Step | Hyp | Ref
| Expression |
1 | | fprodcl2lem.5 |
. . . 4
⊢ (𝜑 → 𝐴 ≠ ∅) |
2 | 1 | a1d 25 |
. . 3
⊢ (𝜑 → (¬ ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → 𝐴 ≠ ∅)) |
3 | 2 | necon4bd 2960 |
. 2
⊢ (𝜑 → (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)) |
4 | | prodfc 15507 |
. . . . . . 7
⊢
∏𝑚 ∈
𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵 |
5 | | fveq2 6717 |
. . . . . . . 8
⊢ (𝑚 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))) |
6 | | simprl 771 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈
ℕ) |
7 | | simprr 773 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) |
8 | | fprodcllem.1 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
9 | 8 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ ℂ) |
10 | | fprodcllem.4 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
11 | 9, 10 | sseldd 3902 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
12 | 11 | fmpttd 6932 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
13 | 12 | ffvelrnda 6904 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ) |
14 | 13 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ) |
15 | | f1of 6661 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴) |
16 | 15 | ad2antll 729 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴) |
17 | | fvco3 6810 |
. . . . . . . . 9
⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))) |
18 | 16, 17 | sylan 583 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))) |
19 | 5, 6, 7, 14, 18 | fprod 15503 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴))) |
20 | 4, 19 | eqtr3id 2792 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑘 ∈ 𝐴 𝐵 = (seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴))) |
21 | | nnuz 12477 |
. . . . . . . 8
⊢ ℕ =
(ℤ≥‘1) |
22 | 6, 21 | eleqtrdi 2848 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈
(ℤ≥‘1)) |
23 | 10 | fmpttd 6932 |
. . . . . . . . 9
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑆) |
24 | | fco 6569 |
. . . . . . . . 9
⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑆 ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶𝑆) |
25 | 23, 16, 24 | syl2an2r 685 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶𝑆) |
26 | 25 | ffvelrnda 6904 |
. . . . . . 7
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ 𝑆) |
27 | | fprodcllem.2 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) |
28 | 27 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) |
29 | 22, 26, 28 | seqcl 13596 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1( · ,
((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴)) ∈ 𝑆) |
30 | 20, 29 | eqeltrd 2838 |
. . . . 5
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
31 | 30 | expr 460 |
. . . 4
⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)) |
32 | 31 | exlimdv 1941 |
. . 3
⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) →
(∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)) |
33 | 32 | expimpd 457 |
. 2
⊢ (𝜑 → (((♯‘𝐴) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)) |
34 | | fprodcllem.3 |
. . 3
⊢ (𝜑 → 𝐴 ∈ Fin) |
35 | | fz1f1o 15274 |
. . 3
⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨
((♯‘𝐴) ∈
ℕ ∧ ∃𝑓
𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) |
36 | 34, 35 | syl 17 |
. 2
⊢ (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) |
37 | 3, 33, 36 | mpjaod 860 |
1
⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |