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Theorem aacllem 42875
Description: Lemma for other theorems about 𝔸. (Contributed by Brendan Leahy, 3-Jan-2020.) (Revised by Alexander van der Vekens and David A. Wheeler, 25-Apr-2020.)
Hypotheses
Ref Expression
aacllem.0 (𝜑𝐴 ∈ ℂ)
aacllem.1 (𝜑𝑁 ∈ ℕ0)
aacllem.2 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
aacllem.3 ((𝜑𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
aacllem.4 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐴𝑘) = Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋))
Assertion
Ref Expression
aacllem (𝜑𝐴 ∈ 𝔸)
Distinct variable groups:   𝐴,𝑘,𝑛   𝑘,𝑁,𝑛   𝑘,𝑋   𝜑,𝑘,𝑛
Allowed substitution hints:   𝐶(𝑘,𝑛)   𝑋(𝑛)

Proof of Theorem aacllem
Dummy variables 𝑤 𝑥 𝑦 𝐵 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aacllem.0 . 2 (𝜑𝐴 ∈ ℂ)
2 aacllem.1 . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
32nn0red 11390 . . . . . 6 (𝜑𝑁 ∈ ℝ)
43ltp1d 10992 . . . . 5 (𝜑𝑁 < (𝑁 + 1))
5 peano2nn0 11371 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
62, 5syl 17 . . . . . . 7 (𝜑 → (𝑁 + 1) ∈ ℕ0)
76nn0red 11390 . . . . . 6 (𝜑 → (𝑁 + 1) ∈ ℝ)
83, 7ltnled 10222 . . . . 5 (𝜑 → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁))
94, 8mpbid 222 . . . 4 (𝜑 → ¬ (𝑁 + 1) ≤ 𝑁)
10 aacllem.3 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
11103expa 1284 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
12 eqid 2651 . . . . . . . . . . 11 (𝑛 ∈ (1...𝑁) ↦ 𝐶) = (𝑛 ∈ (1...𝑁) ↦ 𝐶)
1311, 12fmptd 6425 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶):(1...𝑁)⟶ℚ)
14 qex 11838 . . . . . . . . . . 11 ℚ ∈ V
15 ovex 6718 . . . . . . . . . . 11 (1...𝑁) ∈ V
1614, 15elmap 7928 . . . . . . . . . 10 ((𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑𝑚 (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ 𝐶):(1...𝑁)⟶ℚ)
1713, 16sylibr 224 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑𝑚 (1...𝑁)))
18 eqid 2651 . . . . . . . . 9 (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
1917, 18fmptd 6425 . . . . . . . 8 (𝜑 → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)))
20 eqid 2651 . . . . . . . . . . . 12 (ℂflds ℚ) = (ℂflds ℚ)
2120qdrng 25354 . . . . . . . . . . 11 (ℂflds ℚ) ∈ DivRing
22 drngring 18802 . . . . . . . . . . 11 ((ℂflds ℚ) ∈ DivRing → (ℂflds ℚ) ∈ Ring)
2321, 22ax-mp 5 . . . . . . . . . 10 (ℂflds ℚ) ∈ Ring
24 fzfi 12811 . . . . . . . . . 10 (1...𝑁) ∈ Fin
25 eqid 2651 . . . . . . . . . . 11 ((ℂflds ℚ) freeLMod (1...𝑁)) = ((ℂflds ℚ) freeLMod (1...𝑁))
2625frlmlmod 20141 . . . . . . . . . 10 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod)
2723, 24, 26mp2an 708 . . . . . . . . 9 ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod
28 fzfi 12811 . . . . . . . . 9 (0...𝑁) ∈ Fin
2920qrngbas 25353 . . . . . . . . . . . 12 ℚ = (Base‘(ℂflds ℚ))
3025, 29frlmfibas 20153 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (1...𝑁) ∈ Fin) → (ℚ ↑𝑚 (1...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (1...𝑁))))
3121, 24, 30mp2an 708 . . . . . . . . . 10 (ℚ ↑𝑚 (1...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (1...𝑁)))
3225frlmsca 20145 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (1...𝑁) ∈ Fin) → (ℂflds ℚ) = (Scalar‘((ℂflds ℚ) freeLMod (1...𝑁))))
3321, 24, 32mp2an 708 . . . . . . . . . 10 (ℂflds ℚ) = (Scalar‘((ℂflds ℚ) freeLMod (1...𝑁)))
34 eqid 2651 . . . . . . . . . 10 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁))) = ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))
3520qrng0 25355 . . . . . . . . . . . 12 0 = (0g‘(ℂflds ℚ))
3625, 35frlm0 20146 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) × {0}) = (0g‘((ℂflds ℚ) freeLMod (1...𝑁))))
3723, 24, 36mp2an 708 . . . . . . . . . 10 ((1...𝑁) × {0}) = (0g‘((ℂflds ℚ) freeLMod (1...𝑁)))
38 eqid 2651 . . . . . . . . . . . 12 ((ℂflds ℚ) freeLMod (0...𝑁)) = ((ℂflds ℚ) freeLMod (0...𝑁))
3938, 29frlmfibas 20153 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (0...𝑁) ∈ Fin) → (ℚ ↑𝑚 (0...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (0...𝑁))))
4021, 28, 39mp2an 708 . . . . . . . . . 10 (ℚ ↑𝑚 (0...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (0...𝑁)))
4131, 33, 34, 37, 35, 40islindf4 20225 . . . . . . . . 9 ((((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (0...𝑁) ∈ Fin ∧ (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑𝑚 (1...𝑁))) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0}))))
4227, 28, 41mp3an12 1454 . . . . . . . 8 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0}))))
4319, 42syl 17 . . . . . . 7 (𝜑 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0}))))
44 elmapi 7921 . . . . . . . . 9 (𝑤 ∈ (ℚ ↑𝑚 (0...𝑁)) → 𝑤:(0...𝑁)⟶ℚ)
45 fzfid 12812 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (0...𝑁) ∈ Fin)
46 fvexd 6241 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ V)
4715mptex 6527 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ V
4847a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ V)
49 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑤:(0...𝑁)⟶ℚ)
5049feqmptd 6288 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑤 = (𝑘 ∈ (0...𝑁) ↦ (𝑤𝑘)))
51 eqidd 2652 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
5245, 46, 48, 50, 51offval2 6956 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘)( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶))))
53 fzfid 12812 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
54 ffvelrn 6397 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
5554adantll 750 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
5617adantlr 751 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑𝑚 (1...𝑁)))
57 cnfldmul 19800 . . . . . . . . . . . . . . . . . . . . . 22 · = (.r‘ℂfld)
5820, 57ressmulr 16053 . . . . . . . . . . . . . . . . . . . . 21 (ℚ ∈ V → · = (.r‘(ℂflds ℚ)))
5914, 58ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 · = (.r‘(ℂflds ℚ))
6025, 31, 29, 53, 55, 56, 34, 59frlmvscafval 20157 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘)( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (((1...𝑁) × {(𝑤𝑘)}) ∘𝑓 · (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
61 fvexd 6241 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑤𝑘) ∈ V)
6211adantllr 755 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
63 fconstmpt 5197 . . . . . . . . . . . . . . . . . . . . 21 ((1...𝑁) × {(𝑤𝑘)}) = (𝑛 ∈ (1...𝑁) ↦ (𝑤𝑘))
