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Metamath Proof Explorer

Theorem fprod1p

Description: Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017)

Ref Expression
Hypotheses fprod1p.1 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
fprod1p.2 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → 𝐴 ∈ ℂ )
fprod1p.3 ⊢ ( 𝑘 = 𝑀 → 𝐴 = 𝐵 )
Assertion fprod1p ( 𝜑 → ∏ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = ( 𝐵 · ∏ 𝑘 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) 𝐴 ) )

Proof

Step Hyp Ref Expression
1 fprod1p.1 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
2 fprod1p.2 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → 𝐴 ∈ ℂ )
3 fprod1p.3 ⊢ ( 𝑘 = 𝑀 → 𝐴 = 𝐵 )
4 eluzfz1 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ( 𝑀 ... 𝑁 ) )
5 1 4 syl ⊢ ( 𝜑 → 𝑀 ∈ ( 𝑀 ... 𝑁 ) )
6 5 elfzelzd ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
7 fzsn ⊢ ( 𝑀 ∈ ℤ → ( 𝑀 ... 𝑀 ) = { 𝑀 } )
8 6 7 syl ⊢ ( 𝜑 → ( 𝑀 ... 𝑀 ) = { 𝑀 } )
9 8 ineq1d ⊢ ( 𝜑 → ( ( 𝑀 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ( { 𝑀 } ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) )
10 6 zred ⊢ ( 𝜑 → 𝑀 ∈ ℝ )
11 10 ltp1d ⊢ ( 𝜑 → 𝑀 < ( 𝑀 + 1 ) )
12 fzdisj ⊢ ( 𝑀 < ( 𝑀 + 1 ) → ( ( 𝑀 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ∅ )
13 11 12 syl ⊢ ( 𝜑 → ( ( 𝑀 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ∅ )
14 9 13 eqtr3d ⊢ ( 𝜑 → ( { 𝑀 } ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ∅ )
15 fzsplit ⊢ ( 𝑀 ∈ ( 𝑀 ... 𝑁 ) → ( 𝑀 ... 𝑁 ) = ( ( 𝑀 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) )
16 5 15 syl ⊢ ( 𝜑 → ( 𝑀 ... 𝑁 ) = ( ( 𝑀 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) )
17 8 uneq1d ⊢ ( 𝜑 → ( ( 𝑀 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ( { 𝑀 } ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) )
18 16 17 eqtrd ⊢ ( 𝜑 → ( 𝑀 ... 𝑁 ) = ( { 𝑀 } ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) )
19 fzfid ⊢ ( 𝜑 → ( 𝑀 ... 𝑁 ) ∈ Fin )
20 14 18 19 2 fprodsplit ⊢ ( 𝜑 → ∏ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = ( ∏ 𝑘 ∈ { 𝑀 } 𝐴 · ∏ 𝑘 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) 𝐴 ) )
21 3 eleq1d ⊢ ( 𝑘 = 𝑀 → ( 𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ ) )
22 2 ralrimiva ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 ∈ ℂ )
23 21 22 5 rspcdva ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
24 3 prodsn ⊢ ( ( 𝑀 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝐵 ∈ ℂ ) → ∏ 𝑘 ∈ { 𝑀 } 𝐴 = 𝐵 )
25 5 23 24 syl2anc ⊢ ( 𝜑 → ∏ 𝑘 ∈ { 𝑀 } 𝐴 = 𝐵 )
26 25 oveq1d ⊢ ( 𝜑 → ( ∏ 𝑘 ∈ { 𝑀 } 𝐴 · ∏ 𝑘 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) 𝐴 ) = ( 𝐵 · ∏ 𝑘 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) 𝐴 ) )
27 20 26 eqtrd ⊢ ( 𝜑 → ∏ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = ( 𝐵 · ∏ 𝑘 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) 𝐴 ) )