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Theorem aacllem 49320
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 12588 . . . . . 6 (𝜑𝑁 ∈ ℝ)
43ltp1d 12198 . . . . 5 (𝜑𝑁 < (𝑁 + 1))
5 peano2nn0 12566 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
62, 5syl 17 . . . . . . 7 (𝜑 → (𝑁 + 1) ∈ ℕ0)
76nn0red 12588 . . . . . 6 (𝜑 → (𝑁 + 1) ∈ ℝ)
83, 7ltnled 11408 . . . . 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 1119 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
1211fmpttd 7135 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶):(1...𝑁)⟶ℚ)
13 qex 13003 . . . . . . . . . . 11 ℚ ∈ V
14 ovex 7464 . . . . . . . . . . 11 (1...𝑁) ∈ V
1513, 14elmap 8911 . . . . . . . . . 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 7135 . . . . . . . 8 (𝜑 → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑m (1...𝑁)))
18 eqid 2737 . . . . . . . . . . . 12 (ℂflds ℚ) = (ℂflds ℚ)
1918qdrng 27664 . . . . . . . . . . 11 (ℂflds ℚ) ∈ DivRing
20 drngring 20736 . . . . . . . . . . 11 ((ℂflds ℚ) ∈ DivRing → (ℂflds ℚ) ∈ Ring)
2119, 20ax-mp 5 . . . . . . . . . 10 (ℂflds ℚ) ∈ Ring
22 fzfi 14013 . . . . . . . . . 10 (1...𝑁) ∈ Fin
23 eqid 2737 . . . . . . . . . . 11 ((ℂflds ℚ) freeLMod (1...𝑁)) = ((ℂflds ℚ) freeLMod (1...𝑁))
2423frlmlmod 21769 . . . . . . . . . 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 14013 . . . . . . . . 9 (0...𝑁) ∈ Fin
2718qrngbas 27663 . . . . . . . . . . . 12 ℚ = (Base‘(ℂflds ℚ))
2823, 27frlmfibas 21782 . . . . . . . . . . 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 21773 . . . . . . . . . . 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 2737 . . . . . . . . . 10 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁))) = ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))
3318qrng0 27665 . . . . . . . . . . . 12 0 = (0g‘(ℂflds ℚ))
3423, 33frlm0 21774 . . . . . . . . . . 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 2737 . . . . . . . . . . . 12 ((ℂflds ℚ) freeLMod (0...𝑁)) = ((ℂflds ℚ) freeLMod (0...𝑁))
3736, 27frlmfibas 21782 . . . . . . . . . . 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 21858 . . . . . . . . 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 1453 . . . . . . . 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 8889 . . . . . . . . 9 (𝑤 ∈ (ℚ ↑m (0...𝑁)) → 𝑤:(0...𝑁)⟶ℚ)
43 fzfid 14014 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (0...𝑁) ∈ Fin)
44 fvexd 6921 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ V)
4514mptex 7243 . . . . . . . . . . . . . . . . . . 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 6977 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑤 = (𝑘 ∈ (0...𝑁) ↦ (𝑤𝑘)))
49 eqidd 2738 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
5043, 44, 46, 48, 49offval2 7717 . . . . . . . . . . . . . . . . 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 14014 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
52 ffvelcdm 7101 . . . . . . . . . . . . . . . . . . . . 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 21372 . . . . . . . . . . . . . . . . . . . . . 22 · = (.r‘ℂfld)
5618, 55ressmulr 17351 . . . . . . . . . . . . . . . . . . . . 21 (ℚ ∈ V → · = (.r‘(ℂflds ℚ)))
5713, 56ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 · = (.r‘(ℂflds ℚ))
5823, 29, 27, 51, 53, 54, 32, 57frlmvscafval 21786 . . . . . . . . . . . . . . . . . . 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 5747 . . . . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) = (𝑛 ∈ (1...𝑁) ↦ 𝐶))
6451, 59, 60, 62, 63offval2 7717 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (((1...𝑁) × {(𝑤𝑘)}) ∘f · (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
6558, 64eqtrd 2777 . . . . . . . . . . . . . . . . . 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 5242 . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . . 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 14014 . . . . . . . . . . . . . . . 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 13009 . . . . . . . . . . . . . . . . . . . 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 7135 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
7813, 14elmap 8911 . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
8114mptex 7243 . . . . . . . . . . . . . . . . . 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 5436 . . . . . . . . . . . . . . . . . . 19 {0} ∈ V
8414, 83xpex 7773 . . . . . . . . . . . . . . . . . 18 ((1...𝑁) × {0}) ∈ V
8584a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((1...𝑁) × {0}) ∈ V)
8680, 43, 82, 85fsuppmptdm 9416 . . . . . . . . . . . . . . . 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 21792 . . . . . . . . . . . . . . 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 21368 . . . . . . . . . . . . . . . . . 18 ℂ = (Base‘ℂfld)
89 cnfldadd 21370 . . . . . . . . . . . . . . . . . 18 + = (+g‘ℂfld)
