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Theorem aacllem 49031
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 12585 . . . . . 6 (𝜑𝑁 ∈ ℝ)
43ltp1d 12195 . . . . 5 (𝜑𝑁 < (𝑁 + 1))
5 peano2nn0 12563 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
62, 5syl 17 . . . . . . 7 (𝜑 → (𝑁 + 1) ∈ ℕ0)
76nn0red 12585 . . . . . 6 (𝜑 → (𝑁 + 1) ∈ ℝ)
83, 7ltnled 11405 . . . . 5 (𝜑 → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁))
94, 8mpbid 232 . . . 4 (𝜑 → ¬ (𝑁 + 1) ≤ 𝑁)
10 aacllem.3 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
11103expa 1117 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
1211fmpttd 7134 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶):(1...𝑁)⟶ℚ)
13 qex 13000 . . . . . . . . . . 11 ℚ ∈ V
14 ovex 7463 . . . . . . . . . . 11 (1...𝑁) ∈ V
1513, 14elmap 8909 . . . . . . . . . 10 ((𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑m (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ 𝐶):(1...𝑁)⟶ℚ)
1612, 15sylibr 234 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑m (1...𝑁)))
1716fmpttd 7134 . . . . . . . 8 (𝜑 → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑m (1...𝑁)))
18 eqid 2734 . . . . . . . . . . . 12 (ℂflds ℚ) = (ℂflds ℚ)
1918qdrng 27678 . . . . . . . . . . 11 (ℂflds ℚ) ∈ DivRing
20 drngring 20752 . . . . . . . . . . 11 ((ℂflds ℚ) ∈ DivRing → (ℂflds ℚ) ∈ Ring)
2119, 20ax-mp 5 . . . . . . . . . 10 (ℂflds ℚ) ∈ Ring
22 fzfi 14009 . . . . . . . . . 10 (1...𝑁) ∈ Fin
23 eqid 2734 . . . . . . . . . . 11 ((ℂflds ℚ) freeLMod (1...𝑁)) = ((ℂflds ℚ) freeLMod (1...𝑁))
2423frlmlmod 21786 . . . . . . . . . 10 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod)
2521, 22, 24mp2an 692 . . . . . . . . 9 ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod
26 fzfi 14009 . . . . . . . . 9 (0...𝑁) ∈ Fin
2718qrngbas 27677 . . . . . . . . . . . 12 ℚ = (Base‘(ℂflds ℚ))
2823, 27frlmfibas 21799 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (1...𝑁) ∈ Fin) → (ℚ ↑m (1...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (1...𝑁))))
2919, 22, 28mp2an 692 . . . . . . . . . 10 (ℚ ↑m (1...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (1...𝑁)))
3023frlmsca 21790 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (1...𝑁) ∈ Fin) → (ℂflds ℚ) = (Scalar‘((ℂflds ℚ) freeLMod (1...𝑁))))
3119, 22, 30mp2an 692 . . . . . . . . . 10 (ℂflds ℚ) = (Scalar‘((ℂflds ℚ) freeLMod (1...𝑁)))
32 eqid 2734 . . . . . . . . . 10 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁))) = ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))
3318qrng0 27679 . . . . . . . . . . . 12 0 = (0g‘(ℂflds ℚ))
3423, 33frlm0 21791 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) × {0}) = (0g‘((ℂflds ℚ) freeLMod (1...𝑁))))
3521, 22, 34mp2an 692 . . . . . . . . . 10 ((1...𝑁) × {0}) = (0g‘((ℂflds ℚ) freeLMod (1...𝑁)))
36 eqid 2734 . . . . . . . . . . . 12 ((ℂflds ℚ) freeLMod (0...𝑁)) = ((ℂflds ℚ) freeLMod (0...𝑁))
3736, 27frlmfibas 21799 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (0...𝑁) ∈ Fin) → (ℚ ↑m (0...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (0...𝑁))))
3819, 26, 37mp2an 692 . . . . . . . . . 10 (ℚ ↑m (0...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (0...𝑁)))
3929, 31, 32, 35, 33, 38islindf4 21875 . . . . . . . . 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}))))
4025, 26, 39mp3an12 1450 . . . . . . . 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}))))
4117, 40syl 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}))))
42 elmapi 8887 . . . . . . . . 9 (𝑤 ∈ (ℚ ↑m (0...𝑁)) → 𝑤:(0...𝑁)⟶ℚ)
43 fzfid 14010 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (0...𝑁) ∈ Fin)
44 fvexd 6921 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ V)
4514mptex 7242 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ V
4645a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ V)
47 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑤:(0...𝑁)⟶ℚ)
4847feqmptd 6976 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑤 = (𝑘 ∈ (0...𝑁) ↦ (𝑤𝑘)))
49 eqidd 2735 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
5043, 44, 46, 48, 49offval2 7716 . . . . . . . . . . . . . . . . 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))))
51 fzfid 14010 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
52 ffvelcdm 7100 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
5352adantll 714 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
5416adantlr 715 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑m (1...𝑁)))
55 cnfldmul 21389 . . . . . . . . . . . . . . . . . . . . . 22 · = (.r‘ℂfld)
5618, 55ressmulr 17352 . . . . . . . . . . . . . . . . . . . . 21 (ℚ ∈ V → · = (.r‘(ℂflds ℚ)))
5713, 56ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 · = (.r‘(ℂflds ℚ))
5823, 29, 27, 51, 53, 54, 32, 57frlmvscafval 21803 . . . . . . . . . . . . . . . . . . 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)))
59 fvexd 6921 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑤𝑘) ∈ V)
6011adantllr 719 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
61 fconstmpt 5750 . . . . . . . . . . . . . . . . . . . . 21 ((1...𝑁) × {(𝑤𝑘)}) = (𝑛 ∈ (1...𝑁) ↦ (𝑤𝑘))
6261a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((1...𝑁) × {(𝑤𝑘)}) = (𝑛 ∈ (1...𝑁) ↦ (𝑤𝑘)))
63 eqidd 2735 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) = (𝑛 ∈ (1...𝑁) ↦ 𝐶))
6451, 59, 60, 62, 63offval2 7716 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (((1...𝑁) × {(𝑤𝑘)}) ∘f · (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
6558, 64eqtrd 2774 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘)( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
6665mpteq2dva 5247 . . . . . . . . . . . . . . . . 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))))
6750, 66eqtrd 2774 . . . . . . . . . . . . . . . 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))))
6867oveq2d 7446 . . . . . . . . . . . . . . 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)))))
69 fzfid 14010 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (1...𝑁) ∈ Fin)
7021a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (ℂflds ℚ) ∈ Ring)
7153adantlr 715 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
7211an32s 652 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℚ)
7372adantllr 719 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℚ)
74 qmulcl 13006 . . . . . . . . . . . . . . . . . . . 20 (((𝑤𝑘) ∈ ℚ ∧ 𝐶 ∈ ℚ) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
7571, 73, 74syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
7675an32s 652 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
7776fmpttd 7134 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
7813, 14elmap 8909 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑m (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
7977, 78sylibr 234 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑m (1...𝑁)))
80 eqid 2734 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
8114mptex 7242 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ V
8281a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ V)
83 snex 5441 . . . . . . . . . . . . . . . . . . 19 {0} ∈ V
8414, 83xpex 7771 . . . . . . . . . . . . . . . . . 18 ((1...𝑁) × {0}) ∈ V
8584a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((1...𝑁) × {0}) ∈ V)
8680, 43, 82, 85fsuppmptdm 9413 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))) finSupp ((1...𝑁) × {0}))
8723, 29, 35, 69, 43, 70, 79, 86frlmgsum 21809 . . . . . . . . . . . . . . 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)))))
88 cnfldbas 21385 . . . . . . . . . . . . . . . . . 18 ℂ = (Base‘ℂfld)
89 cnfldadd 21387 . . . . . . . . . . . . . . . . . 18 + = (+g‘ℂfld)
90 cnfldex 21384 . . . . . . . . . . . . . . . . . . 19 fld ∈ V
9190a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℂfld ∈ V)
92 fzfid 14010 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (0...𝑁) ∈ Fin)
93 qsscn 12999 . . . . . . . . . . . . . . . . . . 19 ℚ ⊆ ℂ
9493a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℚ ⊆ ℂ)
9575fmpttd 7134 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(0...𝑁)⟶ℚ)
96 0z 12621 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℤ
97 zq 12993 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℤ → 0 ∈ ℚ)
9896, 97ax-mp 5 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℚ
9998a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 0 ∈ ℚ)
100 addlid 11441 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥)
101 addrid 11438 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → (𝑥 + 0) = 𝑥)
102100, 101jca 511 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℂ → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
103102adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑥 ∈ ℂ) → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
10488, 89, 18, 91, 92, 94, 95, 99, 103gsumress 18707 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (ℂfld Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))))
105 simplr 769 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 𝑤:(0...𝑁)⟶ℚ)
106 qcn 13002 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤𝑘) ∈ ℚ → (𝑤𝑘) ∈ ℂ)
10752, 106syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
108105, 107sylan 580 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
109 qcn 13002 . . . . . . . . . . . . . . . . . . . . . 22 (𝐶 ∈ ℚ → 𝐶 ∈ ℂ)
11011, 109syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℂ)
111110an32s 652 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℂ)
112111adantllr 719 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℂ)
113108, 112mulcld 11278 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℂ)
11492, 113gsumfsum 21469 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (ℂfld Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
115104, 114eqtr3d 2776 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
116115mpteq2dva 5247 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)))) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)))
11768, 87, 1163eqtrd 2778 . . . . . . . . . . . . . 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)))
118 qaddcl 13004 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ) → (𝑥 + 𝑦) ∈ ℚ)
119118adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 + 𝑦) ∈ ℚ)
12094, 119, 92, 75, 99fsumcllem 15764 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ ℚ)
121120fmpttd 7134 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
12213, 14elmap 8909 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑m (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
123121, 122sylibr 234 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑m (1...𝑁)))
124117, 123eqeltrd 2838 . . . . . . . . . . . . 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)))
125 elmapi 8887 . . . . . . . . . . . . 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)
126 ffn 6736 . . . . . . . . . . . . 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))
127124, 125, 1263syl 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))
128 c0ex 11252 . . . . . . . . . . . . 13 0 ∈ V
129 fnconstg 6796 . . . . . . . . . . . . 13 (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
