Metamath Proof Explorer


Theorem esumrnmpt2

Description: Rewrite an extended sum into a sum on the range of a mapping function. (Contributed by Thierry Arnoux, 30-May-2020)

Ref Expression
Hypotheses esumrnmpt2.1 ( 𝑦 = 𝐵𝐶 = 𝐷 )
esumrnmpt2.2 ( 𝜑𝐴𝑉 )
esumrnmpt2.3 ( ( 𝜑𝑘𝐴 ) → 𝐷 ∈ ( 0 [,] +∞ ) )
esumrnmpt2.4 ( ( 𝜑𝑘𝐴 ) → 𝐵𝑊 )
esumrnmpt2.5 ( ( ( 𝜑𝑘𝐴 ) ∧ 𝐵 = ∅ ) → 𝐷 = 0 )
esumrnmpt2.6 ( 𝜑Disj 𝑘𝐴 𝐵 )
Assertion esumrnmpt2 ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘𝐴𝐵 ) 𝐶 = Σ* 𝑘𝐴 𝐷 )

Proof

Step Hyp Ref Expression
1 esumrnmpt2.1 ( 𝑦 = 𝐵𝐶 = 𝐷 )
2 esumrnmpt2.2 ( 𝜑𝐴𝑉 )
3 esumrnmpt2.3 ( ( 𝜑𝑘𝐴 ) → 𝐷 ∈ ( 0 [,] +∞ ) )
4 esumrnmpt2.4 ( ( 𝜑𝑘𝐴 ) → 𝐵𝑊 )
5 esumrnmpt2.5 ( ( ( 𝜑𝑘𝐴 ) ∧ 𝐵 = ∅ ) → 𝐷 = 0 )
6 esumrnmpt2.6 ( 𝜑Disj 𝑘𝐴 𝐵 )
7 nfrab1 𝑘 { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ }
8 ssrab2 { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ⊆ 𝐴
9 8 a1i ( 𝜑 → { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ⊆ 𝐴 )
10 2 9 ssexd ( 𝜑 → { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ∈ V )
11 9 sselda ( ( 𝜑𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) → 𝑘𝐴 )
12 11 3 syldan ( ( 𝜑𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) → 𝐷 ∈ ( 0 [,] +∞ ) )
13 11 4 syldan ( ( 𝜑𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) → 𝐵𝑊 )
14 rabid ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↔ ( 𝑘𝐴 ∧ ¬ 𝐵 = ∅ ) )
15 14 simprbi ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } → ¬ 𝐵 = ∅ )
16 15 adantl ( ( 𝜑𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) → ¬ 𝐵 = ∅ )
17 elsng ( 𝐵𝑊 → ( 𝐵 ∈ { ∅ } ↔ 𝐵 = ∅ ) )
18 13 17 syl ( ( 𝜑𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) → ( 𝐵 ∈ { ∅ } ↔ 𝐵 = ∅ ) )
19 16 18 mtbird ( ( 𝜑𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) → ¬ 𝐵 ∈ { ∅ } )
20 13 19 eldifd ( ( 𝜑𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) → 𝐵 ∈ ( 𝑊 ∖ { ∅ } ) )
21 nfcv 𝑘 𝐴
22 7 21 disjss1f ( { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ⊆ 𝐴 → ( Disj 𝑘𝐴 𝐵Disj 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐵 ) )
23 9 6 22 sylc ( 𝜑Disj 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐵 )
24 7 1 10 12 20 23 esumrnmpt ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 )
25 nfv 𝑦 ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ )
26 snex { ∅ } ∈ V
27 26 a1i ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) → { ∅ } ∈ V )
28 velsn ( 𝑦 ∈ { ∅ } ↔ 𝑦 = ∅ )
29 28 biimpi ( 𝑦 ∈ { ∅ } → 𝑦 = ∅ )
30 29 adantl ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 ∈ { ∅ } ) → 𝑦 = ∅ )
31 nfv 𝑘 𝜑
32 nfre1 𝑘𝑘𝐴 𝐵 = ∅
33 31 32 nfan 𝑘 ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ )
34 nfv 𝑘 𝑦 = ∅
35 33 34 nfan 𝑘 ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ )
36 nfv 𝑘 𝐶 = 0
37 simpllr ( ( ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘𝐴 ) ∧ 𝐵 = ∅ ) → 𝑦 = ∅ )
38 simpr ( ( ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘𝐴 ) ∧ 𝐵 = ∅ ) → 𝐵 = ∅ )
39 37 38 eqtr4d ( ( ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘𝐴 ) ∧ 𝐵 = ∅ ) → 𝑦 = 𝐵 )
40 39 1 syl ( ( ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘𝐴 ) ∧ 𝐵 = ∅ ) → 𝐶 = 𝐷 )
41 simp-4l ( ( ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘𝐴 ) ∧ 𝐵 = ∅ ) → 𝜑 )
42 simplr ( ( ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘𝐴 ) ∧ 𝐵 = ∅ ) → 𝑘𝐴 )
43 41 42 38 5 syl21anc ( ( ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘𝐴 ) ∧ 𝐵 = ∅ ) → 𝐷 = 0 )
44 40 43 eqtrd ( ( ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘𝐴 ) ∧ 𝐵 = ∅ ) → 𝐶 = 0 )
45 simplr ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) → ∃ 𝑘𝐴 𝐵 = ∅ )
46 35 36 44 45 r19.29af2 ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) → 𝐶 = 0 )
47 30 46 syldan ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 ∈ { ∅ } ) → 𝐶 = 0 )
48 0e0iccpnf 0 ∈ ( 0 [,] +∞ )
