Step | Hyp | Ref
| Expression |
1 | | rexnal 3151 |
. . . . 5
⊢
(∃𝑥 ∈
𝐼 ¬ (𝑋‘𝑥) = 0 ↔ ¬ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0) |
2 | | df-ne 2935 |
. . . . . . 7
⊢ ((𝑋‘𝑥) ≠ 0 ↔ ¬ (𝑋‘𝑥) = 0) |
3 | | oveq2 7178 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑋 → (ℂfld
Σg ℎ) = (ℂfld
Σg 𝑋)) |
4 | | tdeglem.h |
. . . . . . . . . . . 12
⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld
Σg ℎ)) |
5 | | ovex 7203 |
. . . . . . . . . . . 12
⊢
(ℂfld Σg 𝑋) ∈ V |
6 | 3, 4, 5 | fvmpt 6775 |
. . . . . . . . . . 11
⊢ (𝑋 ∈ 𝐴 → (𝐻‘𝑋) = (ℂfld
Σg 𝑋)) |
7 | 6 | ad2antlr 727 |
. . . . . . . . . 10
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝐻‘𝑋) = (ℂfld
Σg 𝑋)) |
8 | | tdeglem.a |
. . . . . . . . . . . . . 14
⊢ 𝐴 = {𝑚 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑚 “ ℕ) ∈
Fin} |
9 | 8 | psrbagfOLD 20732 |
. . . . . . . . . . . . 13
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋:𝐼⟶ℕ0) |
10 | 9 | feqmptd 6737 |
. . . . . . . . . . . 12
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 = (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦))) |
11 | 10 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑋 = (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦))) |
12 | 11 | oveq2d 7186 |
. . . . . . . . . 10
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂfld
Σg 𝑋) = (ℂfld
Σg (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)))) |
13 | | cnfldbas 20221 |
. . . . . . . . . . 11
⊢ ℂ =
(Base‘ℂfld) |
14 | | cnfld0 20241 |
. . . . . . . . . . 11
⊢ 0 =
(0g‘ℂfld) |
15 | | cnfldadd 20222 |
. . . . . . . . . . 11
⊢ + =
(+g‘ℂfld) |
16 | | cnring 20239 |
. . . . . . . . . . . 12
⊢
ℂfld ∈ Ring |
17 | | ringcmn 19453 |
. . . . . . . . . . . 12
⊢
(ℂfld ∈ Ring → ℂfld ∈
CMnd) |
18 | 16, 17 | mp1i 13 |
. . . . . . . . . . 11
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ℂfld
∈ CMnd) |
19 | | simpll 767 |
. . . . . . . . . . 11
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝐼 ∈ 𝑉) |
20 | 9 | adantr 484 |
. . . . . . . . . . . . 13
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑋:𝐼⟶ℕ0) |
21 | 20 | ffvelrnda 6861 |
. . . . . . . . . . . 12
⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) ∧ 𝑦 ∈ 𝐼) → (𝑋‘𝑦) ∈
ℕ0) |
22 | 21 | nn0cnd 12038 |
. . . . . . . . . . 11
⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) ∧ 𝑦 ∈ 𝐼) → (𝑋‘𝑦) ∈ ℂ) |
23 | 8 | psrbagfsuppOLD 20734 |
. . . . . . . . . . . . . 14
⊢ ((𝑋 ∈ 𝐴 ∧ 𝐼 ∈ 𝑉) → 𝑋 finSupp 0) |
24 | 23 | ancoms 462 |
. . . . . . . . . . . . 13
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 finSupp 0) |
25 | 24 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑋 finSupp 0) |
26 | 11, 25 | eqbrtrrd 5054 |
. . . . . . . . . . 11
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) finSupp 0) |
27 | | incom 4091 |
. . . . . . . . . . . . 13
⊢ ((𝐼 ∖ {𝑥}) ∩ {𝑥}) = ({𝑥} ∩ (𝐼 ∖ {𝑥})) |
28 | | disjdif 4361 |
. . . . . . . . . . . . 13
⊢ ({𝑥} ∩ (𝐼 ∖ {𝑥})) = ∅ |
29 | 27, 28 | eqtri 2761 |
. . . . . . . . . . . 12
⊢ ((𝐼 ∖ {𝑥}) ∩ {𝑥}) = ∅ |
30 | 29 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((𝐼 ∖ {𝑥}) ∩ {𝑥}) = ∅) |
31 | | difsnid 4698 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝐼 → ((𝐼 ∖ {𝑥}) ∪ {𝑥}) = 𝐼) |
32 | 31 | eqcomd 2744 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝐼 → 𝐼 = ((𝐼 ∖ {𝑥}) ∪ {𝑥})) |
33 | 32 | ad2antrl 728 |
. . . . . . . . . . 11
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝐼 = ((𝐼 ∖ {𝑥}) ∪ {𝑥})) |
34 | 13, 14, 15, 18, 19, 22, 26, 30, 33 | gsumsplit2 19168 |
