Step |
Hyp |
Ref |
Expression |
1 |
|
sge0pr.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
sge0pr.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
3 |
|
sge0pr.d |
⊢ ( 𝜑 → 𝐷 ∈ ( 0 [,] +∞ ) ) |
4 |
|
sge0pr.e |
⊢ ( 𝜑 → 𝐸 ∈ ( 0 [,] +∞ ) ) |
5 |
|
sge0pr.cd |
⊢ ( 𝑘 = 𝐴 → 𝐶 = 𝐷 ) |
6 |
|
sge0pr.ce |
⊢ ( 𝑘 = 𝐵 → 𝐶 = 𝐸 ) |
7 |
|
sge0pr.ab |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
8 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
9 |
8 4
|
sselid |
⊢ ( 𝜑 → 𝐸 ∈ ℝ* ) |
10 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
11 |
10
|
a1i |
⊢ ( 𝜑 → -∞ ∈ ℝ* ) |
12 |
|
0xr |
⊢ 0 ∈ ℝ* |
13 |
12
|
a1i |
⊢ ( 𝜑 → 0 ∈ ℝ* ) |
14 |
|
mnflt0 |
⊢ -∞ < 0 |
15 |
14
|
a1i |
⊢ ( 𝜑 → -∞ < 0 ) |
16 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
17 |
16
|
a1i |
⊢ ( 𝜑 → +∞ ∈ ℝ* ) |
18 |
|
iccgelb |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐸 ∈ ( 0 [,] +∞ ) ) → 0 ≤ 𝐸 ) |
19 |
13 17 4 18
|
syl3anc |
⊢ ( 𝜑 → 0 ≤ 𝐸 ) |
20 |
11 13 9 15 19
|
xrltletrd |
⊢ ( 𝜑 → -∞ < 𝐸 ) |
21 |
11 9 20
|
xrgtned |
⊢ ( 𝜑 → 𝐸 ≠ -∞ ) |
22 |
|
xaddpnf2 |
⊢ ( ( 𝐸 ∈ ℝ* ∧ 𝐸 ≠ -∞ ) → ( +∞ +𝑒 𝐸 ) = +∞ ) |
23 |
9 21 22
|
syl2anc |
⊢ ( 𝜑 → ( +∞ +𝑒 𝐸 ) = +∞ ) |
24 |
23
|
eqcomd |
⊢ ( 𝜑 → +∞ = ( +∞ +𝑒 𝐸 ) ) |
25 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐷 = +∞ ) → +∞ = ( +∞ +𝑒 𝐸 ) ) |
26 |
|
prex |
⊢ { 𝐴 , 𝐵 } ∈ V |
27 |
26
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐷 = +∞ ) → { 𝐴 , 𝐵 } ∈ V ) |
28 |
5
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝐴 ) → 𝐶 = 𝐷 ) |
29 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝐴 ) → 𝐷 ∈ ( 0 [,] +∞ ) ) |
30 |
28 29
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝐴 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
31 |
30
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 , 𝐵 } ) ∧ 𝑘 = 𝐴 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
32 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 , 𝐵 } ) ∧ ¬ 𝑘 = 𝐴 ) → 𝜑 ) |
33 |
|
simpl |
⊢ ( ( 𝑘 ∈ { 𝐴 , 𝐵 } ∧ ¬ 𝑘 = 𝐴 ) → 𝑘 ∈ { 𝐴 , 𝐵 } ) |
34 |
|
neqne |
⊢ ( ¬ 𝑘 = 𝐴 → 𝑘 ≠ 𝐴 ) |
35 |
34
|
adantl |
⊢ ( ( 𝑘 ∈ { 𝐴 , 𝐵 } ∧ ¬ 𝑘 = 𝐴 ) → 𝑘 ≠ 𝐴 ) |
36 |
|
elprn1 |
⊢ ( ( 𝑘 ∈ { 𝐴 , 𝐵 } ∧ 𝑘 ≠ 𝐴 ) → 𝑘 = 𝐵 ) |
37 |
33 35 36
|
syl2anc |
⊢ ( ( 𝑘 ∈ { 𝐴 , 𝐵 } ∧ ¬ 𝑘 = 𝐴 ) → 𝑘 = 𝐵 ) |
38 |
37
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 , 𝐵 } ) ∧ ¬ 𝑘 = 𝐴 ) → 𝑘 = 𝐵 ) |
39 |
6
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝐵 ) → 𝐶 = 𝐸 ) |
40 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝐵 ) → 𝐸 ∈ ( 0 [,] +∞ ) ) |
41 |
39 40
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝐵 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
42 |
32 38 41
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 , 𝐵 } ) ∧ ¬ 𝑘 = 𝐴 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
43 |
31 42
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 , 𝐵 } ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
44 |
|
eqid |
⊢ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) = ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) |
45 |
43 44
|
fmptd |
⊢ ( 𝜑 → ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) : { 𝐴 , 𝐵 } ⟶ ( 0 [,] +∞ ) ) |
46 |
45
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐷 = +∞ ) → ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) : { 𝐴 , 𝐵 } ⟶ ( 0 [,] +∞ ) ) |
47 |
|
id |
⊢ ( 𝐷 = +∞ → 𝐷 = +∞ ) |
48 |
47
|
eqcomd |
⊢ ( 𝐷 = +∞ → +∞ = 𝐷 ) |
49 |
48
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐷 = +∞ ) → +∞ = 𝐷 ) |
50 |
|
prid1g |
⊢ ( 𝐷 ∈ ( 0 [,] +∞ ) → 𝐷 ∈ { 𝐷 , 𝐸 } ) |
51 |
3 50
|
syl |
⊢ ( 𝜑 → 𝐷 ∈ { 𝐷 , 𝐸 } ) |
52 |
1 2 44 5 6
|
rnmptpr |
⊢ ( 𝜑 → ran ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) = { 𝐷 , 𝐸 } ) |
53 |
52
|
eqcomd |
⊢ ( 𝜑 → { 𝐷 , 𝐸 } = ran ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) |
54 |
51 53
|
eleqtrd |
⊢ ( 𝜑 → 𝐷 ∈ ran ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) |
55 |
54
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐷 = +∞ ) → 𝐷 ∈ ran ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) |
56 |
49 55
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝐷 = +∞ ) → +∞ ∈ ran ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) |
57 |
27 46 56
|
sge0pnfval |
⊢ ( ( 𝜑 ∧ 𝐷 = +∞ ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = +∞ ) |
58 |
|
oveq1 |
⊢ ( 𝐷 = +∞ → ( 𝐷 +𝑒 𝐸 ) = ( +∞ +𝑒 𝐸 ) ) |
59 |
58
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐷 = +∞ ) → ( 𝐷 +𝑒 𝐸 ) = ( +∞ +𝑒 𝐸 ) ) |
60 |
25 57 59
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐷 = +∞ ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = ( 𝐷 +𝑒 𝐸 ) ) |
61 |
8 3
|
sselid |
⊢ ( 𝜑 → 𝐷 ∈ ℝ* ) |
62 |
|
iccgelb |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ ( 0 [,] +∞ ) ) → 0 ≤ 𝐷 ) |
63 |
13 17 3 62
|
syl3anc |
⊢ ( 𝜑 → 0 ≤ 𝐷 ) |
64 |
11 13 61 15 63
|
xrltletrd |
⊢ ( 𝜑 → -∞ < 𝐷 ) |
65 |
11 61 64
|
xrgtned |
⊢ ( 𝜑 → 𝐷 ≠ -∞ ) |
66 |
|
xaddpnf1 |
⊢ ( ( 𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞ ) → ( 𝐷 +𝑒 +∞ ) = +∞ ) |
67 |
61 65 66
|
syl2anc |
⊢ ( 𝜑 → ( 𝐷 +𝑒 +∞ ) = +∞ ) |
68 |
67
|
eqcomd |
⊢ ( 𝜑 → +∞ = ( 𝐷 +𝑒 +∞ ) ) |
69 |
68
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐸 = +∞ ) → +∞ = ( 𝐷 +𝑒 +∞ ) ) |
70 |
26
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐸 = +∞ ) → { 𝐴 , 𝐵 } ∈ V ) |
71 |
45
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐸 = +∞ ) → ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) : { 𝐴 , 𝐵 } ⟶ ( 0 [,] +∞ ) ) |
72 |
|
id |
⊢ ( 𝐸 = +∞ → 𝐸 = +∞ ) |
73 |
72
|
eqcomd |
⊢ ( 𝐸 = +∞ → +∞ = 𝐸 ) |
74 |
73
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐸 = +∞ ) → +∞ = 𝐸 ) |
75 |
|
prid2g |
⊢ ( 𝐸 ∈ ( 0 [,] +∞ ) → 𝐸 ∈ { 𝐷 , 𝐸 } ) |
76 |
4 75
|
syl |
⊢ ( 𝜑 → 𝐸 ∈ { 𝐷 , 𝐸 } ) |
77 |
76 53
|
eleqtrd |
⊢ ( 𝜑 → 𝐸 ∈ ran ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) |
78 |
77