49 47 48 eqeltrdi ( ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) ∧ 𝑦 ∈ { ∅ } ) → 𝐶 ∈ ( 0 [,] +∞ ) )
50 nfcv 𝑘 𝑦
51 nfmpt1 𝑘 ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 )
52 51 nfrn 𝑘 ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 )
53 50 52 nfel 𝑘 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 )
54 31 53 nfan 𝑘 ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) )
55 simpr ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 )
56 rabid ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↔ ( 𝑘𝐴𝐵 = ∅ ) )
57 56 simprbi ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } → 𝐵 = ∅ )
58 57 ad2antlr ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐵 = ∅ )
59 55 58 eqtrd ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝑦 = ∅ )
60 59 28 sylibr ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝑦 ∈ { ∅ } )
61 vex 𝑦 ∈ V
62 eqid ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) = ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 )
63 62 elrnmpt ( 𝑦 ∈ V → ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 𝑦 = 𝐵 ) )
64 61 63 ax-mp ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 𝑦 = 𝐵 )
65 64 biimpi ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) → ∃ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 𝑦 = 𝐵 )
66 65 adantl ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) → ∃ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 𝑦 = 𝐵 )
67 54 60 66 r19.29af ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) → 𝑦 ∈ { ∅ } )
68 67 ex ( 𝜑 → ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) → 𝑦 ∈ { ∅ } ) )
69 68 ssrdv ( 𝜑 → ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ⊆ { ∅ } )
70 69 adantr ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) → ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ⊆ { ∅ } )
71 25 27 49 70 esummono ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ Σ* 𝑦 ∈ { ∅ } 𝐶 )
72 0ex ∅ ∈ V
73 72 a1i ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) → ∅ ∈ V )
74 48 a1i ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) → 0 ∈ ( 0 [,] +∞ ) )
75 46 73 74 esumsn ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) → Σ* 𝑦 ∈ { ∅ } 𝐶 = 0 )
76 71 75 breqtrd ( ( 𝜑 ∧ ∃ 𝑘𝐴 𝐵 = ∅ ) → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 )
77 simpr ( ( 𝜑 ∧ ¬ ∃ 𝑘𝐴 𝐵 = ∅ ) → ¬ ∃ 𝑘𝐴 𝐵 = ∅ )
78 nfv 𝑦 ¬ ∃ 𝑘𝐴 𝐵 = ∅
79 32 nfn 𝑘 ¬ ∃ 𝑘𝐴 𝐵 = ∅
80 rabn0 ( { 𝑘𝐴𝐵 = ∅ } ≠ ∅ ↔ ∃ 𝑘𝐴 𝐵 = ∅ )
81 80 biimpi ( { 𝑘𝐴𝐵 = ∅ } ≠ ∅ → ∃ 𝑘𝐴 𝐵 = ∅ )
82 81 necon1bi ( ¬ ∃ 𝑘𝐴 𝐵 = ∅ → { 𝑘𝐴𝐵 = ∅ } = ∅ )
83 eqid 𝐵 = 𝐵
84 83 a1i ( ¬ ∃ 𝑘𝐴 𝐵 = ∅ → 𝐵 = 𝐵 )
85 79 82 84 mpteq12df ( ¬ ∃ 𝑘𝐴 𝐵 = ∅ → ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) = ( 𝑘 ∈ ∅ ↦ 𝐵 ) )
86 mpt0 ( 𝑘 ∈ ∅ ↦ 𝐵 ) = ∅
87 85 86 eqtrdi ( ¬ ∃ 𝑘𝐴 𝐵 = ∅ → ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) = ∅ )
88 87 rneqd ( ¬ ∃ 𝑘𝐴 𝐵 = ∅ → ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) = ran ∅ )
89 rn0 ran ∅ = ∅
90 88 89 eqtrdi ( ¬ ∃ 𝑘𝐴 𝐵 = ∅ → ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) = ∅ )
91 78 90 esumeq1d ( ¬ ∃ 𝑘𝐴 𝐵 = ∅ → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = Σ* 𝑦 ∈ ∅ 𝐶 )
92 esumnul Σ* 𝑦 ∈ ∅ 𝐶 = 0
93 91 92 eqtrdi ( ¬ ∃ 𝑘𝐴 𝐵 = ∅ → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = 0 )
94 0le0 0 ≤ 0
95 93 94 eqbrtrdi ( ¬ ∃ 𝑘𝐴 𝐵 = ∅ → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 )
96 77 95 syl ( ( 𝜑 ∧ ¬ ∃ 𝑘𝐴 𝐵 = ∅ ) → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 )
97 76 96 pm2.61dan ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 )
98 ssrab2 { 𝑘𝐴𝐵 = ∅ } ⊆ 𝐴
99 98 a1i ( 𝜑 → { 𝑘𝐴𝐵 = ∅ } ⊆ 𝐴 )
100 2 99 ssexd ( 𝜑 → { 𝑘𝐴𝐵 = ∅ } ∈ V )
101 nfrab1 𝑘 { 𝑘𝐴𝐵 = ∅ }
102 101 mptexgf ( { 𝑘𝐴𝐵 = ∅ } ∈ V → ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∈ V )
103 rnexg ( ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∈ V → ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∈ V )