. . . . . . . . . 10
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂfld
Σg (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦))) = ((ℂfld
Σg (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) + (ℂfld
Σg (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))))) |
35 | 7, 12, 34 | 3eqtrd 2777 |
. . . . . . . . 9
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝐻‘𝑋) = ((ℂfld
Σg (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) + (ℂfld
Σg (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))))) |
36 | | difexg 5195 |
. . . . . . . . . . . . 13
⊢ (𝐼 ∈ 𝑉 → (𝐼 ∖ {𝑥}) ∈ V) |
37 | 36 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝐼 ∖ {𝑥}) ∈ V) |
38 | | nn0subm 20272 |
. . . . . . . . . . . . 13
⊢
ℕ0 ∈
(SubMnd‘ℂfld) |
39 | 38 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ℕ0 ∈
(SubMnd‘ℂfld)) |
40 | | eldifi 4017 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) → 𝑦 ∈ 𝐼) |
41 | | ffvelrn 6859 |
. . . . . . . . . . . . . 14
⊢ ((𝑋:𝐼⟶ℕ0 ∧ 𝑦 ∈ 𝐼) → (𝑋‘𝑦) ∈
ℕ0) |
42 | 20, 40, 41 | syl2an 599 |
. . . . . . . . . . . . 13
⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) ∧ 𝑦 ∈ (𝐼 ∖ {𝑥})) → (𝑋‘𝑦) ∈
ℕ0) |
43 | 42 | fmpttd 6889 |
. . . . . . . . . . . 12
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)):(𝐼 ∖ {𝑥})⟶ℕ0) |
44 | 36 | mptexd 6997 |
. . . . . . . . . . . . . 14
⊢ (𝐼 ∈ 𝑉 → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ∈ V) |
45 | 44 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ∈ V) |
46 | | funmpt 6377 |
. . . . . . . . . . . . . 14
⊢ Fun
(𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) |
47 | 46 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → Fun (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) |
48 | | funmpt 6377 |
. . . . . . . . . . . . . 14
⊢ Fun
(𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) |
49 | | difss 4022 |
. . . . . . . . . . . . . . . 16
⊢ (𝐼 ∖ {𝑥}) ⊆ 𝐼 |
50 | | resmpt 5879 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐼 ∖ {𝑥}) ⊆ 𝐼 → ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ↾ (𝐼 ∖ {𝑥})) = (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) |
51 | 49, 50 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ↾ (𝐼 ∖ {𝑥})) = (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) |
52 | | resss 5850 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ↾ (𝐼 ∖ {𝑥})) ⊆ (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) |
53 | 51, 52 | eqsstrri 3912 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ⊆ (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) |
54 | | mptexg 6994 |
. . . . . . . . . . . . . . 15
⊢ (𝐼 ∈ 𝑉 → (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∈ V) |
55 | 54 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∈ V) |
56 | | funsssuppss 7885 |
. . . . . . . . . . . . . 14
⊢ ((Fun
(𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∧ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ⊆ (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∧ (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∈ V) → ((𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) supp 0) ⊆ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) supp 0)) |
57 | 48, 53, 55, 56 | mp3an12i 1466 |
. . . . . . . . . . . . 13
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) supp 0) ⊆ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) supp 0)) |
58 | | fsuppsssupp 8922 |
. . . . . . . . . . . . 13
