Step | Hyp | Ref
| Expression |
1 | | cantnfs.s |
. . 3
⊢ 𝑆 = dom (𝐴 CNF 𝐵) |
2 | | cantnfs.a |
. . 3
⊢ (𝜑 → 𝐴 ∈ On) |
3 | | cantnfs.b |
. . 3
⊢ (𝜑 → 𝐵 ∈ On) |
4 | | eqid 2738 |
. . 3
⊢ OrdIso( E
, ((𝐵 × {∅})
supp ∅)) = OrdIso( E , ((𝐵 × {∅}) supp
∅)) |
5 | | cantnf0.a |
. . . . 5
⊢ (𝜑 → ∅ ∈ 𝐴) |
6 | | fconst6g 6567 |
. . . . 5
⊢ (∅
∈ 𝐴 → (𝐵 × {∅}):𝐵⟶𝐴) |
7 | 5, 6 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐵 × {∅}):𝐵⟶𝐴) |
8 | 3, 5 | fczfsuppd 8924 |
. . . 4
⊢ (𝜑 → (𝐵 × {∅}) finSupp
∅) |
9 | 1, 2, 3 | cantnfs 9202 |
. . . 4
⊢ (𝜑 → ((𝐵 × {∅}) ∈ 𝑆 ↔ ((𝐵 × {∅}):𝐵⟶𝐴 ∧ (𝐵 × {∅}) finSupp
∅))) |
10 | 7, 8, 9 | mpbir2and 713 |
. . 3
⊢ (𝜑 → (𝐵 × {∅}) ∈ 𝑆) |
11 | | eqid 2738 |
. . 3
⊢
seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘))
·o ((𝐵
× {∅})‘(OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘)))
+o 𝑧)), ∅)
= seqω((𝑘
∈ V, 𝑧 ∈ V
↦ (((𝐴
↑o (OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘))
·o ((𝐵
× {∅})‘(OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘)))
+o 𝑧)),
∅) |
12 | 1, 2, 3, 4, 10, 11 | cantnfval 9204 |
. 2
⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) =
(seqω((𝑘
∈ V, 𝑧 ∈ V
↦ (((𝐴
↑o (OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘))
·o ((𝐵
× {∅})‘(OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘)))
+o 𝑧)),
∅)‘dom OrdIso( E , ((𝐵 × {∅}) supp
∅)))) |
13 | | eqidd 2739 |
. . . . . . 7
⊢ (𝜑 → (𝐵 × {∅}) = (𝐵 × {∅})) |
14 | | 0ex 5175 |
. . . . . . . . 9
⊢ ∅
∈ V |
15 | | fnconstg 6566 |
. . . . . . . . 9
⊢ (∅
∈ V → (𝐵 ×
{∅}) Fn 𝐵) |
16 | 14, 15 | mp1i 13 |
. . . . . . . 8
⊢ (𝜑 → (𝐵 × {∅}) Fn 𝐵) |
17 | 14 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ∅ ∈
V) |
18 | | fnsuppeq0 7887 |
. . . . . . . 8
⊢ (((𝐵 × {∅}) Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) →
(((𝐵 × {∅})
supp ∅) = ∅ ↔ (𝐵 × {∅}) = (𝐵 × {∅}))) |
19 | 16, 3, 17, 18 | syl3anc 1372 |
. . . . . . 7
⊢ (𝜑 → (((𝐵 × {∅}) supp ∅) = ∅
↔ (𝐵 ×
{∅}) = (𝐵 ×
{∅}))) |
20 | 13, 19 | mpbird 260 |
. . . . . 6
⊢ (𝜑 → ((𝐵 × {∅}) supp ∅) =
∅) |
21 | | oieq2 9050 |
. . . . . 6
⊢ (((𝐵 × {∅}) supp
∅) = ∅ → OrdIso( E , ((𝐵 × {∅}) supp ∅)) = OrdIso(
E , ∅)) |
22 | 20, 21 | syl 17 |
. . . . 5
⊢ (𝜑 → OrdIso( E , ((𝐵 × {∅}) supp
∅)) = OrdIso( E , ∅)) |
23 | 22 | dmeqd 5748 |
. . . 4
⊢ (𝜑 → dom OrdIso( E , ((𝐵 × {∅}) supp
∅)) = dom OrdIso( E , ∅)) |
24 | | we0 5520 |
. . . . . 6
⊢ E We
∅ |
25 | | eqid 2738 |
. . . . . . 7
⊢ OrdIso( E
, ∅) = OrdIso( E , ∅) |
26 | 25 | oien 9075 |
. . . . . 6
⊢ ((∅
∈ V ∧ E We ∅) → dom OrdIso( E , ∅) ≈
∅) |
27 | 14, 24, 26 | mp2an 692 |
. . . . 5
⊢ dom
OrdIso( E , ∅) ≈ ∅ |
28 | | en0 8618 |
. . . . 5
⊢ (dom
OrdIso( E , ∅) ≈ ∅ ↔ dom OrdIso( E , ∅) =
∅) |
29 | 27, 28 | mpbi 233 |
. . . 4
⊢ dom
OrdIso( E , ∅) = ∅ |
30 | 23, 29 | eqtrdi 2789 |
. . 3
⊢ (𝜑 → dom OrdIso( E , ((𝐵 × {∅}) supp
∅)) = ∅) |
31 | 30 | fveq2d 6678 |
. 2
⊢ (𝜑 →
(seqω((𝑘
∈ V, 𝑧 ∈ V
↦ (((𝐴
↑o (OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘))
·o ((𝐵
× {∅})‘(OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘)))
+o 𝑧)),
∅)‘dom OrdIso( E , ((𝐵 × {∅}) supp ∅))) =
(seqω((𝑘
∈ V, 𝑧 ∈ V
↦ (((𝐴
↑o (OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘))
·o ((𝐵
× {∅})‘(OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘)))
+o 𝑧)),
∅)‘∅)) |
32 | 11 | seqom0g 8121 |
. . 3
⊢ (∅
∈ V → (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘))
·o ((𝐵
× {∅})‘(OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘)))
+o 𝑧)),
∅)‘∅) = ∅) |
33 | 14, 32 | mp1i 13 |
. 2
⊢ (𝜑 →
(seqω((𝑘
∈ V, 𝑧 ∈ V
↦ (((𝐴
↑o (OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘))
·o ((𝐵
× {∅})‘(OrdIso( E , ((𝐵 × {∅}) supp
∅))‘𝑘)))
+o 𝑧)),
∅)‘∅) = ∅) |
34 | 12, 31, 33 | 3eqtrd 2777 |
1
⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) =
∅) |