Step |
Hyp |
Ref |
Expression |
1 |
|
0cn |
|- 0 e. CC |
2 |
|
ifcl |
|- ( ( A e. CC /\ 0 e. CC ) -> if ( ps , A , 0 ) e. CC ) |
3 |
1 2
|
mpan2 |
|- ( A e. CC -> if ( ps , A , 0 ) e. CC ) |
4 |
3
|
ad2antrr |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> if ( ps , A , 0 ) e. CC ) |
5 |
|
simpll |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> A e. CC ) |
6 |
4 5 5
|
add12d |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( if ( ps , A , 0 ) + ( A + A ) ) = ( A + ( if ( ps , A , 0 ) + A ) ) ) |
7 |
5 4 5
|
addassd |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( ( A + if ( ps , A , 0 ) ) + A ) = ( A + ( if ( ps , A , 0 ) + A ) ) ) |
8 |
6 7
|
eqtr4d |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( if ( ps , A , 0 ) + ( A + A ) ) = ( ( A + if ( ps , A , 0 ) ) + A ) ) |
9 |
|
pm5.501 |
|- ( ph -> ( ps <-> ( ph <-> ps ) ) ) |
10 |
9
|
adantl |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( ps <-> ( ph <-> ps ) ) ) |
11 |
10
|
bicomd |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( ( ph <-> ps ) <-> ps ) ) |
12 |
11
|
ifbid |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> if ( ( ph <-> ps ) , A , 0 ) = if ( ps , A , 0 ) ) |
13 |
|
animorrl |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( ph \/ ps ) ) |
14 |
|
iftrue |
|- ( ( ph \/ ps ) -> if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) = ( 2 x. A ) ) |
15 |
13 14
|
syl |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) = ( 2 x. A ) ) |
16 |
5
|
2timesd |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( 2 x. A ) = ( A + A ) ) |
17 |
15 16
|
eqtrd |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) = ( A + A ) ) |
18 |
12 17
|
oveq12d |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( if ( ( ph <-> ps ) , A , 0 ) + if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) ) = ( if ( ps , A , 0 ) + ( A + A ) ) ) |
19 |
|
iftrue |
|- ( ph -> if ( ph , A , 0 ) = A ) |
20 |
19
|
adantl |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> if ( ph , A , 0 ) = A ) |
21 |
20
|
oveq1d |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) = ( A + if ( ps , A , 0 ) ) ) |
22 |
21
|
oveq1d |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + A ) = ( ( A + if ( ps , A , 0 ) ) + A ) ) |
23 |
8 18 22
|
3eqtr4d |
|- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( if ( ( ph <-> ps ) , A , 0 ) + if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) ) = ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + A ) ) |
24 |
|
iffalse |
|- ( -. ph -> if ( ph , A , 0 ) = 0 ) |
25 |
24
|
adantl |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> if ( ph , A , 0 ) = 0 ) |
26 |
25
|
oveq1d |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) = ( 0 + if ( ps , A , 0 ) ) ) |
27 |
3
|
ad2antrr |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> if ( ps , A , 0 ) e. CC ) |
28 |
27
|
addid2d |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( 0 + if ( ps , A , 0 ) ) = if ( ps , A , 0 ) ) |
29 |
26 28
|
eqtrd |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) = if ( ps , A , 0 ) ) |
30 |
29
|
oveq1d |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + A ) = ( if ( ps , A , 0 ) + A ) ) |
31 |
|
2cnd |
|- ( A e. CC -> 2 e. CC ) |
32 |
|
id |
|- ( A e. CC -> A e. CC ) |
33 |
31 32
|
mulcld |
|- ( A e. CC -> ( 2 x. A ) e. CC ) |
34 |
33
|
addid2d |
|- ( A e. CC -> ( 0 + ( 2 x. A ) ) = ( 2 x. A ) ) |
35 |
|
2times |
|- ( A e. CC -> ( 2 x. A ) = ( A + A ) ) |
36 |
34 35
|
eqtrd |
|- ( A e. CC -> ( 0 + ( 2 x. A ) ) = ( A + A ) ) |
37 |
36
|
adantr |
|- ( ( A e. CC /\ ps ) -> ( 0 + ( 2 x. A ) ) = ( A + A ) ) |
38 |
|
iftrue |
|- ( ps -> if ( ps , 0 , A ) = 0 ) |
39 |
38
|
adantl |
|- ( ( A e. CC /\ ps ) -> if ( ps , 0 , A ) = 0 ) |
40 |
|
iftrue |