130128, 129ax-mp 5 . . . . . . . . . . . 12 ((1...𝑁) × {0}) Fn (1...𝑁)
131 nfcv 2902 . . . . . . . . . . . . . 14 𝑛((ℂflds ℚ) freeLMod (1...𝑁))
132 nfcv 2902 . . . . . . . . . . . . . 14 𝑛 Σg
133 nfcv 2902 . . . . . . . . . . . . . . 15 𝑛𝑤
134 nfcv 2902 . . . . . . . . . . . . . . 15 𝑛f ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))
135 nfcv 2902 . . . . . . . . . . . . . . . 16 𝑛(0...𝑁)
136 nfmpt1 5255 . . . . . . . . . . . . . . . 16 𝑛(𝑛 ∈ (1...𝑁) ↦ 𝐶)
137135, 136nfmpt 5254 . . . . . . . . . . . . . . 15 𝑛(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
138133, 134, 137nfov 7460 . . . . . . . . . . . . . 14 𝑛(𝑤f ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
139131, 132, 138nfov 7460 . . . . . . . . . . . . 13 𝑛(((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤f ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))
140 nfcv 2902 . . . . . . . . . . . . 13 𝑛((1...𝑁) × {0})
141139, 140eqfnfv2f 7054 . . . . . . . . . . . 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)))
142127, 130, 141sylancl 586 . . . . . . . . . . 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)))
143117fveq1d 6908 . . . . . . . . . . . . . 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))
144 sumex 15720 . . . . . . . . . . . . . . 15 Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ V
145 eqid 2734 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
146145fvmpt2 7026 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑁) ∧ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ V) → ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
147144, 146mpan2 691 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
148143, 147sylan9eq 2794 . . . . . . . . . . . . 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))
149128fvconst2 7223 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
150149adantl 481 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
151148, 150eqeq12d 2750 . . . . . . . . . . . 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))
152151ralbidva 3173 . . . . . . . . . . 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))
153142, 152bitrd 279 . . . . . . . . . 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))
154153imbi1d 341 . . . . . . . . 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}))))
15542, 154sylan2 593 . . . . . . . 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}))))
156155ralbidva 3173 . . . . . . 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}))))
15741, 156bitrd 279 . . . . . 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}))))
158 drngnzr 20764 . . . . . . . . 9 ((ℂflds ℚ) ∈ DivRing → (ℂflds ℚ) ∈ NzRing)
15919, 158ax-mp 5 . . . . . . . 8 (ℂflds ℚ) ∈ NzRing
16031islindf3 21863 . . . . . . . 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))))))
16125, 159, 160mp2an 692 . . . . . . 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)))))
162 eqid 2734 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
16345, 162dmmpti 6712 . . . . . . . . 9 dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (0...𝑁)
164 f1eq2 6800 . . . . . . . . 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))
165163, 164ax-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)
166165anbi1i 624 . . . . . . 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)))))
167161, 166bitri 275 . . . . . 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)))))
168 con34b 316 . . . . . . . . 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))
169 df-nel 3044 . . . . . . . . . . 11 (𝑤 ∉ {((0...𝑁) × {0})} ↔ ¬ 𝑤 ∈ {((0...𝑁) × {0})})
170 velsn 4646 . . . . . . . . . . 11 (𝑤 ∈ {((0...𝑁) × {0})} ↔ 𝑤 = ((0...𝑁) × {0}))
171169, 170xchbinx 334 . . . . . . . . . 10 (𝑤 ∉ {((0...𝑁) × {0})} ↔ ¬ 𝑤 = ((0...𝑁) × {0}))
172171imbi1i 349 . . . . . . . . 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))
173168, 172bitr4i 278 . . . . . . . 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))
174173ralbii 3090 . . . . . . 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))
175 raldifb 4158 . . . . . . 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)
176 ralnex 3069 . . . . . . 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)
177174, 175, 1763bitri 297 . . . . . 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)
178157, 167, 1773bitr3g 313 . . . . 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))
179 eqid 2734 . . . . . . . . . . . . 13 (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))
18029, 179lssmre 20981 . . . . . . . . . . . 12 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod → (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑m (1...𝑁))))
18125, 180ax-mp 5 . . . . . . . . . . 11 (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑m (1...𝑁)))
182181a1i 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))))
183 eqid 2734 . . . . . . . . . . . 12 (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))
184 eqid 2734 . . . . . . . . . . . 12 (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) = (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
185179, 183, 184mrclsp 21004 . . . . . . . . . . 11 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod → (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁))) = (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))))
18625, 185ax-mp 5 . . . . . . . . . 10 (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁))) = (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
187 eqid 2734 . . . . . . . . . 10 (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) = (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
