  ( ph -> if ( N = 0 , [_ N / n ]_ A , if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) ) e. V )
79 27 78 eqeltrd
 |-  ( ph -> [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) e. V )
80 1 fvmpts
 |-  ( ( 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 ) ) ) )
81 3 79 80 syl2anc
 |-  ( ph -> ( G ` N ) = [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
82 5 eqcomd
 |-  ( ( ph /\ N = 0 ) -> [_ N / n ]_ A = Y )
83 35 73 ifeqda
 |-  ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) = Y )
84 31 83 ifeqda
 |-  ( ( ph /\ -. N = 0 ) -> if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) = Y )
85 82 84 ifeqda
 |-  ( ph -> if ( N = 0 , [_ N / n ]_ A , if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) ) = Y )
86 81 27 85 3eqtrd
 |-  ( ph -> ( G ` N ) = Y )