|- ( ps -> if ( ps , ( 2 x. A ) , 0 ) = ( 2 x. A ) ) |
41 |
40
|
adantl |
|- ( ( A e. CC /\ ps ) -> if ( ps , ( 2 x. A ) , 0 ) = ( 2 x. A ) ) |
42 |
39 41
|
oveq12d |
|- ( ( A e. CC /\ ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( 0 + ( 2 x. A ) ) ) |
43 |
|
iftrue |
|- ( ps -> if ( ps , A , 0 ) = A ) |
44 |
43
|
adantl |
|- ( ( A e. CC /\ ps ) -> if ( ps , A , 0 ) = A ) |
45 |
44
|
oveq1d |
|- ( ( A e. CC /\ ps ) -> ( if ( ps , A , 0 ) + A ) = ( A + A ) ) |
46 |
37 42 45
|
3eqtr4d |
|- ( ( A e. CC /\ ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( if ( ps , A , 0 ) + A ) ) |
47 |
|
simpl |
|- ( ( A e. CC /\ -. ps ) -> A e. CC ) |
48 |
|
0cnd |
|- ( ( A e. CC /\ -. ps ) -> 0 e. CC ) |
49 |
47 48
|
addcomd |
|- ( ( A e. CC /\ -. ps ) -> ( A + 0 ) = ( 0 + A ) ) |
50 |
|
iffalse |
|- ( -. ps -> if ( ps , 0 , A ) = A ) |
51 |
50
|
adantl |
|- ( ( A e. CC /\ -. ps ) -> if ( ps , 0 , A ) = A ) |
52 |
|
iffalse |
|- ( -. ps -> if ( ps , ( 2 x. A ) , 0 ) = 0 ) |
53 |
52
|
adantl |
|- ( ( A e. CC /\ -. ps ) -> if ( ps , ( 2 x. A ) , 0 ) = 0 ) |
54 |
51 53
|
oveq12d |
|- ( ( A e. CC /\ -. ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( A + 0 ) ) |
55 |
|
iffalse |
|- ( -. ps -> if ( ps , A , 0 ) = 0 ) |
56 |
55
|
adantl |
|- ( ( A e. CC /\ -. ps ) -> if ( ps , A , 0 ) = 0 ) |
57 |
56
|
oveq1d |
|- ( ( A e. CC /\ -. ps ) -> ( if ( ps , A , 0 ) + A ) = ( 0 + A ) ) |
58 |
49 54 57
|
3eqtr4d |
|- ( ( A e. CC /\ -. ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( if ( ps , A , 0 ) + A ) ) |
59 |
46 58
|
pm2.61dan |
|- ( A e. CC -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( if ( ps , A , 0 ) + A ) ) |
60 |
59
|
ad2antrr |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( if ( ps , A , 0 ) + A ) ) |
61 |
|
ifnot |
|- if ( -. ps , A , 0 ) = if ( ps , 0 , A ) |
62 |
|
nbn2 |
|- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) ) |
63 |
62
|
adantl |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( -. ps <-> ( ph <-> ps ) ) ) |
64 |
63
|
ifbid |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> if ( -. ps , A , 0 ) = if ( ( ph <-> ps ) , A , 0 ) ) |
65 |
61 64
|
syl5eqr |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> if ( ps , 0 , A ) = if ( ( ph <-> ps ) , A , 0 ) ) |
66 |
|
biorf |
|- ( -. ph -> ( ps <-> ( ph \/ ps ) ) ) |
67 |
66
|
adantl |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( ps <-> ( ph \/ ps ) ) ) |
68 |
67
|
ifbid |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> if ( ps , ( 2 x. A ) , 0 ) = if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) ) |
69 |
65 68
|
oveq12d |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( if ( ( ph <-> ps ) , A , 0 ) + if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) ) ) |
70 |
30 60 69
|
3eqtr2rd |
|- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( if ( ( ph <-> ps ) , A , 0 ) + if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) ) = ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + A ) ) |
71 |
23 70
|
pm2.61dan |
|- ( ( A e. CC /\ ch ) -> ( if ( ( ph <-> ps ) , A , 0 ) + if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) ) = ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + A ) ) |
72 |
|
hadrot |
|- ( hadd ( ch , ph , ps ) <-> hadd ( ph , ps , ch ) ) |
73 |
|
had1 |
|- ( ch -> ( hadd ( ch , ph , ps ) <-> ( ph <-> ps ) ) ) |
74 |
72 73
|
bitr3id |
|- ( ch -> ( hadd ( ph , ps , ch ) <-> ( ph <-> ps ) ) ) |
75 |
74
|
adantl |
|- ( ( A e. CC /\ ch ) -> ( hadd ( ph , ps , ch ) <-> ( ph <-> ps ) ) ) |
76 |
75
|
ifbid |
|- ( ( A e. CC /\ ch ) -> if ( hadd ( ph , ps , ch ) , A , 0 ) = if ( ( ph <-> ps ) , A , 0 ) ) |
77 |
|
cad1 |
|- ( ch -> ( cadd ( ph , ps , ch ) <-> ( ph \/ ps ) ) ) |
78 |
77
|
adantl |
|- ( ( A e. CC /\ ch ) -> ( cadd ( ph , ps , ch ) <-> ( ph \/ ps ) ) ) |
79 |