6463a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((1...𝑁) × {(𝑤𝑘)}) = (𝑛 ∈ (1...𝑁) ↦ (𝑤𝑘)))
65 eqidd 2652 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) = (𝑛 ∈ (1...𝑁) ↦ 𝐶))
6653, 61, 62, 64, 65offval2 6956 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (((1...𝑁) × {(𝑤𝑘)}) ∘𝑓 · (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
6760, 66eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘)( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
6867mpteq2dva 4777 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘)( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))))
6952, 68eqtrd 2685 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))))
7069oveq2d 6706 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))))
71 fzfid 12812 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (1...𝑁) ∈ Fin)
7223a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (ℂflds ℚ) ∈ Ring)
7355adantlr 751 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
7411an32s 863 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℚ)
7574adantllr 755 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℚ)
76 qmulcl 11844 . . . . . . . . . . . . . . . . . . . 20 (((𝑤𝑘) ∈ ℚ ∧ 𝐶 ∈ ℚ) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
7773, 75, 76syl2anc 694 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
7877an32s 863 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
79 eqid 2651 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))
8078, 79fmptd 6425 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
8114, 15elmap 7928 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑𝑚 (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
8280, 81sylibr 224 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑𝑚 (1...𝑁)))
83 eqid 2651 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
8415mptex 6527 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ V
8584a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ V)
86 snex 4938 . . . . . . . . . . . . . . . . . . 19 {0} ∈ V
8715, 86xpex 7004 . . . . . . . . . . . . . . . . . 18 ((1...𝑁) × {0}) ∈ V
8887a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((1...𝑁) × {0}) ∈ V)
8983, 45, 85, 88fsuppmptdm 8327 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))) finSupp ((1...𝑁) × {0}))
9025, 31, 37, 71, 45, 72, 82, 89frlmgsum 20159 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))) = (𝑛 ∈ (1...𝑁) ↦ ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)))))
91 cnfldbas 19798 . . . . . . . . . . . . . . . . . 18 ℂ = (Base‘ℂfld)
92 cnfldadd 19799 . . . . . . . . . . . . . . . . . 18 + = (+g‘ℂfld)
93 cnfldex 19797 . . . . . . . . . . . . . . . . . . 19 fld ∈ V
9493a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℂfld ∈ V)
95 fzfid 12812 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (0...𝑁) ∈ Fin)
96 qsscn 11837 . . . . . . . . . . . . . . . . . . 19 ℚ ⊆ ℂ
9796a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℚ ⊆ ℂ)
98 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))
9977, 98fmptd 6425 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(0...𝑁)⟶ℚ)
100 0z 11426 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℤ
101 zq 11832 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℤ → 0 ∈ ℚ)
102100, 101ax-mp 5 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℚ
103102a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 0 ∈ ℚ)
104 addid2 10257 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥)
105 addid1 10254 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → (𝑥 + 0) = 𝑥)
106104, 105jca 553 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℂ → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
107106adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑥 ∈ ℂ) → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
10891, 92, 20, 94, 95, 97, 99, 103, 107gsumress 17323 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (ℂfld Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))))
109 simplr 807 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 𝑤:(0...𝑁)⟶ℚ)
110 qcn 11840 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤𝑘) ∈ ℚ → (𝑤𝑘) ∈ ℂ)
11154, 110syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
112109, 111sylan 487 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
113 qcn 11840 . . . . . . . . . . . . . . . . . . . . . 22 (𝐶 ∈ ℚ → 𝐶 ∈ ℂ)
11411, 113syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℂ)
115114an32s 863 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℂ)
116115adantllr 755 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℂ)
117112, 116mulcld 10098 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℂ)
11895, 117gsumfsum 19861 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (ℂfld Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
119108, 118eqtr3d 2687 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
120119mpteq2dva 4777 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)))) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)))
12170, 90, 1203eqtrd 2689 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)))
122 qaddcl 11842 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ) → (𝑥 + 𝑦) ∈ ℚ)
123122adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 + 𝑦) ∈ ℚ)
12497, 123, 95, 77, 103fsumcllem 14507 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ ℚ)
125 eqid 2651 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
126124, 125fmptd 6425 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
12714, 15elmap 7928 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑𝑚 (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
128126, 127sylibr 224 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑𝑚 (1...𝑁)))
129121, 128eqeltrd 2730 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) ∈ (ℚ ↑𝑚 (1...𝑁)))
130 elmapi 7921 . . . . . . . . . . . . 13 ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) ∈ (ℚ ↑𝑚 (1...𝑁)) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))):(1...𝑁)⟶ℚ)
131 ffn 6083 . . . . . . . . . . . . 13 ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))):(1...𝑁)⟶ℚ → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) Fn (1...𝑁))
132129, 130, 1313syl 18 . . . . . . . . . . . 12 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) Fn (1...𝑁))