⊢ ((((𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ∈ V ∧ Fun (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) ∧ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) finSupp 0 ∧ ((𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) supp 0) ⊆ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) supp 0))) → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) finSupp 0) |
59 | 45, 47, 26, 57, 58 | syl22anc 838 |
. . . . . . . . . . . 12
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) finSupp 0) |
60 | 14, 18, 37, 39, 43, 59 | gsumsubmcl 19158 |
. . . . . . . . . . 11
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂfld
Σg (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) ∈
ℕ0) |
61 | | ringmnd 19426 |
. . . . . . . . . . . . . 14
⊢
(ℂfld ∈ Ring → ℂfld ∈
Mnd) |
62 | 16, 61 | mp1i 13 |
. . . . . . . . . . . . 13
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ℂfld
∈ Mnd) |
63 | | simprl 771 |
. . . . . . . . . . . . 13
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑥 ∈ 𝐼) |
64 | 20, 63 | ffvelrnd 6862 |
. . . . . . . . . . . . . 14
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ∈
ℕ0) |
65 | 64 | nn0cnd 12038 |
. . . . . . . . . . . . 13
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ∈ ℂ) |
66 | | fveq2 6674 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑥 → (𝑋‘𝑦) = (𝑋‘𝑥)) |
67 | 13, 66 | gsumsn 19193 |
. . . . . . . . . . . . 13
⊢
((ℂfld ∈ Mnd ∧ 𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ∈ ℂ) →
(ℂfld Σg (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))) = (𝑋‘𝑥)) |
68 | 62, 63, 65, 67 | syl3anc 1372 |
. . . . . . . . . . . 12
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂfld
Σg (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))) = (𝑋‘𝑥)) |
69 | | simprr 773 |
. . . . . . . . . . . . . 14
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ≠ 0) |
70 | 69, 2 | sylib 221 |
. . . . . . . . . . . . 13
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ¬ (𝑋‘𝑥) = 0) |
71 | | elnn0 11978 |
. . . . . . . . . . . . . 14
⊢ ((𝑋‘𝑥) ∈ ℕ0 ↔ ((𝑋‘𝑥) ∈ ℕ ∨ (𝑋‘𝑥) = 0)) |
72 | 64, 71 | sylib 221 |
. . . . . . . . . . . . 13
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((𝑋‘𝑥) ∈ ℕ ∨ (𝑋‘𝑥) = 0)) |
73 | | orel2 890 |
. . . . . . . . . . . . 13
⊢ (¬
(𝑋‘𝑥) = 0 → (((𝑋‘𝑥) ∈ ℕ ∨ (𝑋‘𝑥) = 0) → (𝑋‘𝑥) ∈ ℕ)) |
74 | 70, 72, 73 | sylc 65 |
. . . . . . . . . . . 12
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ∈ ℕ) |
75 | 68, 74 | eqeltrd 2833 |
. . . . . . . . . . 11
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂfld
Σg (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))) ∈ ℕ) |
76 | | nn0nnaddcl 12007 |
. . . . . . . . . . 11
⊢
(((ℂfld Σg (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) ∈ ℕ0 ∧
(ℂfld Σg (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))) ∈ ℕ) →
((ℂfld Σg (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) + (ℂfld
Σg (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦)))) ∈ ℕ) |
77 | 60, 75, 76 | syl2anc 587 |
. . . . . . . . . 10
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((ℂfld
Σg (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) + (ℂfld
Σg (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦)))) ∈ ℕ) |
78 | 77 | nnne0d 11766 |
. . . . . . . . 9
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((ℂfld
Σg (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) + (ℂfld
Σg (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦)))) ≠ 0) |
79 | 35, 78 | eqnetrd 3001 |
. . . . . . . 8
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝐻‘𝑋) ≠ 0) |
