aacllem - Mathbox for David A. Wheeler

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Theorem aacllem 42356

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	⊢ (𝜑 → 𝐴 ∈ ℂ)
aacllem.1	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
aacllem.2	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
aacllem.3	⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
aacllem.4	⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑𝑘) = Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋))

**Assertion**
Ref	Expression
aacllem	⊢ (𝜑 → 𝐴 ∈ 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.

Step	Hyp	Ref	Expression
1		aacllem.0	. 2 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		aacllem.1	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
3	2	nn0red 11229	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
4	3	ltp1d 10833	. . . . 5 ⊢ (𝜑 → 𝑁 < (𝑁 + 1))
5		peano2nn0 11210	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
6	2, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
7	6	nn0red 11229	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
8	3, 7	ltnled 10063	. . . . 5 ⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁))
9	4, 8	mpbid 221	. . . 4 ⊢ (𝜑 → ¬ (𝑁 + 1) ≤ 𝑁)
10		aacllem.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
11	10	3expa 1257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
12		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) ↦ 𝐶) = (𝑛 ∈ (1...𝑁) ↦ 𝐶)
13	11, 12	fmptd 6292	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶):(1...𝑁)⟶ℚ)
14		qex 11676	. . . . . . . . . . 11 ⊢ ℚ ∈ V
15		ovex 6577	. . . . . . . . . . 11 ⊢ (1...𝑁) ∈ V
16	14, 15	elmap 7772	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑_𝑚 (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ 𝐶):(1...𝑁)⟶ℚ)
17	13, 16	sylibr 223	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑_𝑚 (1...𝑁)))
18		eqid 2610	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
19	17, 18	fmptd 6292	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑_𝑚 (1...𝑁)))
20		eqid 2610	. . . . . . . . . . . 12 ⊢ (ℂ_fld ↾_s ℚ) = (ℂ_fld ↾_s ℚ)
21	20	qdrng 25109	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s ℚ) ∈ DivRing
22		drngring 18577	. . . . . . . . . . 11 ⊢ ((ℂ_fld ↾_s ℚ) ∈ DivRing → (ℂ_fld ↾_s ℚ) ∈ Ring)
23	21, 22	ax-mp 5	. . . . . . . . . 10 ⊢ (ℂ_fld ↾_s ℚ) ∈ Ring
24		fzfi 12633	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ Fin
25		eqid 2610	. . . . . . . . . . 11 ⊢ ((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) = ((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))
26	25	frlmlmod 19912	. . . . . . . . . 10 ⊢ (((ℂ_fld ↾_s ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LMod)
27	23, 24, 26	mp2an 704	. . . . . . . . 9 ⊢ ((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LMod
28		fzfi 12633	. . . . . . . . 9 ⊢ (0...𝑁) ∈ Fin
29	20	qrngbas 25108	. . . . . . . . . . . 12 ⊢ ℚ = (Base‘(ℂ_fld ↾_s ℚ))
30	25, 29	frlmfibas 19924	. . . . . . . . . . 11 ⊢ (((ℂ_fld ↾_s ℚ) ∈ DivRing ∧ (1...𝑁) ∈ Fin) → (ℚ ↑_𝑚 (1...𝑁)) = (Base‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))))
31	21, 24, 30	mp2an 704	. . . . . . . . . 10 ⊢ (ℚ ↑_𝑚 (1...𝑁)) = (Base‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))
32	25	frlmsca 19916	. . . . . . . . . . 11 ⊢ (((ℂ_fld ↾_s ℚ) ∈ DivRing ∧ (1...𝑁) ∈ Fin) → (ℂ_fld ↾_s ℚ) = (Scalar‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))))
33	21, 24, 32	mp2an 704	. . . . . . . . . 10 ⊢ (ℂ_fld ↾_s ℚ) = (Scalar‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))
34		eqid 2610	. . . . . . . . . 10 ⊢ ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) = ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))
35	20	qrng0 25110	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘(ℂ_fld ↾_s ℚ))
36	25, 35	frlm0 19917	. . . . . . . . . . 11 ⊢ (((ℂ_fld ↾_s ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) × {0}) = (0_g‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))))
37	23, 24, 36	mp2an 704	. . . . . . . . . 10 ⊢ ((1...𝑁) × {0}) = (0_g‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))
38		eqid 2610	. . . . . . . . . . . 12 ⊢ ((ℂ_fld ↾_s ℚ) freeLMod (0...𝑁)) = ((ℂ_fld ↾_s ℚ) freeLMod (0...𝑁))
39	38, 29	frlmfibas 19924	. . . . . . . . . . 11 ⊢ (((ℂ_fld ↾_s ℚ) ∈ DivRing ∧ (0...𝑁) ∈ Fin) → (ℚ ↑_𝑚 (0...𝑁)) = (Base‘((ℂ_fld ↾_s ℚ) freeLMod (0...𝑁))))
40	21, 28, 39	mp2an 704	. . . . . . . . . 10 ⊢ (ℚ ↑_𝑚 (0...𝑁)) = (Base‘((ℂ_fld ↾_s ℚ) freeLMod (0...𝑁)))