133 c0ex 10072 . . . . . . . . . . . . 13 0 ∈ V
134 fnconstg 6131 . . . . . . . . . . . . 13 (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
135133, 134ax-mp 5 . . . . . . . . . . . 12 ((1...𝑁) × {0}) Fn (1...𝑁)
136 nfcv 2793 . . . . . . . . . . . . . 14 𝑛((ℂflds ℚ) freeLMod (1...𝑁))
137 nfcv 2793 . . . . . . . . . . . . . 14 𝑛 Σg
138 nfcv 2793 . . . . . . . . . . . . . . 15 𝑛𝑤
139 nfcv 2793 . . . . . . . . . . . . . . 15 𝑛𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))
140 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑛(0...𝑁)
141 nfmpt1 4780 . . . . . . . . . . . . . . . 16 𝑛(𝑛 ∈ (1...𝑁) ↦ 𝐶)
142140, 141nfmpt 4779 . . . . . . . . . . . . . . 15 𝑛(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
143138, 139, 142nfov 6716 . . . . . . . . . . . . . 14 𝑛(𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
144136, 137, 143nfov 6716 . . . . . . . . . . . . 13 𝑛(((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))
145 nfcv 2793 . . . . . . . . . . . . 13 𝑛((1...𝑁) × {0})
146144, 145eqfnfv2f 6355 . . . . . . . . . . . 12 (((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) Fn (1...𝑁) ∧ ((1...𝑁) × {0}) Fn (1...𝑁)) → ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) ↔ ∀𝑛 ∈ (1...𝑁)((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛)))
147132, 135, 146sylancl 695 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) ↔ ∀𝑛 ∈ (1...𝑁)((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛)))
148121fveq1d 6231 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))‘𝑛))
149 sumex 14462 . . . . . . . . . . . . . . 15 Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ V
150125fvmpt2 6330 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑁) ∧ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ V) → ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
151149, 150mpan2 707 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
152148, 151sylan9eq 2705 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
153133fvconst2 6510 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
154153adantl 481 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
155152, 154eqeq12d 2666 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛) ↔ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
156155ralbidva 3014 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (∀𝑛 ∈ (1...𝑁)((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
157147, 156bitrd 268 . . . . . . . . . 10 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) ↔ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
158157imbi1d 330 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0})) ↔ (∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
15944, 158sylan2 490 . . . . . . . 8 ((𝜑𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))) → (((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0})) ↔ (∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
160159ralbidva 3014 . . . . . . 7 (𝜑 → (∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0})) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
16143, 160bitrd 268 . . . . . 6 (𝜑 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
162 drngnzr 19310 . . . . . . . . 9 ((ℂflds ℚ) ∈ DivRing → (ℂflds ℚ) ∈ NzRing)
16321, 162ax-mp 5 . . . . . . . 8 (ℂflds ℚ) ∈ NzRing
16433islindf3 20213 . . . . . . . 8 ((((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ NzRing) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))))))
16527, 163, 164mp2an 708 . . . . . . 7 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))))
16647, 18dmmpti 6061 . . . . . . . . 9 dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (0...𝑁)
167 f1eq2 6135 . . . . . . . . 9 (dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (0...𝑁) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))–1-1→V ↔ (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V))
168166, 167ax-mp 5 . . . . . . . 8 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))–1-1→V ↔ (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V)
169168anbi1i 731 . . . . . . 7 (((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) ↔ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))))
170165, 169bitri 264 . . . . . 6 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))))
171 con34b 305 . . . . . . . . 9 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ (¬ 𝑤 = ((0...𝑁) × {0}) → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
172 df-nel 2927 . . . . . . . . . . 11 (𝑤 ∉ {((0...𝑁) × {0})} ↔ ¬ 𝑤 ∈ {((0...𝑁) × {0})})
173 velsn 4226 . . . . . . . . . . 11 (𝑤 ∈ {((0...𝑁) × {0})} ↔ 𝑤 = ((0...𝑁) × {0}))
174172, 173xchbinx 323 . . . . . . . . . 10 (𝑤 ∉ {((0...𝑁) × {0})} ↔ ¬ 𝑤 = ((0...𝑁) × {0}))
175174imbi1i 338 . . . . . . . . 9 ((𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0) ↔ (¬ 𝑤 = ((0...𝑁) × {0}) → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
176171, 175bitr4i 267 . . . . . . . 8 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ (𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
177176ralbii 3009 . . . . . . 7 (∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
178 raldifb 3783 . . . . . . 7 (∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0) ↔ ∀𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
179 ralnex 3021 . . . . . . 7 (∀𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 ↔ ¬ ∃𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
180177, 178, 1793bitri 286 . . . . . 6 (∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ ¬ ∃𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
181161, 170, 1803bitr3g 302 . . . . 5 (𝜑 → (((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) ↔ ¬ ∃𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
182 eqid 2651 . . . . . . . . . . . . 13 (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))
18331, 182lssmre 19014 . . . . . . . . . . . 12 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod → (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑𝑚 (1...𝑁))))
18427, 183ax-mp 5 . . . . . . . . . . 11 (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑𝑚 (1...𝑁)))
185184a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑𝑚 (1...𝑁))))
186 eqid 2651 . . . . . . . . . . . 12 (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))
187 eqid 2651 . . . . . . . . . . . 12 (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) = (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
188182, 186, 187mrclsp 19037 . . . . . . . . . . 11 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod → (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁))) = (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))))
18927, 188ax-mp 5 . . . . . . . . . 10 (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁))) = (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
190 eqid 2651 . . . . . . . . . 10 (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) = (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
19133islvec 19152 . . . . . . . . . . . . 13 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec ↔ (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ DivRing))
19227, 21, 191mpbir2an 975 . . . . . . . . . . . 12 ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec
193182, 189, 31lssacsex 19192 . . . . . . . . . . . . 13 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec → ((LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (ACS‘(ℚ ↑𝑚 (1...𝑁))) ∧ ∀𝑧 ∈ 𝒫 (ℚ ↑𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦}))))
194193simprd 478 . . . . . . . . . . . 12 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec → ∀𝑧 ∈ 𝒫 (ℚ ↑𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦})))
195192, 194ax-mp 5 . . . . . . . . . . 11 𝑧 ∈ 𝒫 (ℚ ↑𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦}))
196195a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ∀𝑧 ∈ 𝒫 (ℚ ↑𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦})))
197 frn 6091 . . . . . . . . . . . . 13 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ (ℚ ↑𝑚 (1...𝑁)))
19819, 197syl 17 . . . . . . . . . . . 12 (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ (ℚ ↑𝑚 (1...𝑁)))
199 dif0 3983 . . . . . . . . . . . 12 ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅) = (ℚ ↑𝑚 (1...𝑁))
200198, 199syl6sseqr 3685 . . . . . . . . . . 11 (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅))
201200adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅))
202 eqid 2651 . . . . . . . . . . . . . . 15 ((ℂflds ℚ) unitVec (1...𝑁)) = ((ℂflds ℚ) unitVec (1...𝑁))
203202, 25, 31uvcff 20178 . . . . . . . . . . . . . 14 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((ℂflds ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)))
20423, 24, 203mp2an 708 . . . . . . . . . . . . 13 ((ℂflds ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑𝑚 (1...𝑁))
205 frn 6091 . . . . . . . . . . . . 13 (((ℂflds ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)) → ran ((ℂflds ℚ) unitVec (1...𝑁)) ⊆ (ℚ ↑𝑚 (1...𝑁)))
206204, 205ax-mp 5 . . . . . . . . . . . 12 ran ((ℂflds ℚ) unitVec (1...𝑁)) ⊆ (ℚ ↑𝑚 (1...𝑁))
207206, 199sseqtr4i 3671 . . . . . . . . . . 11 ran ((ℂflds ℚ) unitVec (1...𝑁)) ⊆ ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅)
208207a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ran ((ℂflds ℚ) unitVec (1...𝑁)) ⊆ ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅))
209 un0 4000 . . . . . . . . . . . . . 14 (ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅) = ran ((ℂflds ℚ) unitVec (1...𝑁))
210209fveq2i 6232 . . . . . . . . . . . . 13 ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)) = ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘ran ((ℂflds ℚ) unitVec (1...𝑁)))
211 eqid 2651 . . . . . . . . . . . . . . . 16 (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁)))
21225, 202, 211frlmlbs 20184 . . . . . . . . . . . . . . 15 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁))))
21323, 24, 212mp2an 708 . . . . . . . . . . . . . 14 ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁)))
21431, 211, 186lbssp 19127 . . . . . . . . . . . . . 14 (ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁))) → ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘ran ((ℂflds ℚ) unitVec (1...𝑁))) = (ℚ ↑𝑚 (1...𝑁)))
215213, 214ax-mp 5 . . . . . . . . . . . . 13 ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘ran ((ℂflds ℚ) unitVec (1...𝑁))) = (ℚ ↑𝑚 (1...𝑁))
216210, 215eqtri 2673 . . . . . . . . . . . 12 ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)) = (ℚ ↑𝑚 (1...𝑁))
217198, 216syl6sseqr 3685 . . . . . . . . . . 11 (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)))
218217adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)))
219 un0 4000 . . . . . . . . . . 11 (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∪ ∅) = ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
22027, 163pm3.2i 470 . . . . . . . . . . . . . 14 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ NzRing)
221186, 33lindsind2 20206 . . . . . . . . . . . . . 14 (((((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ NzRing) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) ∧ 𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) → ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥})))
222220, 221mp3an1 1451 . . . . . . . . . . . . 13 ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) ∧ 𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) → ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥})))
223222ralrimiva 2995 . . . . . . . . . . . 12 (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) → ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥})))
224189, 190ismri2 16339 . . . . . . . . . . . . . 14 (((LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑𝑚 (1...𝑁))) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ (ℚ ↑𝑚 (1...𝑁))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) ↔ ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥}))))
225184, 198, 224sylancr 696 . . . . . . . . . . . . 13 (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) ↔ ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥}))))
226225biimpar 501 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥}))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))))
227223, 226sylan2 490 . . . . . . . . . . 11 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))))
228219, 227syl5eqel 2734 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))))
229 mptfi 8306 . . . . . . . . . . . . 13 ((0...𝑁) ∈ Fin → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin)
230 rnfi 8290 . . . . . . . . . . . . 13 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin)
23128, 229, 230mp2b 10 . . . . . . . . . . . 12 ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin
232231orci 404 . . . . . . . . . . 11 (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin ∨ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ Fin)
233232a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin ∨ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ Fin))
234185, 189, 190, 196, 201, 208, 218, 228, 233mreexexd 16355 . . . . . . . . 9 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ∃𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁))(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))))
235234ex 449 . . . . . . . 8 (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) → ∃𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁))(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))))))
236 ovex 6718 . . . . . . . . . . . . 13 ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V
237236rnex 7142 . . . . . . . . . . . 12 ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V
238 elpwi 4201 . . . . . . . . . . . 12 (𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁)) → 𝑣 ⊆ ran ((ℂflds ℚ) unitVec (1...𝑁)))
239 ssdomg 8043 . . . . . . . . . . . 12 (ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V → (𝑣 ⊆ ran ((ℂflds ℚ) unitVec (1...𝑁)) → 𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁))))
240237, 238, 239mpsyl 68 . . . . . . . . . . 11 (𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁)) → 𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)))
241 endomtr 8055 . . . . . . . . . . . . . 14 ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)))
242241ancoms 468 . . . . . . . . . . . . 13 ((𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)))
243 f1f1orn 6186 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1-onto→ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
244 ovex 6718 . . . . . . . . . . . . . . . . 17 (0...𝑁) ∈ V
245244f1oen 8018 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1-onto→ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) → (0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
246243, 245syl 17 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
247 endomtr 8055 . . . . . . . . . . . . . . . . 17 (((0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁))) → (0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)))
248202uvcendim 20234 . . . . . . . . . . . . . . . . . . . 20 (((ℂflds ℚ) ∈ NzRing ∧ (1...𝑁) ∈ Fin) → (1...𝑁) ≈ ran ((ℂflds ℚ) unitVec (1...𝑁)))
249163, 24, 248mp2an 708 . . . . . . . . . . . . . . . . . . 19 (1...𝑁) ≈ ran ((ℂflds ℚ) unitVec (1...𝑁))
250249ensymi 8047 . . . . . . . . . . . . . . . . . 18 ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)
251 domentr 8056 . . . . . . . . . . . . . . . . . . 19 (((0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)) → (0...𝑁) ≼ (1...𝑁))
252 hashdom 13206 . . . . . . . . . . . . . . . . . . . . 21 (((0...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((#‘(0...𝑁)) ≤ (#‘(1...𝑁)) ↔ (0...𝑁) ≼ (1...𝑁)))
25328, 24, 252mp2an 708 . . . . . . . . . . . . . . . . . . . 20 ((#‘(0...𝑁)) ≤ (#‘(1...𝑁)) ↔ (0...𝑁) ≼ (1...𝑁))
254 hashfz0 13257 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (#‘(0...𝑁)) = (𝑁 + 1))
2552, 254syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (#‘(0...𝑁)) = (𝑁 + 1))
256 hashfz1 13174 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
2572, 256syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (#‘(1...𝑁)) = 𝑁)
258255, 257breq12d 4698 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((#‘(0...𝑁)) ≤ (#‘(1...𝑁)) ↔ (𝑁 + 1) ≤ 𝑁))
259253, 258syl5bbr 274 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((0...𝑁) ≼ (1...𝑁) ↔ (𝑁 + 1) ≤ 𝑁))
260251, 259syl5ib 234 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)) → (𝑁 + 1) ≤ 𝑁))
261250, 260mpan2i 713 . . . . . . . . . . . . . . . . 17 (𝜑 → ((0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) → (𝑁 + 1) ≤ 𝑁))
262247, 261syl5 34 . . . . . . . . . . . . . . . 16 (𝜑 → (((0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁))) → (𝑁 + 1) ≤ 𝑁))
263262expd 451 . . . . . . . . . . . . . . 15 (𝜑 → ((0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) → (𝑁 + 1) ≤ 𝑁)))
264246, 263syl5 34 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) → (𝑁 + 1) ≤ 𝑁)))
265264com23 86 . . . . . . . . . . . . 13 (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
266242, 265syl5 34 . . . . . . . . . . . 12 (𝜑 → ((𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
267266expdimp 452 . . . . . . . . . . 11 ((𝜑𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
268240, 267sylan2 490 . . . . . . . . . 10 ((𝜑𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
269268adantrd 483 . . . . . . . . 9 ((𝜑𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁))) → ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
270269rexlimdva 3060 . . . . . . . 8 (𝜑 → (∃𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁))(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
271235, 270syld 47 . . . . . . 7 (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
272271impd 446 . . . . . 6 (𝜑 → ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) ∧ (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V) → (𝑁 + 1) ≤ 𝑁))
273272ancomsd 469 . . . . 5 (𝜑 → (((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → (𝑁 + 1) ≤ 𝑁))
274181, 273sylbird 250 . . . 4 (𝜑 → (¬ ∃𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → (𝑁 + 1) ≤ 𝑁))
2759, 274mt3d 140 . . 3 (𝜑 → ∃𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
276 eldifsn 4350 . . . . 5 (𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ↔ (𝑤 ∈ (ℚ ↑𝑚 (0...𝑁)) ∧ 𝑤 ≠ ((0...𝑁) × {0})))
27744anim1i 591 . . . . 5 ((𝑤 ∈ (ℚ ↑𝑚 (0...𝑁)) ∧ 𝑤 ≠ ((0...𝑁) × {0})) → (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})))
278276, 277sylbi 207 . . . 4 (𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) → (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})))
27996a1i 11 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ℚ ⊆ ℂ)
2802adantr 480 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑁 ∈ ℕ0)
281279, 280, 55elplyd 24003 . . . . . . . 8 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ (Poly‘ℚ))
282281adantrr 753 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ (Poly‘ℚ))
283 uzdisj 12451 . . . . . . . . . . . . . . . . . 18 ((0...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ∅
2842nn0cnd 11391 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ ℂ)
285 pncan1 10492 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁)
286284, 285syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
287286oveq2d 6706 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
288287ineq1d 3846 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((0...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))
289283, 288syl5eqr 2699 . . . . . . . . . . . . . . . . 17 (𝜑 → ∅ = ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))
290289eqcomd 2657 . . . . . . . . . . . . . . . 16 (𝜑 → ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅)
291133fconst 6129 . . . . . . . . . . . . . . . . . 18 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0}
292 snssi 4371 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℚ → {0} ⊆ ℚ)
293100, 101, 292mp2b 10 . . . . . . . . . . . . . . . . . . 19 {0} ⊆ ℚ
294293, 96sstri 3645 . . . . . . . . . . . . . . . . . 18 {0} ⊆ ℂ
295 fss 6094 . . . . . . . . . . . . . . . . . 18 ((((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0} ∧ {0} ⊆ ℂ) → ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℂ)
296291, 294, 295mp2an 708 . . . . . . . . . . . . . . . . 17 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℂ
297 fun 6104 . . . . . . . . . . . . . . . . 17 (((𝑤:(0...𝑁)⟶ℚ ∧ ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℂ) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
298296, 297mpanl2 717 . . . . . . . . . . . . . . . 16 ((𝑤:(0...𝑁)⟶ℚ ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
299290, 298sylan2 490 . . . . . . . . . . . . . . 15 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝜑) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
300299ancoms 468 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
301 nn0uz 11760 . . . . . . . . . . . . . . . . . 18 0 = (ℤ‘0)
3026, 301syl6eleq 2740 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ (ℤ‘0))
303 uzsplit 12450 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ (ℤ‘0) → (ℤ‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
304302, 303syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℤ‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
305301, 304syl5eq 2697 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ0 = ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
306287uneq1d 3799 . . . . . . . . . . . . . . . . 17 (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))))
307305, 306eqtr2d 2686 . . . . . . . . . . . . . . . 16 (𝜑 → ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))) = ℕ0)
308 ssequn1 3816 . . . . . . . . . . . . . . . . . 18 (ℚ ⊆ ℂ ↔ (ℚ ∪ ℂ) = ℂ)
30996, 308mpbi 220 . . . . . . . . . . . . . . . . 17 (ℚ ∪ ℂ) = ℂ
310309a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → (ℚ ∪ ℂ) = ℂ)
311307, 310feq23d 6078 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ) ↔ (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):ℕ0⟶ℂ))
312311adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ) ↔ (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):ℕ0⟶ℂ))
313300, 312mpbid 222 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):ℕ0⟶ℂ)
314 ffn 6083 . . . . . . . . . . . . . . . 16 (𝑤:(0...𝑁)⟶ℚ → 𝑤 Fn (0...𝑁))
315 fnimadisj 6050 . . . . . . . . . . . . . . . 16 ((𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) → (𝑤 “ (ℤ‘(𝑁 + 1))) = ∅)
316314, 290, 315syl2anr 494 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 “ (ℤ‘(𝑁 + 1))) = ∅)
3172nn0zd 11518 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℤ)
318317peano2zd 11523 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ ℤ)
319 uzid 11740 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ ℤ → (𝑁 + 1) ∈ (ℤ‘(𝑁 + 1)))
320 ne0i 3954 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ (ℤ‘(𝑁 + 1)) → (ℤ‘(𝑁 + 1)) ≠ ∅)
321318, 319, 3203syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℤ‘(𝑁 + 1)) ≠ ∅)
322 inidm 3855 . . . . . . . . . . . . . . . . . . 19 ((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) = (ℤ‘(𝑁 + 1))
323322neeq1i 2887 . . . . . . . . . . . . . . . . . 18 (((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅ ↔ (ℤ‘(𝑁 + 1)) ≠ ∅)
324321, 323sylibr 224 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅)
325 xpima2 5613 . . . . . . . . . . . . . . . . 17 (((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅ → (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))) = {0})
326324, 325syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))) = {0})
327326adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))) = {0})
328316, 327uneq12d 3801 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 “ (ℤ‘(𝑁 + 1))) ∪ (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1)))) = (∅ ∪ {0}))
329 imaundir 5581 . . . . . . . . . . . . . 14 ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) “ (ℤ‘(𝑁 + 1))) = ((𝑤 “ (ℤ‘(𝑁 + 1))) ∪ (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))))
330 uncom 3790 . . . . . . . . . . . . . . 15 (∅ ∪ {0}) = ({0} ∪ ∅)
331 un0 4000 . . . . . . . . . . . . . . 15 ({0} ∪ ∅) = {0}
332330, 331eqtr2i 2674 . . . . . . . . . . . . . 14 {0} = (∅ ∪ {0})
333328, 329, 3323eqtr4g 2710 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) “ (ℤ‘(𝑁 + 1))) = {0})
334290, 314anim12ci 590 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅))
335 fnconstg 6131 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ V → ((ℤ‘(𝑁 + 1)) × {0}) Fn (ℤ‘(𝑁 + 1)))
336133, 335ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((ℤ‘(𝑁 + 1)) × {0}) Fn (ℤ‘(𝑁 + 1))
337 fvun1 6308 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 Fn (0...𝑁) ∧ ((ℤ‘(𝑁 + 1)) × {0}) Fn (ℤ‘(𝑁 + 1)) ∧ (((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅ ∧ 𝑘 ∈ (0...𝑁))) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
338336, 337mp3an2 1452 . . . . . . . . . . . . . . . . . . 19 ((𝑤 Fn (0...𝑁) ∧ (((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅ ∧ 𝑘 ∈ (0...𝑁))) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
339338anassrs 681 . . . . . . . . . . . . . . . . . 18 (((𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