104 100 102 103 3syl ( 𝜑 → ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∈ V )
105 1 adantl ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐶 = 𝐷 )
106 simplll ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝜑 )
107 99 sselda ( ( 𝜑𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) → 𝑘𝐴 )
108 107 adantlr ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) → 𝑘𝐴 )
109 108 adantr ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝑘𝐴 )
110 106 109 3 syl2anc ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐷 ∈ ( 0 [,] +∞ ) )
111 105 110 eqeltrd ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐶 ∈ ( 0 [,] +∞ ) )
112 54 111 66 r19.29af ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ) → 𝐶 ∈ ( 0 [,] +∞ ) )
113 112 ralrimiva ( 𝜑 → ∀ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) )
114 nfcv 𝑦 ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 )
115 114 esumcl ( ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∈ V ∧ ∀ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) ) → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) )
116 104 113 115 syl2anc ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) )
117 elxrge0 ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) ↔ ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ℝ* ∧ 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) )
118 117 simprbi ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) → 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 )
119 116 118 syl ( 𝜑 → 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 )
120 97 119 jca ( 𝜑 → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 ∧ 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) )
121 iccssxr ( 0 [,] +∞ ) ⊆ ℝ*
122 121 116 sselid ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ℝ* )
123 121 48 sselii 0 ∈ ℝ*
124 123 a1i ( 𝜑 → 0 ∈ ℝ* )
125 xrletri3 ( ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = 0 ↔ ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 ∧ 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) ) )
126 122 124 125 syl2anc ( 𝜑 → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = 0 ↔ ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 ∧ 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) ) )
127 120 126 mpbird ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = 0 )
128 127 oveq1d ( 𝜑 → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) = ( 0 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) )
129 12 ralrimiva ( 𝜑 → ∀ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ( 0 [,] +∞ ) )
130 7 esumcl ( ( { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ∈ V ∧ ∀ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ( 0 [,] +∞ ) ) → Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ( 0 [,] +∞ ) )
131 10 129 130 syl2anc ( 𝜑 → Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ( 0 [,] +∞ ) )
132 121 131 sselid ( 𝜑 → Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ℝ* )
133 24 132 eqeltrd ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ℝ* )
134 xaddid2 ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ℝ* → ( 0 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) = Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 )
135 133 134 syl ( 𝜑 → ( 0 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) = Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 )
136 128 135 eqtrd ( 𝜑 → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) = Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 )
137 simpl ( ( 𝜑𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) → 𝜑 )
138 57 adantl ( ( 𝜑𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) → 𝐵 = ∅ )
139 137 107 138 5 syl21anc ( ( 𝜑𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) → 𝐷 = 0 )