41	31, 33, 34, 37, 35, 40	islindf4 19996	. . . . . . . . 9 ⊢ ((((ℂ_fld ↾_s 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 ((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁))((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0}))))
42	27, 28, 41	mp3an12 1406	. . . . . . . 8 ⊢ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑_𝑚 (1...𝑁)) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁))((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0}))))
43	19, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁))((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0}))))
44		elmapi 7765	. . . . . . . . 9 ⊢ (𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁)) → 𝑤:(0...𝑁)⟶ℚ)
45		fzfid 12634	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (0...𝑁) ∈ Fin)
46		fvex 6113	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤‘𝑘) ∈ V
47	46	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤‘𝑘) ∈ V)
48	15	mptex 6390	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ V
49	48	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ V)
50		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → 𝑤:(0...𝑁)⟶ℚ)
51	50	feqmptd 6159	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → 𝑤 = (𝑘 ∈ (0...𝑁) ↦ (𝑤‘𝑘)))
52		eqidd 2611	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
53	45, 47, 49, 51, 52	offval2 6812	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ ((𝑤‘𝑘)( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶))))
54		fzfid 12634	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
55		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑘 ∈ (0...𝑁)) → (𝑤‘𝑘) ∈ ℚ)
56	55	adantll 746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤‘𝑘) ∈ ℚ)
57	17	adantlr 747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑_𝑚 (1...𝑁)))
58		cnfldmul 19573	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ · = (._r‘ℂ_fld)
59	20, 58	ressmulr 15829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℚ ∈ V → · = (._r‘(ℂ_fld ↾_s ℚ)))
60	14, 59	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ · = (._r‘(ℂ_fld ↾_s ℚ))
61	25, 31, 29, 54, 56, 57, 34, 60	frlmvscafval 19928	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤‘𝑘)( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (((1...𝑁) × {(𝑤‘𝑘)}) ∘_𝑓 · (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
62	46	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑤‘𝑘) ∈ V)
63	11	adantllr 751	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
64		fconstmpt 5085	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑁) × {(𝑤‘𝑘)}) = (𝑛 ∈ (1...𝑁) ↦ (𝑤‘𝑘))
65	64	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((1...𝑁) × {(𝑤‘𝑘)}) = (𝑛 ∈ (1...𝑁) ↦ (𝑤‘𝑘)))
66		eqidd 2611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) = (𝑛 ∈ (1...𝑁) ↦ 𝐶))
67	54, 62, 63, 65, 66	offval2 6812	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (((1...𝑁) × {(𝑤‘𝑘)}) ∘_𝑓 · (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)))
68	61, 67	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤‘𝑘)( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)))
69	68	mpteq2dva 4672	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ ((𝑤‘𝑘)( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶))))
70	53, 69	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶))))
71	70	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)))))
72		fzfid 12634	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (1...𝑁) ∈ Fin)
73	23	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (ℂ_fld ↾_s ℚ) ∈ Ring)
74	56	adantlr 747	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤‘𝑘) ∈ ℚ)
75	11	an32s 842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℚ)
76	75	adantllr 751	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℚ)
77		qmulcl 11682	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑤‘𝑘) ∈ ℚ ∧ 𝐶 ∈ ℚ) → ((𝑤‘𝑘) · 𝐶) ∈ ℚ)
78	74, 76, 77	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤‘𝑘) · 𝐶) ∈ ℚ)
79	78	an32s 842	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑤‘𝑘) · 𝐶) ∈ ℚ)
80		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶))
81	79, 80	fmptd 6292	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
82	14, 15	elmap 7772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)) ∈ (ℚ ↑_𝑚 (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
83	81, 82	sylibr 223	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)) ∈ (ℚ ↑_𝑚 (1...𝑁)))
84		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)))