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐸 = +∞ ) → 𝐸 ∈ ran ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) |
79 |
74 78
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝐸 = +∞ ) → +∞ ∈ ran ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) |
80 |
70 71 79
|
sge0pnfval |
⊢ ( ( 𝜑 ∧ 𝐸 = +∞ ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = +∞ ) |
81 |
|
oveq2 |
⊢ ( 𝐸 = +∞ → ( 𝐷 +𝑒 𝐸 ) = ( 𝐷 +𝑒 +∞ ) ) |
82 |
81
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐸 = +∞ ) → ( 𝐷 +𝑒 𝐸 ) = ( 𝐷 +𝑒 +∞ ) ) |
83 |
69 80 82
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐸 = +∞ ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = ( 𝐷 +𝑒 𝐸 ) ) |
84 |
83
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ 𝐸 = +∞ ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = ( 𝐷 +𝑒 𝐸 ) ) |
85 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
86 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
87 |
85 86
|
sstri |
⊢ ( 0 [,) +∞ ) ⊆ ℂ |
88 |
12
|
a1i |
⊢ ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) → 0 ∈ ℝ* ) |
89 |
16
|
a1i |
⊢ ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) → +∞ ∈ ℝ* ) |
90 |
61
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) → 𝐷 ∈ ℝ* ) |
91 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) → 0 ≤ 𝐷 ) |
92 |
|
pnfge |
⊢ ( 𝐷 ∈ ℝ* → 𝐷 ≤ +∞ ) |
93 |
61 92
|
syl |
⊢ ( 𝜑 → 𝐷 ≤ +∞ ) |
94 |
93
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) → 𝐷 ≤ +∞ ) |
95 |
47
|
necon3bi |
⊢ ( ¬ 𝐷 = +∞ → 𝐷 ≠ +∞ ) |
96 |
95
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) → 𝐷 ≠ +∞ ) |
97 |
90 89 94 96
|
xrleneltd |
⊢ ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) → 𝐷 < +∞ ) |
98 |
88 89 90 91 97
|
elicod |
⊢ ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) → 𝐷 ∈ ( 0 [,) +∞ ) ) |
99 |
98
|
adantr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → 𝐷 ∈ ( 0 [,) +∞ ) ) |
100 |
87 99
|
sselid |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → 𝐷 ∈ ℂ ) |
101 |
12
|
a1i |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ) → 0 ∈ ℝ* ) |
102 |
16
|
a1i |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ) → +∞ ∈ ℝ* ) |
103 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ) → 𝐸 ∈ ℝ* ) |
104 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ) → 0 ≤ 𝐸 ) |
105 |
|
pnfge |
⊢ ( 𝐸 ∈ ℝ* → 𝐸 ≤ +∞ ) |
106 |
9 105
|
syl |
⊢ ( 𝜑 → 𝐸 ≤ +∞ ) |
107 |
106
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ) → 𝐸 ≤ +∞ ) |
108 |
72
|
necon3bi |
⊢ ( ¬ 𝐸 = +∞ → 𝐸 ≠ +∞ ) |
109 |
108
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ) → 𝐸 ≠ +∞ ) |
110 |
103 102 107 109
|
xrleneltd |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ) → 𝐸 < +∞ ) |
111 |
101 102 103 104 110
|
elicod |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ) → 𝐸 ∈ ( 0 [,) +∞ ) ) |
112 |
87 111
|
sselid |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ) → 𝐸 ∈ ℂ ) |
113 |
112