90 cnfldex 21367 . . . . . . . . . . . . . . . . . . 19 fld ∈ V
9190a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℂfld ∈ V)
92 fzfid 14014 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (0...𝑁) ∈ Fin)
93 qsscn 13002 . . . . . . . . . . . . . . . . . . 19 ℚ ⊆ ℂ
9493a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℚ ⊆ ℂ)
9575fmpttd 7135 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(0...𝑁)⟶ℚ)
96 0z 12624 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℤ
97 zq 12996 . . . . . . . . . . . . . . . . . . . 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 11444 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥)
101 addrid 11441 . . . . . . . . . . . . . . . . . . . 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 18695 . . . . . . . . . . . . . . . . 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 13005 . . . . . . . . . . . . . . . . . . . . 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 13005 . . . . . . . . . . . . . . . . . . . . . 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 11281 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℂ)
11492, 113gsumfsum 21452 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (ℂfld Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
115104, 114eqtr3d 2779 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
116115mpteq2dva 5242 . . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . . . 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 13007 . . . . . . . . . . . . . . . . . 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 15768 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ ℚ)
121120fmpttd 7135 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
12213, 14elmap 8911 . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . 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 8889 . . . . . . . . . . . . 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 11255 . . . . . . . . . . . . 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 2905 . . . . . . . . . . . . . 14 𝑛((ℂflds ℚ) freeLMod (1...𝑁))
132 nfcv 2905 . . . . . . . . . . . . . 14 𝑛 Σg
133 nfcv 2905 . . . . . . . . . . . . . . 15 𝑛𝑤
134 nfcv 2905 . . . . . . . . . . . . . . 15 𝑛f ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))
135 nfcv 2905 . . . . . . . . . . . . . . . 16 𝑛(0...𝑁)
136 nfmpt1 5250 . . . . . . . . . . . . . . . 16 𝑛(𝑛 ∈ (1...𝑁) ↦ 𝐶)
137135, 136nfmpt 5249 . . . . . . . . . . . . . . 15 𝑛(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
138133, 134, 137nfov 7461 . . . . . . . . . . . . . 14 𝑛(𝑤f ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
139131, 132, 138nfov 7461 . . . . . . . . . . . . 13 𝑛(((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤f ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))
140 nfcv 2905 . . . . . . . . . . . . 13 𝑛((1...𝑁) × {0})
141139, 140eqfnfv2f 7055 . . . . . . . . . . . 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 15724 . . . . . . . . . . . . . . 15 Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ V
145 eqid 2737 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
146145fvmpt2 7027 . . . . . . . . . . . . . . 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 2797 . . . . . . . . . . . . 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 7224 . . . . . . . . . . . . . 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 2753 . . . . . . . . . . . 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 3176 . . . . . . . . . . 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 3176 . . . . . . 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 20748 . . . . . . . . 9 ((ℂflds ℚ) ∈ DivRing → (ℂflds ℚ) ∈ NzRing)
15919, 158ax-mp 5 . . . . . . . 8 (ℂflds ℚ) ∈ NzRing
16031islindf3 21846 . . . . . . . 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 2737 . . . . . . . . . 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 3047 . . . . . . . . . . 11 (𝑤 ∉ {((0...𝑁) × {0})} ↔ ¬ 𝑤 ∈ {((0...𝑁) × {0})})
170 velsn 4642 . . . . . . . . . . 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 3093 . . . . . . 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 4149 . . . . . . 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 3072 . . . . . . 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 2737 . . . . . . . . . . . . 13 (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))
18029, 179lssmre 20964 . . . . . . . . . . . 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 2737 . . . . . . . . . . . 12 (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))
184 eqid 2737 . . . . . . . . . . . 12 (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) = (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
185179, 183, 184mrclsp 20987 . . . . . . . . . . 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 2737 . . . . . . . . . 10 (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) = (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