85	15	mptex 6390	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)) ∈ V
86	85	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)) ∈ V)
87		snex 4835	. . . . . . . . . . . . . . . . . . 19 ⊢ {0} ∈ V
88	15, 87	xpex 6860	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...𝑁) × {0}) ∈ V
89	88	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ((1...𝑁) × {0}) ∈ V)
90	84, 45, 86, 89	fsuppmptdm 8169	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶))) finSupp ((1...𝑁) × {0}))
91	25, 31, 37, 72, 45, 73, 83, 90	frlmgsum 19930	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)))) = (𝑛 ∈ (1...𝑁) ↦ ((ℂ_fld ↾_s ℚ) Σ_g (𝑘 ∈ (0...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)))))
92		cnfldbas 19571	. . . . . . . . . . . . . . . . . 18 ⊢ ℂ = (Base‘ℂ_fld)
93		cnfldadd 19572	. . . . . . . . . . . . . . . . . 18 ⊢ + = (+_g‘ℂ_fld)
94		cnfldex 19570	. . . . . . . . . . . . . . . . . . 19 ⊢ ℂ_fld ∈ V
95	94	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℂ_fld ∈ V)
96		fzfid 12634	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (0...𝑁) ∈ Fin)
97		qsscn 11675	. . . . . . . . . . . . . . . . . . 19 ⊢ ℚ ⊆ ℂ
98	97	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℚ ⊆ ℂ)
99		eqid 2610	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ ((𝑤‘𝑘) · 𝐶))
100	78, 99	fmptd 6292	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ (0...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)):(0...𝑁)⟶ℚ)
101		0z 11265	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ ℤ
102		zq 11670	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
103	101, 102	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℚ
104	103	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 0 ∈ ℚ)
105		addid2 10098	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥)
106		addid1 10095	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → (𝑥 + 0) = 𝑥)
107	105, 106	jca 553	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
108	107	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑥 ∈ ℂ) → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
109	92, 93, 20, 95, 96, 98, 100, 104, 108	gsumress 17099	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (ℂ_fld Σ_g (𝑘 ∈ (0...𝑁) ↦ ((𝑤‘𝑘) · 𝐶))) = ((ℂ_fld ↾_s ℚ) Σ_g (𝑘 ∈ (0...𝑁) ↦ ((𝑤‘𝑘) · 𝐶))))
110		simplr 788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 𝑤:(0...𝑁)⟶ℚ)
111		qcn 11678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤‘𝑘) ∈ ℚ → (𝑤‘𝑘) ∈ ℂ)
112	55, 111	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑘 ∈ (0...𝑁)) → (𝑤‘𝑘) ∈ ℂ)
113	110, 112	sylan 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤‘𝑘) ∈ ℂ)
114		qcn 11678	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐶 ∈ ℚ → 𝐶 ∈ ℂ)
115	11, 114	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℂ)
116	115	an32s 842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℂ)
117	116	adantllr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℂ)
118	113, 117	mulcld 9939	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤‘𝑘) · 𝐶) ∈ ℂ)
119	96, 118	gsumfsum 19632	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (ℂ_fld Σ_g (𝑘 ∈ (0...𝑁) ↦ ((𝑤‘𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶))
120	109, 119	eqtr3d 2646	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ((ℂ_fld ↾_s ℚ) Σ_g (𝑘 ∈ (0...𝑁) ↦ ((𝑤‘𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶))
121	120	mpteq2dva 4672	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ ((ℂ_fld ↾_s ℚ) Σ_g (𝑘 ∈ (0...𝑁) ↦ ((𝑤‘𝑘) · 𝐶)))) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶)))
122	71, 91, 121	3eqtrd 2648	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶)))
123		qaddcl 11680	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ) → (𝑥 + 𝑦) ∈ ℚ)
124	123	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 + 𝑦) ∈ ℚ)
125	98, 124, 96, 78, 104	fsumcllem 14310	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) ∈ ℚ)
126		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶))
127	125, 126	fmptd 6292	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
128	14, 15	elmap 7772	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶)) ∈ (ℚ ↑_𝑚 (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
129	127, 128	sylibr 223	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶)) ∈ (ℚ ↑_𝑚 (1...𝑁)))
130	122, 129	eqeltrd 2688	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) ∈ (ℚ ↑_𝑚 (1...𝑁)))