140 139 ralrimiva ( 𝜑 → ∀ 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 𝐷 = 0 )
141 31 140 esumeq2d ( 𝜑 → Σ* 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 𝐷 = Σ* 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 0 )
142 101 esum0 ( { 𝑘𝐴𝐵 = ∅ } ∈ V → Σ* 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 0 = 0 )
143 100 142 syl ( 𝜑 → Σ* 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 0 = 0 )
144 141 143 eqtrd ( 𝜑 → Σ* 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 𝐷 = 0 )
145 144 oveq1d ( 𝜑 → ( Σ* 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 𝐷 +𝑒 Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) = ( 0 +𝑒 Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) )
146 xaddid2 ( Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ℝ* → ( 0 +𝑒 Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) = Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 )
147 132 146 syl ( 𝜑 → ( 0 +𝑒 Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) = Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 )
148 145 147 eqtrd ( 𝜑 → ( Σ* 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 𝐷 +𝑒 Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) = Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 )
149 24 136 148 3eqtr4d ( 𝜑 → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) = ( Σ* 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 𝐷 +𝑒 Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) )
150 nfv 𝑦 𝜑
151 nfcv 𝑦 ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 )
152 7 mptexgf ( { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ∈ V → ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V )
153 rnexg ( ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V → ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V )
154 10 152 153 3syl ( 𝜑 → ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V )
155 69 ssrind ( 𝜑 → ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∩ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ⊆ ( { ∅ } ∩ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) )
156 incom ( ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∩ { ∅ } ) = ( { ∅ } ∩ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) )
157 15 neqned ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } → 𝐵 ≠ ∅ )
158 157 necomd ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } → ∅ ≠ 𝐵 )
159 158 neneqd ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } → ¬ ∅ = 𝐵 )
160 159 nrex ¬ ∃ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ∅ = 𝐵
161 eqid ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) = ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 )
162 161 elrnmpt ( ∅ ∈ V → ( ∅ ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ∅ = 𝐵 ) )
163 72 162 ax-mp ( ∅ ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ∅ = 𝐵 )
164 160 163 mtbir ¬ ∅ ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 )
165 disjsn ( ( ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∩ { ∅ } ) = ∅ ↔ ¬ ∅ ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) )
166 164 165 mpbir ( ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∩ { ∅ } ) = ∅
167 156 166 eqtr3i ( { ∅ } ∩ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) = ∅
168 155 167 sseqtrdi ( 𝜑 → ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∩ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ⊆ ∅ )
169 ss0 ( ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∩ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ⊆ ∅ → ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∩ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) = ∅ )
170 168 169 syl ( 𝜑 → ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∩ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) = ∅ )