18831islvec 21120 . . . . . . . . . . . . 13 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec ↔ (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ DivRing))
18925, 19, 188mpbir2an 711 . . . . . . . . . . . 12 ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec
190179, 186, 29lssacsex 21163 . . . . . . . . . . . . 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}))))
191190simprd 495 . . . . . . . . . . . 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})))
192189, 191ax-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}))
193192a1i 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})))
19417frnd 6744 . . . . . . . . . . . 12 (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ (ℚ ↑m (1...𝑁)))
195 dif0 4383 . . . . . . . . . . . 12 ((ℚ ↑m (1...𝑁)) ∖ ∅) = (ℚ ↑m (1...𝑁))
196194, 195sseqtrrdi 4046 . . . . . . . . . . 11 (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((ℚ ↑m (1...𝑁)) ∖ ∅))
197196adantr 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)) ∖ ∅))
198 eqid 2734 . . . . . . . . . . . . . . 15 ((ℂflds ℚ) unitVec (1...𝑁)) = ((ℂflds ℚ) unitVec (1...𝑁))
199198, 23, 29uvcff 21828 . . . . . . . . . . . . . 14 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((ℂflds ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑m (1...𝑁)))
20021, 22, 199mp2an 692 . . . . . . . . . . . . 13 ((ℂflds ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑m (1...𝑁))
201 frn 6743 . . . . . . . . . . . . 13 (((ℂflds ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑m (1...𝑁)) → ran ((ℂflds ℚ) unitVec (1...𝑁)) ⊆ (ℚ ↑m (1...𝑁)))
202200, 201ax-mp 5 . . . . . . . . . . . 12 ran ((ℂflds ℚ) unitVec (1...𝑁)) ⊆ (ℚ ↑m (1...𝑁))
203202, 195sseqtrri 4032 . . . . . . . . . . 11 ran ((ℂflds ℚ) unitVec (1...𝑁)) ⊆ ((ℚ ↑m (1...𝑁)) ∖ ∅)
204203a1i 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)) ∖ ∅))
205 un0 4399 . . . . . . . . . . . . . 14 (ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅) = ran ((ℂflds ℚ) unitVec (1...𝑁))
206205fveq2i 6909 . . . . . . . . . . . . 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)))
207 eqid 2734 . . . . . . . . . . . . . . . 16 (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁)))
20823, 198, 207frlmlbs 21834 . . . . . . . . . . . . . . 15 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁))))
20921, 22, 208mp2an 692 . . . . . . . . . . . . . 14 ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁)))
21029, 207, 183lbssp 21095 . . . . . . . . . . . . . 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)))
211209, 210ax-mp 5 . . . . . . . . . . . . 13 ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘ran ((ℂflds ℚ) unitVec (1...𝑁))) = (ℚ ↑m (1...𝑁))
212206, 211eqtri 2762 . . . . . . . . . . . 12 ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)) = (ℚ ↑m (1...𝑁))
213194, 212sseqtrrdi 4046 . . . . . . . . . . 11 (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)))
214213adantr 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)) ∪ ∅)))
215 un0 4399 . . . . . . . . . . 11 (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∪ ∅) = ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
21625, 159pm3.2i 470 . . . . . . . . . . . . . 14 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ NzRing)
217183, 31lindsind2 21856 . . . . . . . . . . . . . 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})))
218216, 217mp3an1 1447 . . . . . . . . . . . . 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})))
219218ralrimiva 3143 . . . . . . . . . . . 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})))
220186, 187ismri2 17676 . . . . . . . . . . . . . 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}))))
221181, 194, 220sylancr 587 . . . . . . . . . . . . 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}))))
222221biimpar 477 . . . . . . . . . . . 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)))))
223219, 222sylan2 593 . . . . . . . . . . 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)))))
224215, 223eqeltrid 2842 . . . . . . . . . 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)))))
225 mptfi 9388 . . . . . . . . . . . . 13 ((0...𝑁) ∈ Fin → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin)
226 rnfi 9377 . . . . . . . . . . . . 13 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin)
22726, 225, 226mp2b 10 . . . . . . . . . . . 12 ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin
228227orci 865 . . . . . . . . . . 11 (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin ∨ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ Fin)
229228a1i 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))
230182, 186, 187, 193, 197, 204, 214, 224, 229mreexexd 17692 . . . . . . . . 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))))))
231230ex 412 . . . . . . . 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)))))))
232 ovex 7463 . . . . . . . . . . . . 13 ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V
233232rnex 7932 . . . . . . . . . . . 12 ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V
234 elpwi 4611 . . . . . . . . . . . 12 (𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁)) → 𝑣 ⊆ ran ((ℂflds ℚ) unitVec (1...𝑁)))
235 ssdomg 9038 . . . . . . . . . . . 12 (ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V → (𝑣 ⊆ ran ((ℂflds ℚ) unitVec (1...𝑁)) → 𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁))))
236233, 234, 235mpsyl 68 . . . . . . . . . . 11 (𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁)) → 𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)))
