Theorem sumeq1

Description: Equality theorem for a sum. (Contributed by NM, 11-Dec-2005) (Revised by Mario Carneiro, 13-Jun-2019)

	Ref	Expression
Assertion	sumeq1	⊢ ( 𝐴 = 𝐵 → Σ 𝑘 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sseq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ↔ 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ) )
2		simpl	⊢ ( ( 𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ ) → 𝐴 = 𝐵 )
3	2	eleq2d	⊢ ( ( 𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ ) → ( 𝑛 ∈ 𝐴 ↔ 𝑛 ∈ 𝐵 ) )
4	3	ifbid	⊢ ( ( 𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ ) → if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) = if ( 𝑛 ∈ 𝐵 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) )
5	4	mpteq2dva	⊢ ( 𝐴 = 𝐵 → ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) = ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐵 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) )
6	5	seqeq3d	⊢ ( 𝐴 = 𝐵 → seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) = seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐵 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) )
7	6	breq1d	⊢ ( 𝐴 = 𝐵 → ( seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ↔ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐵 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) )
8	1 7	anbi12d	⊢ ( 𝐴 = 𝐵 → ( ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ↔ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐵 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ) )
9	8	rexbidv	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ↔ ∃ 𝑚 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐵 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ) )
10		f1oeq3	⊢ ( 𝐴 = 𝐵 → ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ↔ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ) )
11	10	anbi1d	⊢ ( 𝐴 = 𝐵 → ( ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ↔ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
12	11	exbidv	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
13	12	rexbidv	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
14	9 13	orbi12d	⊢ ( 𝐴 = 𝐵 → ( ( ∃ 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 ) ) ) ↔ ( ∃ 𝑚 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐵 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) ) )
15	14	iotabidv	⊢ ( 𝐴 = 𝐵 → ( ℩ 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 ) ) ) ) = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐵 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) ) )
16		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 ) ) ) )
17		df-sum	⊢ Σ 𝑘 ∈ 𝐵 𝐶 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐵 , ⦋ 𝑛 / 𝑘 ⦌ 𝐶 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
18	15 16 17	3eqtr4g	⊢ ( 𝐴 = 𝐵 → Σ 𝑘 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝐵 𝐶 )