171 nfmpt1 𝑘 ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 )
172 171 nfrn 𝑘 ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 )
173 50 172 nfel 𝑘 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 )
174 31 173 nfan 𝑘 ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) )
175 1 adantl ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐶 = 𝐷 )
176 simplll ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝜑 )
177 11 adantlr ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) → 𝑘𝐴 )
178 177 adantr ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝑘𝐴 )
179 176 178 3 syl2anc ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐷 ∈ ( 0 [,] +∞ ) )
180 175 179 eqeltrd ( ( ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐶 ∈ ( 0 [,] +∞ ) )
181 161 elrnmpt ( 𝑦 ∈ V → ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝑦 = 𝐵 ) )
182 61 181 ax-mp ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝑦 = 𝐵 )
183 182 biimpi ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) → ∃ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝑦 = 𝐵 )
184 183 adantl ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) → ∃ 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝑦 = 𝐵 )
185 174 180 184 r19.29af ( ( 𝜑𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) → 𝐶 ∈ ( 0 [,] +∞ ) )
186 150 114 151 104 154 170 112 185 esumsplit ( 𝜑 → Σ* 𝑦 ∈ ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) 𝐶 = ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) 𝐶 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) )
187 rabnc ( { 𝑘𝐴𝐵 = ∅ } ∩ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) = ∅
188 187 a1i ( 𝜑 → ( { 𝑘𝐴𝐵 = ∅ } ∩ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) = ∅ )
189 107 3 syldan ( ( 𝜑𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ) → 𝐷 ∈ ( 0 [,] +∞ ) )
190 31 101 7 100 10 188 189 12 esumsplit ( 𝜑 → Σ* 𝑘 ∈ ( { 𝑘𝐴𝐵 = ∅ } ∪ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) 𝐷 = ( Σ* 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } 𝐷 +𝑒 Σ* 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) )
191 149 186 190 3eqtr4d ( 𝜑 → Σ* 𝑦 ∈ ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) 𝐶 = Σ* 𝑘 ∈ ( { 𝑘𝐴𝐵 = ∅ } ∪ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) 𝐷 )
192 rabxm 𝐴 = ( { 𝑘𝐴𝐵 = ∅ } ∪ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } )
193 192 83 mpteq12i ( 𝑘𝐴𝐵 ) = ( 𝑘 ∈ ( { 𝑘𝐴𝐵 = ∅ } ∪ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) ↦ 𝐵 )
194 mptun ( 𝑘 ∈ ( { 𝑘𝐴𝐵 = ∅ } ∪ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) ↦ 𝐵 ) = ( ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∪ ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) )
195 193 194 eqtri ( 𝑘𝐴𝐵 ) = ( ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∪ ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) )
196 195 rneqi ran ( 𝑘𝐴𝐵 ) = ran ( ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∪ ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) )
197 rnun ran ( ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∪ ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) = ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) )
198 196 197 eqtri ran ( 𝑘𝐴𝐵 ) = ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) )
199 198 a1i ( 𝜑 → ran ( 𝑘𝐴𝐵 ) = ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) )
200 150 199 esumeq1d ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘𝐴𝐵 ) 𝐶 = Σ* 𝑦 ∈ ( ran ( 𝑘 ∈ { 𝑘𝐴𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) 𝐶 )
201 192 a1i ( 𝜑𝐴 = ( { 𝑘𝐴𝐵 = ∅ } ∪ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) )
202 31 201 esumeq1d ( 𝜑 → Σ* 𝑘𝐴 𝐷 = Σ* 𝑘 ∈ ( { 𝑘𝐴𝐵 = ∅ } ∪ { 𝑘𝐴 ∣ ¬ 𝐵 = ∅ } ) 𝐷 )
203 191 200 202 3eqtr4d ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘𝐴𝐵 ) 𝐶 = Σ* 𝑘𝐴 𝐷 )