Metamath Proof Explorer


Theorem sumeq2sdv

Description: Equality deduction for sum. (Contributed by NM, 3-Jan-2006) (Proof shortened by Glauco Siliprandi, 5-Apr-2020) Avoid axioms. (Revised by GG, 14-Aug-2025)

Ref Expression
Hypothesis sumeq2sdv.1 ( 𝜑𝐵 = 𝐶 )
Assertion sumeq2sdv ( 𝜑 → Σ 𝑘𝐴 𝐵 = Σ 𝑘𝐴 𝐶 )

Proof

Step Hyp Ref Expression
1 sumeq2sdv.1 ( 𝜑𝐵 = 𝐶 )
2 1 csbeq2dv ( 𝜑 𝑛 / 𝑘 𝐵 = 𝑛 / 𝑘 𝐶 )
3 2 ifeq1d ( 𝜑 → if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐵 , 0 ) = if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) )
4 3 mpteq2dv ( 𝜑 → ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐵 , 0 ) ) = ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) )
5 4 seqeq3d ( 𝜑 → seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐵 , 0 ) ) ) = seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) ) )
6 5 breq1d ( 𝜑 → ( seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐵 , 0 ) ) ) ⇝ 𝑥 ↔ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) ) ⇝ 𝑥 ) )
7 6 anbi2d ( 𝜑 → ( ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ↔ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ) )
8 7 rexbidv ( 𝜑 → ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ↔ ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ) )
9 1 csbeq2dv ( 𝜑 ( 𝑓𝑛 ) / 𝑘 𝐵 = ( 𝑓𝑛 ) / 𝑘 𝐶 )
10 9 mpteq2dv ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐵 ) = ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) )
11 10 seqeq3d ( 𝜑 → seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐵 ) ) = seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) )
12 11 fveq1d ( 𝜑 → ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐵 ) ) ‘ 𝑚 ) = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) )
13 12 eqeq2d ( 𝜑 → ( 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐵 ) ) ‘ 𝑚 ) ↔ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) ) )
14 13 anbi2d ( 𝜑 → ( ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐵 ) ) ‘ 𝑚 ) ) ↔ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) ) ) )
15 14 exbidv ( 𝜑 → ( ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐵 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) ) ) )
16 15 rexbidv ( 𝜑 → ( ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐵 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐶 ) ) ‘ 𝑚 ) ) ) )
17 8 16 orbi12d ( 𝜑 → ( ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 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 ) ) ) ) )
18 17 iotabidv ( 𝜑 → ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 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 ) ) ) ) )
19 df-sum Σ 𝑘𝐴 𝐵 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛𝐴 , 𝑛 / 𝑘 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto𝐴𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ( 𝑓𝑛 ) / 𝑘 𝐵 ) ) ‘ 𝑚 ) ) ) )
20 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 ) ) ) )
21 18 19 20 3eqtr4g ( 𝜑 → Σ 𝑘𝐴 𝐵 = Σ 𝑘𝐴 𝐶 )