131		elmapi 7765	. . . . . . . . . . . . 13 ⊢ ((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) ∈ (ℚ ↑_𝑚 (1...𝑁)) → (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))):(1...𝑁)⟶ℚ)
132		ffn 5958	. . . . . . . . . . . . 13 ⊢ ((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))):(1...𝑁)⟶ℚ → (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) Fn (1...𝑁))
133	130, 131, 132	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) Fn (1...𝑁))
134		c0ex 9913	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
135		fnconstg 6006	. . . . . . . . . . . . 13 ⊢ (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
136	134, 135	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((1...𝑁) × {0}) Fn (1...𝑁)
137		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))
138		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 Σ_g
139		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝑤
140		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))
141		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛(0...𝑁)
142		nfmpt1 4675	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛(𝑛 ∈ (1...𝑁) ↦ 𝐶)
143	141, 142	nfmpt 4674	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
144	139, 140, 143	nfov 6575	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
145	137, 138, 144	nfov 6575	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))
146		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((1...𝑁) × {0})
147	145, 146	eqfnfv2f 6223	. . . . . . . . . . . 12 ⊢ (((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) Fn (1...𝑁) ∧ ((1...𝑁) × {0}) Fn (1...𝑁)) → ((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) ↔ ∀𝑛 ∈ (1...𝑁)((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛)))
148	133, 136, 147	sylancl 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) ↔ ∀𝑛 ∈ (1...𝑁)((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛)))
149	122	fveq1d 6105	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶))‘𝑛))
150		sumex 14266	. . . . . . . . . . . . . . 15 ⊢ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) ∈ V
151	126	fvmpt2 6200	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (1...𝑁) ∧ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) ∈ V) → ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶))
152	150, 151	mpan2 703	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶))
153	149, 152	sylan9eq 2664	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶))
154	134	fvconst2 6374	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
155	154	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
156	153, 155	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛) ↔ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0))
157	156	ralbidva 2968	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (∀𝑛 ∈ (1...𝑁)((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0))
158	148, 157	bitrd 267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) ↔ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0))
159	158	imbi1d 330	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0})) ↔ (∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
160	44, 159	sylan2 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁))) → (((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0})) ↔ (∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
161	160	ralbidva 2968	. . . . . . 7 ⊢ (𝜑 → (∀𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁))((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) Σ_g (𝑤 ∘_𝑓 ( ·_𝑠 ‘((ℂ_fld ↾_s 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...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
162	43, 161	bitrd 267	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁))(∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
163		drngnzr 19083	. . . . . . . . 9 ⊢ ((ℂ_fld ↾_s ℚ) ∈ DivRing → (ℂ_fld ↾_s ℚ) ∈ NzRing)
164	21, 163	ax-mp 5	. . . . . . . 8 ⊢ (ℂ_fld ↾_s ℚ) ∈ NzRing
165	33	islindf3 19984	. . . . . . . 8 ⊢ ((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂ_fld ↾_s ℚ) ∈ NzRing) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂ_fld ↾_s 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‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))))))
166	27, 164, 165	mp2an 704	. . . . . . 7 ⊢ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂ_fld ↾_s 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‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))))
167	48, 18	dmmpti 5936	. . . . . . . . 9 ⊢ dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (0...𝑁)