80 | 79 | expr 460 |
. . . . . . 7
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ 𝐼) → ((𝑋‘𝑥) ≠ 0 → (𝐻‘𝑋) ≠ 0)) |
81 | 2, 80 | syl5bir 246 |
. . . . . 6
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ 𝐼) → (¬ (𝑋‘𝑥) = 0 → (𝐻‘𝑋) ≠ 0)) |
82 | 81 | rexlimdva 3194 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (∃𝑥 ∈ 𝐼 ¬ (𝑋‘𝑥) = 0 → (𝐻‘𝑋) ≠ 0)) |
83 | 1, 82 | syl5bir 246 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (¬ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0 → (𝐻‘𝑋) ≠ 0)) |
84 | 83 | necon4bd 2954 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 → ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0)) |
85 | 9 | ffnd 6505 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 Fn 𝐼) |
86 | | 0nn0 11991 |
. . . . . 6
⊢ 0 ∈
ℕ0 |
87 | | fnconstg 6566 |
. . . . . 6
⊢ (0 ∈
ℕ0 → (𝐼 × {0}) Fn 𝐼) |
88 | 86, 87 | mp1i 13 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐼 × {0}) Fn 𝐼) |
89 | | eqfnfv 6809 |
. . . . 5
⊢ ((𝑋 Fn 𝐼 ∧ (𝐼 × {0}) Fn 𝐼) → (𝑋 = (𝐼 × {0}) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = ((𝐼 × {0})‘𝑥))) |
90 | 85, 88, 89 | syl2anc 587 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑋 = (𝐼 × {0}) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = ((𝐼 × {0})‘𝑥))) |
91 | | c0ex 10713 |
. . . . . . 7
⊢ 0 ∈
V |
92 | 91 | fvconst2 6976 |
. . . . . 6
⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {0})‘𝑥) = 0) |
93 | 92 | eqeq2d 2749 |
. . . . 5
⊢ (𝑥 ∈ 𝐼 → ((𝑋‘𝑥) = ((𝐼 × {0})‘𝑥) ↔ (𝑋‘𝑥) = 0)) |
94 | 93 | ralbiia 3079 |
. . . 4
⊢
(∀𝑥 ∈
𝐼 (𝑋‘𝑥) = ((𝐼 × {0})‘𝑥) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0) |
95 | 90, 94 | bitrdi 290 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑋 = (𝐼 × {0}) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0)) |
96 | 84, 95 | sylibrd 262 |
. 2
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 → 𝑋 = (𝐼 × {0}))) |
97 | 8 | psrbag0 20874 |
. . . . . 6
⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ 𝐴) |
98 | 97 | adantr 484 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐼 × {0}) ∈ 𝐴) |
99 | | oveq2 7178 |
. . . . . 6
⊢ (ℎ = (𝐼 × {0}) → (ℂfld
Σg ℎ) = (ℂfld
Σg (𝐼 × {0}))) |
100 | | ovex 7203 |
. . . . . 6
⊢
(ℂfld Σg (𝐼 × {0})) ∈ V |
101 | 99, 4, 100 | fvmpt 6775 |
. . . . 5
⊢ ((𝐼 × {0}) ∈ 𝐴 → (𝐻‘(𝐼 × {0})) = (ℂfld
Σg (𝐼 × {0}))) |
102 | 98, 101 | syl 17 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐻‘(𝐼 × {0})) = (ℂfld
Σg (𝐼 × {0}))) |
103 | | fconstmpt 5585 |
. . . . . 6
⊢ (𝐼 × {0}) = (𝑥 ∈ 𝐼 ↦ 0) |
104 | 103 | oveq2i 7181 |
. . . . 5
⊢
(ℂfld Σg (𝐼 × {0})) = (ℂfld
Σg (𝑥 ∈ 𝐼 ↦ 0)) |
105 | 16, 61 | ax-mp 5 |
. . . . . . 7
⊢
ℂfld ∈ Mnd |
106 | 14 | gsumz 18116 |
. . . . . . 7
⊢
((ℂfld ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (ℂfld
Σg (𝑥 ∈ 𝐼 ↦ 0)) = 0) |
107 | 105, 106 | mpan 690 |
. . . . . 6
⊢ (𝐼 ∈ 𝑉 → (ℂfld
Σg (𝑥 ∈ 𝐼 ↦ 0)) = 0) |
108 | 107 | adantr 484 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (ℂfld
Σg (𝑥 ∈ 𝐼 ↦ 0)) = 0) |
109 | 104, 108 | syl5eq 2785 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (ℂfld
Σg (𝐼 × {0})) = 0) |
110 | 102, 109 | eqtrd 2773 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐻‘(𝐼 × {0})) = 0) |
111 | | fveqeq2 6683 |
. . 3
⊢ (𝑋 = (𝐼 × {0}) → ((𝐻‘𝑋) = 0 ↔ (𝐻‘(𝐼 × {0})) = 0)) |
112 | 110, 111 | syl5ibrcom 250 |
. 2
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑋 = (𝐼 × {0}) → (𝐻‘𝑋) = 0)) |
113 | 96, 112 | impbid 215 |
1
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 ↔ 𝑋 = (𝐼 × {0}))) |