78
|
ifbid |
|- ( ( A e. CC /\ ch ) -> if ( cadd ( ph , ps , ch ) , ( 2 x. A ) , 0 ) = if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) ) |
80 |
76 79
|
oveq12d |
|- ( ( A e. CC /\ ch ) -> ( if ( hadd ( ph , ps , ch ) , A , 0 ) + if ( cadd ( ph , ps , ch ) , ( 2 x. A ) , 0 ) ) = ( if ( ( ph <-> ps ) , A , 0 ) + if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) ) ) |
81 |
|
iftrue |
|- ( ch -> if ( ch , A , 0 ) = A ) |
82 |
81
|
adantl |
|- ( ( A e. CC /\ ch ) -> if ( ch , A , 0 ) = A ) |
83 |
82
|
oveq2d |
|- ( ( A e. CC /\ ch ) -> ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + if ( ch , A , 0 ) ) = ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + A ) ) |
84 |
71 80 83
|
3eqtr4d |
|- ( ( A e. CC /\ ch ) -> ( if ( hadd ( ph , ps , ch ) , A , 0 ) + if ( cadd ( ph , ps , ch ) , ( 2 x. A ) , 0 ) ) = ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + if ( ch , A , 0 ) ) ) |
85 |
19
|
adantl |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> if ( ph , A , 0 ) = A ) |
86 |
85
|
oveq1d |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) = ( A + if ( ps , A , 0 ) ) ) |
87 |
44
|
oveq2d |
|- ( ( A e. CC /\ ps ) -> ( A + if ( ps , A , 0 ) ) = ( A + A ) ) |
88 |
37 42 87
|
3eqtr4d |
|- ( ( A e. CC /\ ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( A + if ( ps , A , 0 ) ) ) |
89 |
53 56
|
eqtr4d |
|- ( ( A e. CC /\ -. ps ) -> if ( ps , ( 2 x. A ) , 0 ) = if ( ps , A , 0 ) ) |
90 |
51 89
|
oveq12d |
|- ( ( A e. CC /\ -. ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( A + if ( ps , A , 0 ) ) ) |
91 |
88 90
|
pm2.61dan |
|- ( A e. CC -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( A + if ( ps , A , 0 ) ) ) |
92 |
91
|
ad2antrr |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( A + if ( ps , A , 0 ) ) ) |
93 |
9
|
adantl |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( ps <-> ( ph <-> ps ) ) ) |
94 |
93
|
notbid |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( -. ps <-> -. ( ph <-> ps ) ) ) |
95 |
|
df-xor |
|- ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) |
96 |
94 95
|
bitr4di |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( -. ps <-> ( ph \/_ ps ) ) ) |
97 |
96
|
ifbid |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> if ( -. ps , A , 0 ) = if ( ( ph \/_ ps ) , A , 0 ) ) |
98 |
61 97
|
syl5eqr |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> if ( ps , 0 , A ) = if ( ( ph \/_ ps ) , A , 0 ) ) |
99 |
|
ibar |
|- ( ph -> ( ps <-> ( ph /\ ps ) ) ) |
100 |
99
|
adantl |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( ps <-> ( ph /\ ps ) ) ) |
101 |
100
|
ifbid |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> if ( ps , ( 2 x. A ) , 0 ) = if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) ) |
102 |
98 101
|
oveq12d |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( if ( ( ph \/_ ps ) , A , 0 ) + if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) ) ) |
103 |
86 92 102
|
3eqtr2rd |
|- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( if ( ( ph \/_ ps ) , A , 0 ) + if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) ) = ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) ) |
104 |
|
simplll |
|- ( ( ( ( A e. CC /\ -. ch ) /\ -. ph ) /\ ps ) -> A e. CC ) |
105 |
|
0cnd |
|- ( ( ( ( A e. CC /\ -. ch ) /\ -. ph ) /\ -. ps ) -> 0 e. CC ) |
106 |
104 105
|
ifclda |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> if ( ps , A , 0 ) e. CC ) |
107 |
|
0cnd |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> 0 e. CC ) |
108 |
106 107
|
addcomd |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( if ( ps , A , 0 ) + 0 ) = ( 0 + if ( ps , A , 0 ) ) ) |
109 |
62
|
adantl |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( -. ps <-> ( ph <-> ps ) ) ) |
110 |
109
|
con1bid |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( -. ( ph <-> ps ) <-> ps ) ) |
111 |
95 110
|
syl5bb |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( ( ph \/_ ps ) <-> ps ) ) |
112 |
111
|
ifbid |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> if ( ( ph \/_ ps ) , A , 0 ) = if ( ps , A , 0 ) ) |
113 |
|
simpr |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> -. ph ) |
114 |
113
|
intnanrd |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> -. ( ph /\ ps ) ) |
115 |
|
iffalse |
|- ( -. ( ph /\ ps ) -> if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) = 0 ) |
116 |
114 115
|
syl |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) = 0 ) |
117 |
112 116
|
oveq12d |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( if ( ( ph \/_ ps ) , A , 0 ) + if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) ) = ( if ( ps , A , 0 ) + 0 ) ) |
118 |
24
|
adantl |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> if ( ph , A , 0 ) = 0 ) |
119 |
118
|
oveq1d |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) = ( 0 + if ( ps , A , 0 ) ) ) |
120 |
108 117 119
|
3eqtr4d |
|- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( if ( ( ph \/_ ps ) , A , 0 ) + if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) ) = ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) ) |
121 |
103 120
|
pm2.61dan |
|- ( ( A e. CC /\ -. ch ) -> ( if ( ( ph \/_ ps ) , A , 0 ) + if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) ) = ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) ) |
122 |
|
had0 |
|- ( -. ch -> ( hadd ( ch , ph , ps ) <-> ( ph \/_ ps ) ) ) |
123 |
72 122
|
bitr3id |
|- ( -. ch -> ( hadd ( ph , ps , ch ) <-> ( ph \/_ ps ) ) ) |
124 |
123
|
adantl |
|- ( ( A e. CC /\ -. ch ) -> ( hadd ( ph , ps , ch ) <-> ( ph \/_ ps ) ) ) |
125 |
124
|
ifbid |
|- ( ( A e. CC /\ -. ch ) -> if ( hadd ( ph , ps , ch ) , A , 0 ) = if ( ( ph \/_ ps ) , A , 0 ) ) |
126 |
|
cad0 |
|- ( -. ch -> ( cadd ( ph , ps , ch ) <-> ( ph /\ ps ) ) ) |
127 |
126
|
adantl |
|- ( ( A e. CC /\ -. ch ) -> ( cadd ( ph , ps , ch ) <-> ( ph /\ ps ) ) ) |
128 |
127
|
ifbid |
|- ( ( A e. CC /\ -. ch ) -> if ( cadd ( ph , ps , ch ) , ( 2 x. A ) , 0 ) = if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) ) |
129 |
125 128
|
oveq12d |
|- ( ( A e. CC /\ -. ch ) -> ( if ( hadd ( ph , ps , ch ) , A , 0 ) + if ( cadd ( ph , ps , ch ) , ( 2 x. A ) , 0 ) ) = ( if ( ( ph \/_ ps ) , A , 0 ) + if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) ) ) |
130 |
|
iffalse |
|- ( -. ch -> if ( ch , A , 0 ) = 0 ) |
131 |
130
|
oveq2d |
|- ( -. ch -> ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + if ( ch , A , 0 ) ) = ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + 0 ) ) |
132 |
|
ifcl |
|- ( ( A e. CC /\ 0 e. CC ) -> if ( ph , A , 0 ) e. CC ) |
133 |
1 132
|
mpan2 |
|- ( A e. CC -> if ( ph , A , 0 ) e. CC ) |
134 |
133 3
|
addcld |
|- ( A e. CC -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) e. CC ) |
135 |
134
|
addid1d |
|- ( A e. CC -> ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + 0 ) = ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) ) |
136 |
131 135
|
sylan9eqr |
|- ( ( A e. CC /\ -. ch ) -> ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + if ( ch , A , 0 ) ) = ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) ) |
137 |
121 129 136
|
3eqtr4d |
|- ( ( A e. CC /\ -. ch ) -> ( if ( hadd ( ph , ps , ch ) , A , 0 ) + if ( cadd ( ph , ps , ch ) , ( 2 x. A ) , 0 ) ) = ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + if ( ch , A , 0 ) ) ) |
138 |
84 137
|
pm2.61dan |
|- ( A e. CC -> ( if ( hadd ( ph , ps , ch ) , A , 0 ) + if ( cadd ( ph , ps , ch ) , ( 2 x. A ) , 0 ) ) = ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + if ( ch , A , 0 ) ) ) |