168		f1eq2 6010	. . . . . . . . 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))
169	167, 168	ax-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)
170	169	anbi1i 727	. . . . . . 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‘((ℂ_fld ↾_s 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‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))))
171	166, 170	bitri 263	. . . . . 6 ⊢ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂ_fld ↾_s 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‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))))
172		con34b 305	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ (¬ 𝑤 = ((0...𝑁) × {0}) → ¬ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0))
173		df-nel 2783	. . . . . . . . . . 11 ⊢ (𝑤 ∉ {((0...𝑁) × {0})} ↔ ¬ 𝑤 ∈ {((0...𝑁) × {0})})
174		velsn 4141	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {((0...𝑁) × {0})} ↔ 𝑤 = ((0...𝑁) × {0}))
175	173, 174	xchbinx 323	. . . . . . . . . 10 ⊢ (𝑤 ∉ {((0...𝑁) × {0})} ↔ ¬ 𝑤 = ((0...𝑁) × {0}))
176	175	imbi1i 338	. . . . . . . . 9 ⊢ ((𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0) ↔ (¬ 𝑤 = ((0...𝑁) × {0}) → ¬ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0))
177	172, 176	bitr4i 266	. . . . . . . 8 ⊢ ((∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ (𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0))
178	177	ralbii 2963	. . . . . . 7 ⊢ (∀𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁))(∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ ∀𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁))(𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0))
179		raldifb 3712	. . . . . . 7 ⊢ (∀𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁))(𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0) ↔ ∀𝑤 ∈ ((ℚ ↑_𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ¬ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)
180		ralnex 2975	. . . . . . 7 ⊢ (∀𝑤 ∈ ((ℚ ↑_𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ¬ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 ↔ ¬ ∃𝑤 ∈ ((ℚ ↑_𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)
181	178, 179, 180	3bitri 285	. . . . . 6 ⊢ (∀𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁))(∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ ¬ ∃𝑤 ∈ ((ℚ ↑_𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)
182	162, 171, 181	3bitr3g 301	. . . . 5 ⊢ (𝜑 → (((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) ↔ ¬ ∃𝑤 ∈ ((ℚ ↑_𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0))
183		eqid 2610	. . . . . . . . . . . . 13 ⊢ (LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) = (LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))
184	31, 183	lssmre 18787	. . . . . . . . . . . 12 ⊢ (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LMod → (LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑_𝑚 (1...𝑁))))
185	27, 184	ax-mp 5	. . . . . . . . . . 11 ⊢ (LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑_𝑚 (1...𝑁)))
186	185	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) → (LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑_𝑚 (1...𝑁))))
187		eqid 2610	. . . . . . . . . . . 12 ⊢ (LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) = (LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))
188		eqid 2610	. . . . . . . . . . . 12 ⊢ (mrCls‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) = (mrCls‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))))
189	183, 187, 188	mrclsp 18810	. . . . . . . . . . 11 ⊢ (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LMod → (LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) = (mrCls‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))))
190	27, 189	ax-mp 5	. . . . . . . . . 10 ⊢ (LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) = (mrCls‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))))
191		eqid 2610	. . . . . . . . . 10 ⊢ (mrInd‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) = (mrInd‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))))
192	33	islvec 18925	. . . . . . . . . . . . 13 ⊢ (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LVec ↔ (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing))
193	27, 21, 192	mpbir2an 957	. . . . . . . . . . . 12 ⊢ ((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LVec