340334, 339sylan 487 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
341340eqcomd 2657 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) = ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘))
342341oveq1d 6705 . . . . . . . . . . . . . . 15 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝑦𝑘)) = (((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘)))
343342sumeq2dv 14477 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)) = Σ𝑘 ∈ (0...𝑁)(((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘)))
344343mpteq2dv 4778 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘))))
345281, 280, 313, 333, 344coeeq 24028 . . . . . . . . . . . 12 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) = (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})))
346345reseq1d 5427 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)))
347 res0 5432 . . . . . . . . . . . . . 14 (𝑤 ↾ ∅) = ∅
348289reseq2d 5428 . . . . . . . . . . . . . 14 (𝜑 → (𝑤 ↾ ∅) = (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
349 res0 5432 . . . . . . . . . . . . . . 15 (((ℤ‘(𝑁 + 1)) × {0}) ↾ ∅) = ∅
350289reseq2d 5428 . . . . . . . . . . . . . . 15 (𝜑 → (((ℤ‘(𝑁 + 1)) × {0}) ↾ ∅) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
351349, 350syl5eqr 2699 . . . . . . . . . . . . . 14 (𝜑 → ∅ = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
352347, 348, 3513eqtr3a 2709 . . . . . . . . . . . . 13 (𝜑 → (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
353 fss 6094 . . . . . . . . . . . . . . 15 ((((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0} ∧ {0} ⊆ ℚ) → ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℚ)
354291, 293, 353mp2an 708 . . . . . . . . . . . . . 14 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℚ
355 fresaunres1 6115 . . . . . . . . . . . . . 14 ((𝑤:(0...𝑁)⟶ℚ ∧ ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℚ ∧ (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
356354, 355mp3an2 1452 . . . . . . . . . . . . 13 ((𝑤:(0...𝑁)⟶ℚ ∧ (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
357352, 356sylan2 490 . . . . . . . . . . . 12 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝜑) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
358357ancoms 468 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
359346, 358eqtrd 2685 . . . . . . . . . 10 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = 𝑤)
360 fveq2 6229 . . . . . . . . . . 11 ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = 0𝑝 → (coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) = (coeff‘0𝑝))
361360reseq1d 5427 . . . . . . . . . 10 ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = 0𝑝 → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((coeff‘0𝑝) ↾ (0...𝑁)))
362 eqtr2 2671 . . . . . . . . . . . 12 ((((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = 𝑤 ∧ ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((coeff‘0𝑝) ↾ (0...𝑁))) → 𝑤 = ((coeff‘0𝑝) ↾ (0...𝑁)))
363 coe0 24057 . . . . . . . . . . . . . 14 (coeff‘0𝑝) = (ℕ0 × {0})
364363reseq1i 5424 . . . . . . . . . . . . 13 ((coeff‘0𝑝) ↾ (0...𝑁)) = ((ℕ0 × {0}) ↾ (0...𝑁))
365 elfznn0 12471 . . . . . . . . . . . . . . 15 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℕ0)
366365ssriv 3640 . . . . . . . . . . . . . 14 (0...𝑁) ⊆ ℕ0
367 xpssres 5469 . . . . . . . . . . . . . 14 ((0...𝑁) ⊆ ℕ0 → ((ℕ0 × {0}) ↾ (0...𝑁)) = ((0...𝑁) × {0}))
368366, 367ax-mp 5 . . . . . . . . . . . . 13 ((ℕ0 × {0}) ↾ (0...𝑁)) = ((0...𝑁) × {0})
369364, 368eqtri 2673 . . . . . . . . . . . 12 ((coeff‘0𝑝) ↾ (0...𝑁)) = ((0...𝑁) × {0})
370362, 369syl6eq 2701 . . . . . . . . . . 11 ((((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = 𝑤 ∧ ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((coeff‘0𝑝) ↾ (0...𝑁))) → 𝑤 = ((0...𝑁) × {0}))
371370ex 449 . . . . . . . . . 10 (((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = 𝑤 → (((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((coeff‘0𝑝) ↾ (0...𝑁)) → 𝑤 = ((0...𝑁) × {0})))
372359, 361, 371syl2im 40 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = 0𝑝𝑤 = ((0...𝑁) × {0})))
373372necon3d 2844 . . . . . . . 8 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 ≠ ((0...𝑁) × {0}) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ≠ 0𝑝))
374373impr 648 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ≠ 0𝑝)
375 eldifsn 4350 . . . . . . 7 ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ (Poly‘ℚ) ∧ (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ≠ 0𝑝))
376282, 374, 375sylanbrc 699 . . . . . 6 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ ((Poly‘ℚ) ∖ {0𝑝}))
377376adantrr 753 . . . . 5 ((𝜑 ∧ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})) ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ ((Poly‘ℚ) ∖ {0𝑝}))
378 oveq1 6697 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝑦𝑘) = (𝐴𝑘))
379378oveq2d 6706 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝑤𝑘) · (𝑦𝑘)) = ((𝑤𝑘) · (𝐴𝑘)))
380379sumeq2sdv 14479 . . . . . . . . . 10 (𝑦 = 𝐴 → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
381 eqid 2651 . . . . . . . . . 10 (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))
382 sumex 14462 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) ∈ V
383380, 381, 382fvmpt 6321 . . . . . . . . 9 (𝐴 ∈ ℂ → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
3841, 383syl 17 . . . . . . . 8 (𝜑 → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
385384adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
386111adantll 750 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
387 aacllem.2 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
388387adantlr 751 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
389114, 388mulcld 10098 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝐶 · 𝑋) ∈ ℂ)
390389adantllr 755 . . . . . . . . . . . . . 14 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝐶 · 𝑋) ∈ ℂ)
39153, 386, 390fsummulc2 14560 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)) = Σ𝑛 ∈ (1...𝑁)((𝑤𝑘) · (𝐶 · 𝑋)))
392 aacllem.4 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐴𝑘) = Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋))
393392oveq2d 6706 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = ((𝑤𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)))
394393adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = ((𝑤𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)))