237 endomtr 9050 . . . . . . . . . . . . . 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)))
238237ancoms 458 . . . . . . . . . . . . 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)))
239 f1f1orn 6859 . . . . . . . . . . . . . . . 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)))
240 ovex 7463 . . . . . . . . . . . . . . . . 17 (0...𝑁) ∈ V
241240f1oen 9011 . . . . . . . . . . . . . . . 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)))
242239, 241syl 17 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
243 endomtr 9050 . . . . . . . . . . . . . . . . 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)))
244198uvcendim 21884 . . . . . . . . . . . . . . . . . . . 20 (((ℂflds ℚ) ∈ NzRing ∧ (1...𝑁) ∈ Fin) → (1...𝑁) ≈ ran ((ℂflds ℚ) unitVec (1...𝑁)))
245159, 22, 244mp2an 692 . . . . . . . . . . . . . . . . . . 19 (1...𝑁) ≈ ran ((ℂflds ℚ) unitVec (1...𝑁))
246245ensymi 9042 . . . . . . . . . . . . . . . . . 18 ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)
247 domentr 9051 . . . . . . . . . . . . . . . . . . 19 (((0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)) → (0...𝑁) ≼ (1...𝑁))
248 hashdom 14414 . . . . . . . . . . . . . . . . . . . . 21 (((0...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((♯‘(0...𝑁)) ≤ (♯‘(1...𝑁)) ↔ (0...𝑁) ≼ (1...𝑁)))
24926, 22, 248mp2an 692 . . . . . . . . . . . . . . . . . . . 20 ((♯‘(0...𝑁)) ≤ (♯‘(1...𝑁)) ↔ (0...𝑁) ≼ (1...𝑁))
250 hashfz0 14467 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (♯‘(0...𝑁)) = (𝑁 + 1))
2512, 250syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘(0...𝑁)) = (𝑁 + 1))
252 hashfz1 14381 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
2532, 252syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘(1...𝑁)) = 𝑁)
254251, 253breq12d 5160 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((♯‘(0...𝑁)) ≤ (♯‘(1...𝑁)) ↔ (𝑁 + 1) ≤ 𝑁))
255249, 254bitr3id 285 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((0...𝑁) ≼ (1...𝑁) ↔ (𝑁 + 1) ≤ 𝑁))
256247, 255imbitrid 244 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)) → (𝑁 + 1) ≤ 𝑁))
257246, 256mpan2i 697 . . . . . . . . . . . . . . . . 17 (𝜑 → ((0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) → (𝑁 + 1) ≤ 𝑁))
258243, 257syl5 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))
259258expd 415 . . . . . . . . . . . . . . 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)))
260242, 259syl5 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)))
261260com23 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)))
262238, 261syl5 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)))
263262expdimp 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)))
264236, 263sylan2 593 . . . . . . . . . 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)))
265264adantrd 491 . . . . . . . . 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)))
266265rexlimdva 3152 . . . . . . . 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)))
267231, 266syld 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)))
268267impd 410 . . . . . 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))
269268ancomsd 465 . . . . 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))
270178, 269sylbird 260 . . . 4 (𝜑 → (¬ ∃𝑤 ∈ ((ℚ ↑m (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → (𝑁 + 1) ≤ 𝑁))
2719, 270mt3d 148 . . 3 (𝜑 → ∃𝑤 ∈ ((ℚ ↑m (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
272 eldifsn 4790 . . . . 5 (𝑤 ∈ ((ℚ ↑m (0...𝑁)) ∖ {((0...𝑁) × {0})}) ↔ (𝑤 ∈ (ℚ ↑m (0...𝑁)) ∧ 𝑤 ≠ ((0...𝑁) × {0})))
27342anim1i 615 . . . . 5 ((𝑤 ∈ (ℚ ↑m (0...𝑁)) ∧ 𝑤 ≠ ((0...𝑁) × {0})) → (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})))
274272, 273sylbi 217 . . . 4 (𝑤 ∈ ((ℚ ↑m (0...𝑁)) ∖ {((0...𝑁) × {0})}) → (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})))
27593a1i 11 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ℚ ⊆ ℂ)
2762adantr 480 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑁 ∈ ℕ0)
277275, 276, 53elplyd 26255 . . . . . . . 8 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ (Poly‘ℚ))
278277adantrr 717 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ (Poly‘ℚ))
279 uzdisj 13633 . . . . . . . . . . . . . . . . . 18 ((0...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ∅
2802nn0cnd 12586 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ ℂ)
281 pncan1 11684 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁)
282280, 281syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
283282oveq2d 7446 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
284283ineq1d 4226 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((0...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))
285279, 284eqtr3id 2788 . . . . . . . . . . . . . . . . 17 (𝜑 → ∅ = ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))
286285eqcomd 2740 . . . . . . . . . . . . . . . 16 (𝜑 → ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅)
287128fconst 6794 . . . . . . . . . . . . . . . . . 18 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0}
288 snssi 4812 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℚ → {0} ⊆ ℚ)
28996, 97, 288mp2b 10 . . . . . . . . . . . . . . . . . . 19 {0} ⊆ ℚ
290289, 93sstri 4004 . . . . . . . . . . . . . . . . . 18 {0} ⊆ ℂ
291 fss 6752 . . . . . . . . . . . . . . . . . 18 ((((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0} ∧ {0} ⊆ ℂ) → ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℂ)
292287, 290, 291mp2an 692 . . . . . . . . . . . . . . . . 17 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℂ
293 fun 6770 . . . . . . . . . . . . . . . . 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))
294292, 293mpanl2 701 . . . . . . . . . . . . . . . 16 ((𝑤:(0...𝑁)⟶ℚ ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
295286, 294sylan2 593 . . . . . . . . . . . . . . 15 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝜑) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
296295ancoms 458 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
297 nn0uz 12917 . . . . . . . . . . . . . . . . . 18 0 = (ℤ‘0)
2986, 297eleqtrdi 2848 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ (ℤ‘0))
299 uzsplit 13632 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ (ℤ‘0) → (ℤ‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
300298, 299syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℤ‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
301297, 300eqtrid 2786 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ0 = ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
302283uneq1d 4176 . . . . . . . . . . . . . . . . 17 (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))))
303301, 302eqtr2d 2775 . . . . . . . . . . . . . . . 16 (𝜑 → ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))) = ℕ0)
304 ssequn1 4195 . . . . . . . . . . . . . . . . . 18 (ℚ ⊆ ℂ ↔ (ℚ ∪ ℂ) = ℂ)
30593, 304mpbi 230 . . . . . . . . . . . . . . . . 17 (ℚ ∪ ℂ) = ℂ
306305a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → (ℚ ∪ ℂ) = ℂ)
307303, 306feq23d 6731 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ) ↔ (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):ℕ0⟶ℂ))
308307adantr 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))
309296, 308mpbid 232 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):ℕ0⟶ℂ)
310 ffn 6736 . . . . . . . . . . . . . . . 16 (𝑤:(0...𝑁)⟶ℚ → 𝑤 Fn (0...𝑁))
311 fnimadisj 6700 . . . . . . . . . . . . . . . 16 ((𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) → (𝑤 “ (ℤ‘(𝑁 + 1))) = ∅)
312310, 286, 311syl2anr 597 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 “ (ℤ‘(𝑁 + 1))) = ∅)
3132nn0zd 12636 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℤ)
314313peano2zd 12722 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ ℤ)
315 uzid 12890 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ ℤ → (𝑁 + 1) ∈ (ℤ‘(𝑁 + 1)))
316 ne0i 4346 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ (ℤ‘(𝑁 + 1)) → (ℤ‘(𝑁 + 1)) ≠ ∅)
317314, 315, 3163syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℤ‘(𝑁 + 1)) ≠ ∅)
318 inidm 4234 . . . . . . . . . . . . . . . . . . 19 ((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) = (ℤ‘(𝑁 + 1))
319318neeq1i 3002 . . . . . . . . . . . . . . . . . 18 (((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅ ↔ (ℤ‘(𝑁 + 1)) ≠ ∅)
320317, 319sylibr 234 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅)
321 xpima2 6205 . . . . . . . . . . . . . . . . 17 (((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅ → (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))) = {0})
322320, 321syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))) = {0})
323322adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))) = {0})
324312, 323uneq12d 4178 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 “ (ℤ‘(𝑁 + 1))) ∪ (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1)))) = (∅ ∪ {0}))
325 imaundir 6172 . . . . . . . . . . . . . 14 ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) “ (ℤ‘(𝑁 + 1))) = ((𝑤 “ (ℤ‘(𝑁 + 1))) ∪ (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))))
326 uncom 4167 . . . . . . . . . . . . . . 15 (∅ ∪ {0}) = ({0} ∪ ∅)
327 un0 4399 . . . . . . . . . . . . . . 15 ({0} ∪ ∅) = {0}
328326, 327eqtr2i 2763 . . . . . . . . . . . . . 14 {0} = (∅ ∪ {0})
329324, 325, 3283eqtr4g 2799 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) “ (ℤ‘(𝑁 + 1))) = {0})
330286, 310anim12ci 614 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅))
331 fnconstg 6796 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ V → ((ℤ‘(𝑁 + 1)) × {0}) Fn (ℤ‘(𝑁 + 1)))
332128, 331ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((ℤ‘(𝑁 + 1)) × {0}) Fn (ℤ‘(𝑁 + 1))
333 fvun1 6999 . . . . . . . . . . . . . . . . . . . 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))
334332, 333mp3an2 1448 . . . . . . . . . . . . . . . . . . 19 ((𝑤 Fn (0...𝑁) ∧ (((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅ ∧ 𝑘 ∈ (0...𝑁))) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
335334anassrs 467 . . . . . . . . . . . . . . . . . 18 (((𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
336330, 335sylan 580 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
337336eqcomd 2740 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) = ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘))
338337oveq1d 7445 . . . . . . . . . . . . . . 15 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝑦𝑘)) = (((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘)))
339338sumeq2dv 15734 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)) = Σ𝑘 ∈ (0...𝑁)(((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘)))
340339mpteq2dv 5249 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘))))
341277, 276, 309, 329, 340coeeq 26280 . . . . . . . . . . . 12 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) = (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})))
342341reseq1d 5998 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)))
343 res0 6003 . . . . . . . . . . . . . 14 (𝑤 ↾ ∅) = ∅
344285reseq2d 5999 . . . . . . . . . . . . . 14 (𝜑 → (𝑤 ↾ ∅) = (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
