Metamath Proof Explorer


Theorem cbvsum

Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005) (Revised by Mario Carneiro, 13-Jun-2019)

Ref Expression
Hypotheses cbvsum.1 ( 𝑗 = 𝑘𝐵 = 𝐶 )
cbvsum.2 𝑘 𝐵
cbvsum.3 𝑗 𝐶
Assertion cbvsum Σ 𝑗𝐴 𝐵 = Σ 𝑘𝐴 𝐶

Proof

Step Hyp Ref Expression
1 cbvsum.1 ( 𝑗 = 𝑘𝐵 = 𝐶 )
2 cbvsum.2 𝑘 𝐵
3 cbvsum.3 𝑗 𝐶
4 2 3 1 cbvcsbw 𝑛 / 𝑗 𝐵 = 𝑛 / 𝑘 𝐶
5 4 a1i ( ⊤ → 𝑛 / 𝑗 𝐵 = 𝑛 / 𝑘 𝐶 )
6 5 ifeq1d ( ⊤ → if ( 𝑛𝐴 , 𝑛 / 𝑗 𝐵 , 0 ) = if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) )
7 6 mpteq2dv ( ⊤ → ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑗 𝐵 , 0 ) ) = ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) )
8 7 seqeq3d ( ⊤ → seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑗 𝐵 , 0 ) ) ) = seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) ) )
9 8 mptru seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑗 𝐵 , 0 ) ) ) = seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) )
10 9 breq1i ( seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑗 𝐵 , 0 ) ) ) ⇝ 𝑥 ↔ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) ) ⇝ 𝑥 )
11 10 anbi2i ( ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑗 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ↔ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) ) ⇝ 𝑥 ) )
12 11 rexbii ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑗 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ↔ ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) ) ⇝ 𝑥 ) )
13 2 3 1 cbvcsbw ( 𝑓𝑛 ) / 𝑗 𝐵 = ( 𝑓𝑛 ) / 𝑘 𝐶
14 13 a1i ( ⊤ → ( 𝑓𝑛 ) / 𝑗 𝐵 = ( 𝑓𝑛 ) / 𝑘 𝐶 )
15 14 mpteq2dv ( ⊤ → ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑗 𝐵 ) = ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) )
16 15 seqeq3d ( ⊤ → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑗 𝐵 ) ) = seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) )
17 16 mptru seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑗 𝐵 ) ) = seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) )
18 17 fveq1i ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑗 𝐵 ) ) ‘ 𝑚 ) = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 )
19 18 eqeq2i ( 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑗 𝐵 ) ) ‘ 𝑚 ) ↔ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) )
20 19 anbi2i ( ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑗 𝐵 ) ) ‘ 𝑚 ) ) ↔ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) ) )
21 20 exbii ( ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑗 𝐵 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) ) )
22 21 rexbii ( ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑗 𝐵 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) ) )
23 12 22 orbi12i ( ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑗 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑗 𝐵 ) ) ‘ 𝑚 ) ) ) ↔ ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) ) ) )
24 23 iotabii ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑗 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑗 𝐵 ) ) ‘ 𝑚 ) ) ) ) = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) ) ) )
25 df-sum Σ 𝑗𝐴 𝐵 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑗 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑗 𝐵 ) ) ‘ 𝑚 ) ) ) )
26 df-sum Σ 𝑘𝐴 𝐶 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) ) ) )
27 24 25 26 3eqtr4i Σ 𝑗𝐴 𝐵 = Σ 𝑘𝐴 𝐶