194	183, 190, 31	lssacsex 18965	. . . . . . . . . . . . 13 ⊢ (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LVec → ((LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) ∈ (ACS‘(ℚ ↑_𝑚 (1...𝑁))) ∧ ∀𝑧 ∈ 𝒫 (ℚ ↑_𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑_𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦}))))
195	194	simprd 478	. . . . . . . . . . . 12 ⊢ (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LVec → ∀𝑧 ∈ 𝒫 (ℚ ↑_𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑_𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦})))
196	193, 195	ax-mp 5	. . . . . . . . . . 11 ⊢ ∀𝑧 ∈ 𝒫 (ℚ ↑_𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑_𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦}))
197	196	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) → ∀𝑧 ∈ 𝒫 (ℚ ↑_𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑_𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦})))
198		frn 5966	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑_𝑚 (1...𝑁)) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ (ℚ ↑_𝑚 (1...𝑁)))
199	19, 198	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ (ℚ ↑_𝑚 (1...𝑁)))
200		dif0 3904	. . . . . . . . . . . 12 ⊢ ((ℚ ↑_𝑚 (1...𝑁)) ∖ ∅) = (ℚ ↑_𝑚 (1...𝑁))
201	199, 200	syl6sseqr 3615	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((ℚ ↑_𝑚 (1...𝑁)) ∖ ∅))
202	201	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((ℚ ↑_𝑚 (1...𝑁)) ∖ ∅))
203		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) = ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))
204	203, 25, 31	uvcff 19949	. . . . . . . . . . . . . 14 ⊢ (((ℂ_fld ↾_s ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑_𝑚 (1...𝑁)))
205	23, 24, 204	mp2an 704	. . . . . . . . . . . . 13 ⊢ ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑_𝑚 (1...𝑁))
206		frn 5966	. . . . . . . . . . . . 13 ⊢ (((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑_𝑚 (1...𝑁)) → ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ⊆ (ℚ ↑_𝑚 (1...𝑁)))
207	205, 206	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ⊆ (ℚ ↑_𝑚 (1...𝑁))
208	207, 200	sseqtr4i 3601	. . . . . . . . . . 11 ⊢ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ⊆ ((ℚ ↑_𝑚 (1...𝑁)) ∖ ∅)
209	208	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) → ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ⊆ ((ℚ ↑_𝑚 (1...𝑁)) ∖ ∅))
210		un0 3919	. . . . . . . . . . . . . 14 ⊢ (ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∪ ∅) = ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))
211	210	fveq2i 6106	. . . . . . . . . . . . 13 ⊢ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∪ ∅)) = ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)))
212		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (LBasis‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) = (LBasis‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))
213	25, 203, 212	frlmlbs 19955	. . . . . . . . . . . . . . 15 ⊢ (((ℂ_fld ↾_s ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))))
214	23, 24, 213	mp2an 704	. . . . . . . . . . . . . 14 ⊢ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))
215	31, 212, 187	lbssp 18900	. . . . . . . . . . . . . 14 ⊢ (ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) → ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))) = (ℚ ↑_𝑚 (1...𝑁)))
216	214, 215	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))) = (ℚ ↑_𝑚 (1...𝑁))
217	211, 216	eqtri 2632	. . . . . . . . . . . 12 ⊢ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∪ ∅)) = (ℚ ↑_𝑚 (1...𝑁))
218	199, 217	syl6sseqr 3615	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∪ ∅)))
219	218	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∪ ∅)))
220		un0 3919	. . . . . . . . . . 11 ⊢ (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∪ ∅) = ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
221	27, 164	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂ_fld ↾_s ℚ) ∈ NzRing)
222	187, 33	lindsind2 19977	. . . . . . . . . . . . . 14 ⊢ (((((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂ_fld ↾_s ℚ) ∈ NzRing) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) ∧ 𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) → ¬ 𝑥 ∈ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥})))
223	221, 222	mp3an1 1403	. . . . . . . . . . . . 13 ⊢ ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) ∧ 𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) → ¬ 𝑥 ∈ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥})))