345 res0 6003 . . . . . . . . . . . . . . 15 (((ℤ‘(𝑁 + 1)) × {0}) ↾ ∅) = ∅
346285reseq2d 5999 . . . . . . . . . . . . . . 15 (𝜑 → (((ℤ‘(𝑁 + 1)) × {0}) ↾ ∅) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
347345, 346eqtr3id 2788 . . . . . . . . . . . . . 14 (𝜑 → ∅ = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
348343, 344, 3473eqtr3a 2798 . . . . . . . . . . . . 13 (𝜑 → (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
349 fss 6752 . . . . . . . . . . . . . . 15 ((((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0} ∧ {0} ⊆ ℚ) → ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℚ)
350287, 289, 349mp2an 692 . . . . . . . . . . . . . 14 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℚ
351 fresaunres1 6781 . . . . . . . . . . . . . 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)
352350, 351mp3an2 1448 . . . . . . . . . . . . 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)
353348, 352sylan2 593 . . . . . . . . . . . 12 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝜑) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
354353ancoms 458 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
355342, 354eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = 𝑤)
356 fveq2 6906 . . . . . . . . . . 11 ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = 0𝑝 → (coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) = (coeff‘0𝑝))
357356reseq1d 5998 . . . . . . . . . 10 ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = 0𝑝 → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((coeff‘0𝑝) ↾ (0...𝑁)))
358 eqtr2 2758 . . . . . . . . . . . 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)))
359 coe0 26309 . . . . . . . . . . . . . 14 (coeff‘0𝑝) = (ℕ0 × {0})
360359reseq1i 5995 . . . . . . . . . . . . 13 ((coeff‘0𝑝) ↾ (0...𝑁)) = ((ℕ0 × {0}) ↾ (0...𝑁))
361 elfznn0 13656 . . . . . . . . . . . . . . 15 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℕ0)
362361ssriv 3998 . . . . . . . . . . . . . 14 (0...𝑁) ⊆ ℕ0
363 xpssres 6037 . . . . . . . . . . . . . 14 ((0...𝑁) ⊆ ℕ0 → ((ℕ0 × {0}) ↾ (0...𝑁)) = ((0...𝑁) × {0}))
364362, 363ax-mp 5 . . . . . . . . . . . . 13 ((ℕ0 × {0}) ↾ (0...𝑁)) = ((0...𝑁) × {0})
365360, 364eqtri 2762 . . . . . . . . . . . 12 ((coeff‘0𝑝) ↾ (0...𝑁)) = ((0...𝑁) × {0})
366358, 365eqtrdi 2790 . . . . . . . . . . 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}))
367366ex 412 . . . . . . . . . 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})))
368355, 357, 367syl2im 40 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = 0𝑝𝑤 = ((0...𝑁) × {0})))
369368necon3d 2958 . . . . . . . 8 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 ≠ ((0...𝑁) × {0}) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ≠ 0𝑝))
370369impr 454 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ≠ 0𝑝)
371 eldifsn 4790 . . . . . . 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))
372278, 370, 371sylanbrc 583 . . . . . 6 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ ((Poly‘ℚ) ∖ {0𝑝}))
373372adantrr 717 . . . . 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}))
374 oveq1 7437 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝑦𝑘) = (𝐴𝑘))
375374oveq2d 7446 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝑤𝑘) · (𝑦𝑘)) = ((𝑤𝑘) · (𝐴𝑘)))
376375sumeq2sdv 15735 . . . . . . . . . 10 (𝑦 = 𝐴 → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
377 eqid 2734 . . . . . . . . . 10 (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))
378 sumex 15720 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) ∈ V
379376, 377, 378fvmpt 7015 . . . . . . . . 9 (𝐴 ∈ ℂ → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
3801, 379syl 17 . . . . . . . 8 (𝜑 → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
381380adantr 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)))
382107adantll 714 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
383 aacllem.2 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
384383adantlr 715 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
385110, 384mulcld 11278 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝐶 · 𝑋) ∈ ℂ)
386385adantllr 719 . . . . . . . . . . . . . 14 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝐶 · 𝑋) ∈ ℂ)
38751, 382, 386fsummulc2 15816 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)) = Σ𝑛 ∈ (1...𝑁)((𝑤𝑘) · (𝐶 · 𝑋)))
388 aacllem.4 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐴𝑘) = Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋))
389388oveq2d 7446 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = ((𝑤𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)))
390389adantlr 715 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = ((𝑤𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)))
391382adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑤𝑘) ∈ ℂ)
392110adantllr 719 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℂ)
393 simpll 767 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → 𝜑)
394393, 383sylan 580 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
395391, 392, 394mulassd 11281 . . . . . . . . . . . . . 14 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((𝑤𝑘) · 𝐶) · 𝑋) = ((𝑤𝑘) · (𝐶 · 𝑋)))
396395sumeq2dv 15734 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁)((𝑤𝑘) · (𝐶 · 𝑋)))
397387, 390, 3963eqtr4d 2784 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
398397sumeq2dv 15734 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑘 ∈ (0...𝑁𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