395386adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑤𝑘) ∈ ℂ)
396114adantllr 755 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℂ)
397 simpll 805 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → 𝜑)
398397, 387sylan 487 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
399395, 396, 398mulassd 10101 . . . . . . . . . . . . . 14 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((𝑤𝑘) · 𝐶) · 𝑋) = ((𝑤𝑘) · (𝐶 · 𝑋)))
400399sumeq2dv 14477 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁)((𝑤𝑘) · (𝐶 · 𝑋)))
401391, 394, 4003eqtr4d 2695 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
402401sumeq2dv 14477 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑘 ∈ (0...𝑁𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
403111ad2ant2lr 799 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → (𝑤𝑘) ∈ ℂ)
404114anasss 680 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝐶 ∈ ℂ)
405404adantlr 751 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝐶 ∈ ℂ)
406403, 405mulcld 10098 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → ((𝑤𝑘) · 𝐶) ∈ ℂ)
407387ad2ant2rl 800 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝑋 ∈ ℂ)
408406, 407mulcld 10098 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → (((𝑤𝑘) · 𝐶) · 𝑋) ∈ ℂ)
40945, 71, 408fsumcom 14551 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
410402, 409eqtrd 2685 . . . . . . . . . 10 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
411410adantrr 753 . . . . . . . . 9 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
412 nfv 1883 . . . . . . . . . . . 12 𝑛𝜑
413 nfv 1883 . . . . . . . . . . . . 13 𝑛 𝑤:(0...𝑁)⟶ℚ
414 nfra1 2970 . . . . . . . . . . . . 13 𝑛𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0
415413, 414nfan 1868 . . . . . . . . . . . 12 𝑛(𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
416412, 415nfan 1868 . . . . . . . . . . 11 𝑛(𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
417 rspa 2959 . . . . . . . . . . . . . . . 16 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
418417oveq1d 6705 . . . . . . . . . . . . . . 15 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
419418adantll 750 . . . . . . . . . . . . . 14 (((𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
420419adantll 750 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
421387adantlr 751 . . . . . . . . . . . . . . 15 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
42295, 421, 117fsummulc1 14561 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
423422adantlrr 757 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
424387mul02d 10272 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → (0 · 𝑋) = 0)
425424adantlr 751 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (0 · 𝑋) = 0)
426420, 423, 4253eqtr3d 2693 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = 0)
427426ex 449 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → (𝑛 ∈ (1...𝑁) → Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = 0))
428416, 427ralrimi 2986 . . . . . . . . . 10 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = 0)
429428sumeq2d 14476 . . . . . . . . 9 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁)0)
430411, 429eqtrd 2685 . . . . . . . 8 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁)0)
43124olci 405 . . . . . . . . 9 ((1...𝑁) ⊆ (ℤ𝐵) ∨ (1...𝑁) ∈ Fin)
432 sumz 14497 . . . . . . . . 9 (((1...𝑁) ⊆ (ℤ𝐵) ∨ (1...𝑁) ∈ Fin) → Σ𝑛 ∈ (1...𝑁)0 = 0)
433431, 432ax-mp 5 . . . . . . . 8 Σ𝑛 ∈ (1...𝑁)0 = 0
434430, 433syl6eq 2701 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = 0)
435385, 434eqtrd 2685 . . . . . 6 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0)
436435adantrlr 759 . . . . 5 ((𝜑 ∧ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})) ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0)
437 fveq1 6228 . . . . . . 7 (𝑥 = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) → (𝑥𝐴) = ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴))
438437eqeq1d 2653 . . . . . 6 (𝑥 = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) → ((𝑥𝐴) = 0 ↔ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0))
439438rspcev 3340 . . . . 5 (((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
440377, 436, 439syl2anc 694 . . . 4 ((𝜑 ∧ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})) ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
441278, 440sylanr1 685 . . 3 ((𝜑 ∧ (𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
442275, 441rexlimddv 3064 . 2 (𝜑 → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
443 elqaa 24122 . 2 (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0))
4441, 442, 443sylanbrc 699 1 (𝜑𝐴 ∈ 𝔸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wnel 2926  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  cres 5145  cima 5146   Fn wfn 5921  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  𝑓 cof 6937  𝑚 cmap 7899  cen 7994  cdom 7995  Fincfn 7997  cc 9972  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  cmin 10304  0cn0 11330  cz 11415  cuz 11725  cq 11826  ...cfz 12364  cexp 12900  #chash 13157  Σcsu 14460  Basecbs 15904  s cress 15905  .rcmulr 15989  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147   Σg cgsu 16148  Moorecmre 16289  mrClscmrc 16290  mrIndcmri 16291  ACScacs 16292  Ringcrg 18593  DivRingcdr 18795  LModclmod 18911  LSubSpclss 18980  LSpanclspn 19019  LBasisclbs 19122  LVecclvec 19150  NzRingcnzr 19305  fldccnfld 19794   freeLMod cfrlm 20138   unitVec cuvc 20169   LIndF clindf 20191  LIndSclinds 20192  0𝑝c0p 23481  Polycply 23985  coeffccoe 23987  𝔸caa 24114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-xnn0 11402  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-0g 16149  df-gsum 16150  df-prds 16155  df-pws 16157  df-mre 16293  df-mrc 16294  df-mri 16295  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-subrg 18826  df-lmod 18913  df-lss 18981  df-lsp 19020  df-lmhm 19070  df-lbs 19123  df-lvec 19151  df-sra 19220  df-rgmod 19221  df-nzr 19306  df-cnfld 19795  df-dsmm 20124  df-frlm 20139  df-uvc 20170  df-lindf 20193  df-linds 20194  df-0p 23482  df-ply 23989  df-coe 23991  df-dgr 23992  df-aa 24115
This theorem is referenced by: (None)
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