18831islvec 21103 . . . . . . . . . . . . 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 21146 . . . . . . . . . . . . 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 4378 . . . . . . . . . . . 12 ((ℚ ↑m (1...𝑁)) ∖ ∅) = (ℚ ↑m (1...𝑁))
196194, 195sseqtrrdi 4025 . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . 15 ((ℂflds ℚ) unitVec (1...𝑁)) = ((ℂflds ℚ) unitVec (1...𝑁))
199198, 23, 29uvcff 21811 . . . . . . . . . . . . . 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 4033 . . . . . . . . . . 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 4394 . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . 16 (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁)))
20823, 198, 207frlmlbs 21817 . . . . . . . . . . . . . . 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 21078 . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . 12 ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)) = (ℚ ↑m (1...𝑁))
213194, 212sseqtrrdi 4025 . . . . . . . . . . 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 4394 . . . . . . . . . . 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 21839 . . . . . . . . . . . . . 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 1450 . . . . . . . . . . . . 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 3146 . . . . . . . . . . . 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 17675 . . . . . . . . . . . . . 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 2845 . . . . . . . . . 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 9391 . . . . . . . . . . . . 13 ((0...𝑁) ∈ Fin → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin)
226 rnfi 9380 . . . . . . . . . . . . 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 866 . . . . . . . . . . 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 17691 . . . . . . . . 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 7464 . . . . . . . . . . . . 13 ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V
233232rnex 7932 . . . . . . . . . . . 12 ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V
234 elpwi 4607 . . . . . . . . . . . 12 (𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁)) → 𝑣 ⊆ ran ((ℂflds ℚ) unitVec (1...𝑁)))
235 ssdomg 9040 . . . . . . . . . . . 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 9052 . . . . . . . . . . . . . 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 7464 . . . . . . . . . . . . . . . . 17 (0...𝑁) ∈ V
241240f1oen 9013 . . . . . . . . . . . . . . . 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 9052 . . . . . . . . . . . . . . . . 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 21867 . . . . . . . . . . . . . . . . . . . 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 9044 . . . . . . . . . . . . . . . . . 18 ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)
247 domentr 9053 . . . . . . . . . . . . . . . . . . 19 (((0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)) → (0...𝑁) ≼ (1...𝑁))
248 hashdom 14418 . . . . . . . . . . . . . . . . . . . . 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 14471 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (♯‘(0...𝑁)) = (𝑁 + 1))
2512, 250syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘(0...𝑁)) = (𝑁 + 1))
252 hashfz1 14385 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
2532, 252syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘(1...𝑁)) = 𝑁)
254251, 253breq12d 5156 . . . . . . . . . . . . . . . . . . . 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 3155 . . . . . . . 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 4786 . . . . 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 26241 . . . . . . . 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 13637 . . . . . . . . . . . . . . . . . 18 ((0...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ∅
2802nn0cnd 12589 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ ℂ)
281 pncan1 11687 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁)
282280, 281syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
283282oveq2d 7447 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
284283ineq1d 4219 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((0...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))
285279, 284eqtr3id 2791 . . . . . . . . . . . . . . . . 17 (𝜑 → ∅ = ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))
286285eqcomd 2743 . . . . . . . . . . . . . . . 16 (𝜑 → ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅)
287128fconst 6794 . . . . . . . . . . . . . . . . . 18 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0}
288 snssi 4808 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℚ → {0} ⊆ ℚ)
28996, 97, 288mp2b 10 . . . . . . . . . . . . . . . . . . 19 {0} ⊆ ℚ
