Metamath Proof Explorer

Theorem sumeq2si

Description: Equality inference for sum. (Contributed by GG, 1-Sep-2025)

Ref Expression
Hypothesis sumeq2si.1 ⊢ 𝐵 = 𝐶
Assertion sumeq2si Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶

Proof

Step Hyp Ref Expression
1 sumeq2si.1 ⊢ 𝐵 = 𝐶
2 1 csbeq2i ⊢ ⦋ 𝑛 / 𝑘 ⦌ 𝐵 = ⦋ 𝑛 / 𝑘 ⦌ 𝐶
3 ifeq1 ⊢ ( ⦋ 𝑛 / 𝑘 ⦌ 𝐵 = ⦋ 𝑛 / 𝑘 ⦌ 𝐶 → if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) = if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) )
4 2 3 ax-mp ⊢ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) = if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 )
5 4 mpteq2i ⊢ ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) = ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) )
6 seqeq3 ⊢ ( ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) = ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) → seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) = seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) )
7 5 6 ax-mp ⊢ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) = seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) )
8 7 breq1i ⊢ ( seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ↔ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 )
9 8 anbi2i ⊢ ( ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ↔ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) )
10 9 rexbii ⊢ ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ↔ ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) )
11 1 csbeq2i ⊢ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 = ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶
12 11 mpteq2i ⊢ ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) = ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 )
13 seqeq3 ⊢ ( ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) = ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) = seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) )
14 12 13 ax-mp ⊢ seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) = seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) )
15 14 fveq1i ⊢ ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 )
16 15 eqeq2i ⊢ ( 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ↔ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) )
17 16 anbi2i ⊢ ( ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ↔ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) )
18 17 exbii ⊢ ( ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) )
19 18 rexbii ⊢ ( ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) )
20 10 19 orbi12i ⊢ ( ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) ↔ ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
21 20 iotabii ⊢ ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) ) = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
22 df-sum ⊢ Σ 𝑘 ∈ 𝐴 𝐵 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) )
23 df-sum ⊢ Σ 𝑘 ∈ 𝐴 𝐶 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
24 21 22 23 3eqtr4i ⊢ Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