Metamath Proof Explorer


Theorem esumrnmpt

Description: Rewrite an extended sum into a sum on the range of a mapping function. (Contributed by Thierry Arnoux, 27-May-2020)

Ref Expression
Hypotheses esumrnmpt.0 𝑘 𝐴
esumrnmpt.1 ( 𝑦 = 𝐵𝐶 = 𝐷 )
esumrnmpt.2 ( 𝜑𝐴𝑉 )
esumrnmpt.3 ( ( 𝜑𝑘𝐴 ) → 𝐷 ∈ ( 0 [,] +∞ ) )
esumrnmpt.4 ( ( 𝜑𝑘𝐴 ) → 𝐵 ∈ ( 𝑊 ∖ { ∅ } ) )
esumrnmpt.5 ( 𝜑Disj 𝑘𝐴 𝐵 )
Assertion esumrnmpt ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘𝐴𝐵 ) 𝐶 = Σ* 𝑘𝐴 𝐷 )

Proof

Step Hyp Ref Expression
1 esumrnmpt.0 𝑘 𝐴
2 esumrnmpt.1 ( 𝑦 = 𝐵𝐶 = 𝐷 )
3 esumrnmpt.2 ( 𝜑𝐴𝑉 )
4 esumrnmpt.3 ( ( 𝜑𝑘𝐴 ) → 𝐷 ∈ ( 0 [,] +∞ ) )
5 esumrnmpt.4 ( ( 𝜑𝑘𝐴 ) → 𝐵 ∈ ( 𝑊 ∖ { ∅ } ) )
6 esumrnmpt.5 ( 𝜑Disj 𝑘𝐴 𝐵 )
7 eqid ( 𝑘𝐴𝐵 ) = ( 𝑘𝐴𝐵 )
8 7 rnmpt ran ( 𝑘𝐴𝐵 ) = { 𝑧 ∣ ∃ 𝑘𝐴 𝑧 = 𝐵 }
9 esumeq1 ( ran ( 𝑘𝐴𝐵 ) = { 𝑧 ∣ ∃ 𝑘𝐴 𝑧 = 𝐵 } → Σ* 𝑦 ∈ ran ( 𝑘𝐴𝐵 ) 𝐶 = Σ* 𝑦 ∈ { 𝑧 ∣ ∃ 𝑘𝐴 𝑧 = 𝐵 } 𝐶 )
10 8 9 ax-mp Σ* 𝑦 ∈ ran ( 𝑘𝐴𝐵 ) 𝐶 = Σ* 𝑦 ∈ { 𝑧 ∣ ∃ 𝑘𝐴 𝑧 = 𝐵 } 𝐶
11 nfcv 𝑘 𝐶
12 nfv 𝑘 𝜑
13 12 1 5 6 disjdsct ( 𝜑 → Fun ( 𝑘𝐴𝐵 ) )
14 11 12 1 2 3 13 4 5 esumc ( 𝜑 → Σ* 𝑘𝐴 𝐷 = Σ* 𝑦 ∈ { 𝑧 ∣ ∃ 𝑘𝐴 𝑧 = 𝐵 } 𝐶 )
15 10 14 eqtr4id ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘𝐴𝐵 ) 𝐶 = Σ* 𝑘𝐴 𝐷 )