290289, 93sstri 3993 . . . . . . . . . . . . . . . . . 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 12920 . . . . . . . . . . . . . . . . . 18 0 = (ℤ‘0)
2986, 297eleqtrdi 2851 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ (ℤ‘0))
299 uzsplit 13636 . . . . . . . . . . . . . . . . . . 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 2789 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ0 = ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
302283uneq1d 4167 . . . . . . . . . . . . . . . . 17 (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))))
303301, 302eqtr2d 2778 . . . . . . . . . . . . . . . 16 (𝜑 → ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))) = ℕ0)
304 ssequn1 4186 . . . . . . . . . . . . . . . . . 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 12639 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℤ)
314313peano2zd 12725 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ ℤ)
315 uzid 12893 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ ℤ → (𝑁 + 1) ∈ (ℤ‘(𝑁 + 1)))
316 ne0i 4341 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ (ℤ‘(𝑁 + 1)) → (ℤ‘(𝑁 + 1)) ≠ ∅)
317314, 315, 3163syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℤ‘(𝑁 + 1)) ≠ ∅)
318 inidm 4227 . . . . . . . . . . . . . . . . . . 19 ((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) = (ℤ‘(𝑁 + 1))
319318neeq1i 3005 . . . . . . . . . . . . . . . . . 18 (((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅ ↔ (ℤ‘(𝑁 + 1)) ≠ ∅)
320317, 319sylibr 234 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅)
321 xpima2 6204 . . . . . . . . . . . . . . . . 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 4169 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 “ (ℤ‘(𝑁 + 1))) ∪ (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1)))) = (∅ ∪ {0}))
325 imaundir 6170 . . . . . . . . . . . . . 14 ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) “ (ℤ‘(𝑁 + 1))) = ((𝑤 “ (ℤ‘(𝑁 + 1))) ∪ (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))))
326 uncom 4158 . . . . . . . . . . . . . . 15 (∅ ∪ {0}) = ({0} ∪ ∅)
327 un0 4394 . . . . . . . . . . . . . . 15 ({0} ∪ ∅) = {0}
328326, 327eqtr2i 2766 . . . . . . . . . . . . . 14 {0} = (∅ ∪ {0})
329324, 325, 3283eqtr4g 2802 . . . . . . . . . . . . 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 7000 . . . . . . . . . . . . . . . . . . . 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 1451 . . . . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) = ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘))
338337oveq1d 7446 . . . . . . . . . . . . . . 15 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝑦𝑘)) = (((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘)))
339338sumeq2dv 15738 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)) = Σ𝑘 ∈ (0...𝑁)(((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘)))
340339mpteq2dv 5244 . . . . . . . . . . . . 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 26266 . . . . . . . . . . . 12 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) = (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})))
342341reseq1d 5996 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)))
343 res0 6001 . . . . . . . . . . . . . 14 (𝑤 ↾ ∅) = ∅
344285reseq2d 5997 . . . . . . . . . . . . . 14 (𝜑 → (𝑤 ↾ ∅) = (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
345 res0 6001 . . . . . . . . . . . . . . 15 (((ℤ‘(𝑁 + 1)) × {0}) ↾ ∅) = ∅
346285reseq2d 5997 . . . . . . . . . . . . . . 15 (𝜑 → (((ℤ‘(𝑁 + 1)) × {0}) ↾ ∅) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
347345, 346eqtr3id 2791 . . . . . . . . . . . . . 14 (𝜑 → ∅ = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
348343, 344, 3473eqtr3a 2801 . . . . . . . . . . . . 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 1451 . . . . . . . . . . . . 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 2777 . . . . . . . . . 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 5996 . . . . . . . . . 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 2761 . . . . . . . . . . . 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 26295 . . . . . . . . . . . . . 14 (coeff‘0𝑝) = (ℕ0 × {0})
360359reseq1i 5993 . . . . . . . . . . . . 13 ((coeff‘0𝑝) ↾ (0...𝑁)) = ((ℕ0 × {0}) ↾ (0...𝑁))
361 elfznn0 13660 . . . . . . . . . . . . . . 15 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℕ0)
362361ssriv 3987 . . . . . . . . . . . . . 14 (0...𝑁) ⊆ ℕ0
363 xpssres 6036 . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . 12 ((coeff‘0𝑝) ↾ (0...𝑁)) = ((0...𝑁) × {0})
366358, 365eqtrdi 2793 . . . . . . . . . . 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 2961 . . . . . . . 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 4786 . . . . . . 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 7438 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝑦𝑘) = (𝐴𝑘))