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → 𝐸 ∈ ℂ ) |
114 |
100 113
|
jca |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → ( 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ) ) |
115 |
1 2
|
jca |
⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ) |
116 |
115
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ) |
117 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → 𝐴 ≠ 𝐵 ) |
118 |
5 6 114 116 117
|
sumpr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → Σ 𝑘 ∈ { 𝐴 , 𝐵 } 𝐶 = ( 𝐷 + 𝐸 ) ) |
119 |
|
prfi |
⊢ { 𝐴 , 𝐵 } ∈ Fin |
120 |
119
|
a1i |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → { 𝐴 , 𝐵 } ∈ Fin ) |
121 |
5
|
adantl |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ 𝑘 = 𝐴 ) → 𝐶 = 𝐷 ) |
122 |
98
|
adantr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ 𝑘 = 𝐴 ) → 𝐷 ∈ ( 0 [,) +∞ ) ) |
123 |
121 122
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ 𝑘 = 𝐴 ) → 𝐶 ∈ ( 0 [,) +∞ ) ) |
124 |
123
|
ad4ant14 |
⊢ ( ( ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) ∧ 𝑘 ∈ { 𝐴 , 𝐵 } ) ∧ 𝑘 = 𝐴 ) → 𝐶 ∈ ( 0 [,) +∞ ) ) |
125 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) ∧ 𝑘 ∈ { 𝐴 , 𝐵 } ) ∧ ¬ 𝑘 = 𝐴 ) → 𝜑 ) |
126 |
|
simpllr |
⊢ ( ( ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) ∧ 𝑘 ∈ { 𝐴 , 𝐵 } ) ∧ ¬ 𝑘 = 𝐴 ) → ¬ 𝐸 = +∞ ) |
127 |
37
|
adantll |
⊢ ( ( ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) ∧ 𝑘 ∈ { 𝐴 , 𝐵 } ) ∧ ¬ 𝑘 = 𝐴 ) → 𝑘 = 𝐵 ) |
128 |
39
|
3adant2 |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵 ) → 𝐶 = 𝐸 ) |
129 |
111
|
3adant3 |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵 ) → 𝐸 ∈ ( 0 [,) +∞ ) ) |
130 |
128 129
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵 ) → 𝐶 ∈ ( 0 [,) +∞ ) ) |
131 |
125 126 127 130
|
syl3anc |
⊢ ( ( ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) ∧ 𝑘 ∈ { 𝐴 , 𝐵 } ) ∧ ¬ 𝑘 = 𝐴 ) → 𝐶 ∈ ( 0 [,) +∞ ) ) |
132 |
124 131
|
pm2.61dan |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) ∧ 𝑘 ∈ { 𝐴 , 𝐵 } ) → 𝐶 ∈ ( 0 [,) +∞ ) ) |
133 |
120 132
|
sge0fsummpt |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = Σ 𝑘 ∈ { 𝐴 , 𝐵 } 𝐶 ) |
134 |
85 99
|
sselid |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → 𝐷 ∈ ℝ ) |
135 |
85 111
|
sselid |
⊢ ( ( 𝜑 ∧ ¬ 𝐸 = +∞ ) → 𝐸 ∈ ℝ ) |
136 |
135
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → 𝐸 ∈ ℝ ) |
137 |
|
rexadd |
⊢ ( ( 𝐷 ∈ ℝ ∧ 𝐸 ∈ ℝ ) → ( 𝐷 +𝑒 𝐸 ) = ( 𝐷 + 𝐸 ) ) |
138 |
134 136 137
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → ( 𝐷 +𝑒 𝐸 ) = ( 𝐷 + 𝐸 ) ) |
139 |
118 133 138
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) ∧ ¬ 𝐸 = +∞ ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = ( 𝐷 +𝑒 𝐸 ) ) |
140 |
84 139
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ ¬ 𝐷 = +∞ ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = ( 𝐷 +𝑒 𝐸 ) ) |
141 |
60 140
|
pm2.61dan |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = ( 𝐷 +𝑒 𝐸 ) ) |