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
|
eqtrdi |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐵 = 1 ) |
6 |
|
fveq2 |
⊢ ( 𝐴 = ∅ → ( ♯ ‘ 𝐴 ) = ( ♯ ‘ ∅ ) ) |
7 |
|
hash0 |
⊢ ( ♯ ‘ ∅ ) = 0 |
8 |
6 7
|
eqtrdi |
⊢ ( 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 ) ) ) |