375374oveq2d 7447 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝑤𝑘) · (𝑦𝑘)) = ((𝑤𝑘) · (𝐴𝑘)))
376375sumeq2sdv 15739 . . . . . . . . . 10 (𝑦 = 𝐴 → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
377 eqid 2737 . . . . . . . . . 10 (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))
378 sumex 15724 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) ∈ V
379376, 377, 378fvmpt 7016 . . . . . . . . 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 11281 . . . . . . . . . . . . . . 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 15820 . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . 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 11284 . . . . . . . . . . . . . 14 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((𝑤𝑘) · 𝐶) · 𝑋) = ((𝑤𝑘) · (𝐶 · 𝑋)))
396395sumeq2dv 15738 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁)((𝑤𝑘) · (𝐶 · 𝑋)))
397387, 390, 3963eqtr4d 2787 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
398397sumeq2dv 15738 . . . . . . . . . . 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 11281 . . . . . . . . . . . . 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 11281 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → (((𝑤𝑘) · 𝐶) · 𝑋) ∈ ℂ)
40543, 69, 404fsumcom 15811 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
406398, 405eqtrd 2777 . . . . . . . . . 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 1914 . . . . . . . . . . . 12 𝑛𝜑
409 nfv 1914 . . . . . . . . . . . . 13 𝑛 𝑤:(0...𝑁)⟶ℚ
410 nfra1 3284 . . . . . . . . . . . . 13 𝑛𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0
411409, 410nfan 1899 . . . . . . . . . . . 12 𝑛(𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
412408, 411nfan 1899 . . . . . . . . . . 11 𝑛(𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
413 rspa 3248 . . . . . . . . . . . . . . . 16 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
414413oveq1d 7446 . . . . . . . . . . . . . . 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 15821 . . . . . . . . . . . . . 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 11459 . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . 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 3257 . . . . . . . . . 10 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = 0)
425424sumeq2d 15737 . . . . . . . . 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 2777 . . . . . . . 8 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁)0)
42722olci 867 . . . . . . . . 9 ((1...𝑁) ⊆ (ℤ𝐵) ∨ (1...𝑁) ∈ Fin)
428 sumz 15758 . . . . . . . . 9 (((1...𝑁) ⊆ (ℤ𝐵) ∨ (1...𝑁) ∈ Fin) → Σ𝑛 ∈ (1...𝑁)0 = 0)
429427, 428ax-mp 5 . . . . . . . 8 Σ𝑛 ∈ (1...𝑁)0 = 0
430426, 429eqtrdi 2793 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = 0)
431381, 430eqtrd 2777 . . . . . 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 2739 . . . . . 6 (𝑥 = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) → ((𝑥𝐴) = 0 ↔ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0))
435434rspcev 3622 . . . . 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 3161 . 2 (𝜑 → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
439 elqaa 26364 . 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 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wnel 3046  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686  cres 5687  cima 5688   Fn wfn 6556  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  f cof 7695  m cmap 8866  cen 8982  cdom 8983  Fincfn 8985  cc 11153  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cle 11296  cmin 11492  0cn0 12526  cz 12613  cuz 12878  cq 12990  ...cfz 13547  cexp 14102  chash 14369  Σcsu 15722  Basecbs 17247  s cress 17274  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484   Σg cgsu 17485  Moorecmre 17625  mrClscmrc 17626  mrIndcmri 17627  ACScacs 17628  Ringcrg 20230  NzRingcnzr 20512  DivRingcdr 20729  LModclmod 20858  LSubSpclss 20929  LSpanclspn 20969  LBasisclbs 21073  LVecclvec 21101  fldccnfld 21364   freeLMod cfrlm 21766   unitVec cuvc 21802   LIndF clindf 21824  LIndSclinds 21825  0𝑝c0p 25704  Polycply 26223  coeffccoe 26225  𝔸caa 26356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234  ax-mulf 11235
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  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 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-mri 17631  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-nzr 20513  df-subrng 20546  df-subrg 20570  df-drng 20731  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lmhm 21021  df-lbs 21074  df-lvec 21102  df-sra 21172  df-rgmod 21173  df-cnfld 21365  df-dsmm 21752  df-frlm 21767  df-uvc 21803  df-lindf 21826  df-linds 21827  df-0p 25705  df-ply 26227  df-coe 26229  df-dgr 26230  df-aa 26357
This theorem is referenced by: (None)
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