Metamath Proof Explorer

Theorem fprodconst

Description: The product of constant terms ( k is not free in B .) (Contributed by Scott Fenton, 12-Jan-2018)

Ref Expression
Assertion fprodconst ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) → ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 exp0 ⊢ ( 𝐵 ∈ ℂ → ( 𝐵 ↑ 0 ) = 1 )
2 1 eqcomd ⊢ ( 𝐵 ∈ ℂ → 1 = ( 𝐵 ↑ 0 ) )
3 prodeq1 ⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ ∅ 𝐵 )
4 prod0 ⊢ ∏ 𝑘 ∈ ∅ 𝐵 = 1
5 3 4 syl6eq ⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐵 = 1 )
6 fveq2 ⊢ ( 𝐴 = ∅ → ( ♯ ‘ 𝐴 ) = ( ♯ ‘ ∅ ) )
7 hash0 ⊢ ( ♯ ‘ ∅ ) = 0
8 6 7 syl6eq ⊢ ( 𝐴 = ∅ → ( ♯ ‘ 𝐴 ) = 0 )
9 8 oveq2d ⊢ ( 𝐴 = ∅ → ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) = ( 𝐵 ↑ 0 ) )
10 5 9 eqeq12d ⊢ ( 𝐴 = ∅ → ( ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) ↔ 1 = ( 𝐵 ↑ 0 ) ) )
11 2 10 syl5ibrcom ⊢ ( 𝐵 ∈ ℂ → ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) ) )
12 11 adantl ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) → ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) ) )
13 eqidd ⊢ ( 𝑘 = ( 𝑓 ‘ 𝑛 ) → 𝐵 = 𝐵 )
14 simprl ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ℕ )
15 simprr ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 )
16 simpllr ⊢ ( ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
17 simpllr ⊢ ( ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → 𝐵 ∈ ℂ )
18 elfznn ⊢ ( 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) → 𝑛 ∈ ℕ )
19 18 adantl ⊢ ( ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → 𝑛 ∈ ℕ )
20 fvconst2g ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝑛 ∈ ℕ ) → ( ( ℕ × { 𝐵 } ) ‘ 𝑛 ) = 𝐵 )
21 17 19 20 syl2anc ⊢ ( ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ℕ × { 𝐵 } ) ‘ 𝑛 ) = 𝐵 )
22 13 14 15 16 21 fprod ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ∏ 𝑘 ∈ 𝐴 𝐵 = ( seq 1 ( · , ( ℕ × { 𝐵 } ) ) ‘ ( ♯ ‘ 𝐴 ) ) )
23 expnnval ⊢ ( ( 𝐵 ∈ ℂ ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) = ( seq 1 ( · , ( ℕ × { 𝐵 } ) ) ‘ ( ♯ ‘ 𝐴 ) ) )
24 23 ad2ant2lr ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) = ( seq 1 ( · , ( ℕ × { 𝐵 } ) ) ‘ ( ♯ ‘ 𝐴 ) ) )
25 22 24 eqtr4d ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) )
26 25 expr ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) ) )
27 26 exlimdv ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) ) )
28 27 expimpd ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) → ( ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) ) )
29 fz1f1o ⊢ ( 𝐴 ∈ Fin → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) )
30 29 adantr ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) )
31 12 28 30 mpjaod ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ ) → ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐵 ↑ ( ♯ ‘ 𝐴 ) ) )