399107ad2ant2lr 748 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → (𝑤𝑘) ∈ ℂ)
400110anasss 466 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝐶 ∈ ℂ)
401400adantlr 715 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝐶 ∈ ℂ)
402399, 401mulcld 11278 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → ((𝑤𝑘) · 𝐶) ∈ ℂ)
403383ad2ant2rl 749 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝑋 ∈ ℂ)
404402, 403mulcld 11278 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → (((𝑤𝑘) · 𝐶) · 𝑋) ∈ ℂ)
40543, 69, 404fsumcom 15807 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
406398, 405eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
407406adantrr 717 . . . . . . . . 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))
408 nfv 1911 . . . . . . . . . . . 12 𝑛𝜑
409 nfv 1911 . . . . . . . . . . . . 13 𝑛 𝑤:(0...𝑁)⟶ℚ
410 nfra1 3281 . . . . . . . . . . . . 13 𝑛𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0
411409, 410nfan 1896 . . . . . . . . . . . 12 𝑛(𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
412408, 411nfan 1896 . . . . . . . . . . 11 𝑛(𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
413 rspa 3245 . . . . . . . . . . . . . . . 16 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
414413oveq1d 7445 . . . . . . . . . . . . . . 15 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
415414adantll 714 . . . . . . . . . . . . . 14 (((𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
416415adantll 714 . . . . . . . . . . . . 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))
417383adantlr 715 . . . . . . . . . . . . . . 15 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
41892, 417, 113fsummulc1 15817 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
419418adantlrr 721 . . . . . . . . . . . . 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))
420383mul02d 11456 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → (0 · 𝑋) = 0)
421420adantlr 715 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (0 · 𝑋) = 0)
422416, 419, 4213eqtr3d 2782 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = 0)
423422ex 412 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → (𝑛 ∈ (1...𝑁) → Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = 0))
424412, 423ralrimi 3254 . . . . . . . . . 10 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = 0)
425424sumeq2d 15733 . . . . . . . . 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)
426407, 425eqtrd 2774 . . . . . . . 8 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁)0)
42722olci 866 . . . . . . . . 9 ((1...𝑁) ⊆ (ℤ𝐵) ∨ (1...𝑁) ∈ Fin)
428 sumz 15754 . . . . . . . . 9 (((1...𝑁) ⊆ (ℤ𝐵) ∨ (1...𝑁) ∈ Fin) → Σ𝑛 ∈ (1...𝑁)0 = 0)
429427, 428ax-mp 5 . . . . . . . 8 Σ𝑛 ∈ (1...𝑁)0 = 0
430426, 429eqtrdi 2790 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = 0)
431381, 430eqtrd 2774 . . . . . 6 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0)
432431adantrlr 723 . . . . 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)
433 fveq1 6905 . . . . . . 7 (𝑥 = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) → (𝑥𝐴) = ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴))
434433eqeq1d 2736 . . . . . 6 (𝑥 = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) → ((𝑥𝐴) = 0 ↔ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0))
435434rspcev 3621 . . . . 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)
436373, 432, 435syl2anc 584 . . . 4 ((𝜑 ∧ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})) ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
437274, 436sylanr1 682 . . 3 ((𝜑 ∧ (𝑤 ∈ ((ℚ ↑m (0...𝑁)) ∖ {((0...𝑁) × {0})}) ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
438271, 437rexlimddv 3158 . 2 (𝜑 → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
439 elqaa 26378 . 2 (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0))
4401, 438, 439sylanbrc 583 1 (𝜑𝐴 ∈ 𝔸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wnel 3043  wral 3058  wrex 3067  Vcvv 3477  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338  𝒫 cpw 4604  {csn 4630   class class class wbr 5147  cmpt 5230   × cxp 5686  dom cdm 5688  ran crn 5689  cres 5690  cima 5691   Fn wfn 6557  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  f cof 7694  m cmap 8864  cen 8980  cdom 8981  Fincfn 8983  cc 11150  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157   < clt 11292  cle 11293  cmin 11489  0cn0 12523  cz 12610  cuz 12875  cq 12987  ...cfz 13543  cexp 14098  chash 14365  Σcsu 15718  Basecbs 17244  s cress 17273  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17485   Σg cgsu 17486  Moorecmre 17626  mrClscmrc 17627  mrIndcmri 17628  ACScacs 17629  Ringcrg 20250  NzRingcnzr 20528  DivRingcdr 20745  LModclmod 20874  LSubSpclss 20946  LSpanclspn 20986  LBasisclbs 21090  LVecclvec 21118  fldccnfld 21381   freeLMod cfrlm 21783   unitVec cuvc 21819   LIndF clindf 21841  LIndSclinds 21842  0𝑝c0p 25717  Polycply 26237  coeffccoe 26239  𝔸caa 26370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231  ax-mulf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-mri 17632  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-dvr 20417  df-nzr 20529  df-subrng 20562  df-subrg 20586  df-drng 20747  df-lmod 20876  df-lss 20947  df-lsp 20987  df-lmhm 21038  df-lbs 21091  df-lvec 21119  df-sra 21189  df-rgmod 21190  df-cnfld 21382  df-dsmm 21769  df-frlm 21784  df-uvc 21820  df-lindf 21843  df-linds 21844  df-0p 25718  df-ply 26241  df-coe 26243  df-dgr 26244  df-aa 26371
This theorem is referenced by: (None)
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