224	223	ralrimiva 2949	. . . . . . . . . . . 12 ⊢ (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) → ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥})))
225	190, 191	ismri2 16115	. . . . . . . . . . . . . 14 ⊢ (((LSubSp‘((ℂ_fld ↾_s 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‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) ↔ ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥}))))
226	185, 199, 225	sylancr 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (mrInd‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) ↔ ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥}))))
227	226	biimpar 501	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥}))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (mrInd‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))))
228	224, 227	sylan2 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (mrInd‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))))
229	220, 228	syl5eqel 2692	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))))
230		mptfi 8148	. . . . . . . . . . . . 13 ⊢ ((0...𝑁) ∈ Fin → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin)
231		rnfi 8132	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin)
232	28, 230, 231	mp2b 10	. . . . . . . . . . . 12 ⊢ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin
233	232	orci 404	. . . . . . . . . . 11 ⊢ (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin ∨ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∈ Fin)
234	233	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin ∨ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∈ Fin))
235	186, 190, 191, 197, 202, 209, 219, 229, 234	mreexexd 16131	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) → ∃𝑣 ∈ 𝒫 ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))))))
236	235	ex 449	. . . . . . . 8 ⊢ (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) → ∃𝑣 ∈ 𝒫 ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))))))
237		ovex 6577	. . . . . . . . . . . . 13 ⊢ ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∈ V
238	237	rnex 6992	. . . . . . . . . . . 12 ⊢ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∈ V
239		elpwi 4117	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝒫 ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) → 𝑣 ⊆ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)))
240		ssdomg 7887	. . . . . . . . . . . 12 ⊢ (ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∈ V → (𝑣 ⊆ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) → 𝑣 ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))))
241	238, 239, 240	mpsyl 66	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝒫 ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) → 𝑣 ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)))
242		endomtr 7900	. . . . . . . . . . . . . 14 ⊢ ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ 𝑣 ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)))
243	242	ancoms 468	. . . . . . . . . . . . 13 ⊢ ((𝑣 ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)))
244		f1f1orn 6061	. . . . . . . . . . . . . . . 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)))
245		ovex 6577	. . . . . . . . . . . . . . . . 17 ⊢ (0...𝑁) ∈ V
246	245	f1oen 7862	. . . . . . . . . . . . . . . 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)))
247	244, 246	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
248		endomtr 7900	. . . . . . . . . . . . . . . . 17 ⊢ (((0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))) → (0...𝑁) ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)))
249	203	uvcendim 20005	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℂ_fld ↾_s ℚ) ∈ NzRing ∧ (1...𝑁) ∈ Fin) → (1...𝑁) ≈ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)))
250	164, 24, 249	mp2an 704	. . . . . . . . . . . . . . . . . . 19 ⊢ (1...𝑁) ≈ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))
251	250	ensymi 7892	. . . . . . . . . . . . . . . . . 18 ⊢ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)
252		domentr 7901	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0...𝑁) ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∧ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)) → (0...𝑁) ≼ (1...𝑁))
253		hashdom 13029	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((0...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((#‘(0...𝑁)) ≤ (#‘(1...𝑁)) ↔ (0...𝑁) ≼ (1...𝑁)))
254	28, 24, 253	mp2an 704	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘(0...𝑁)) ≤ (#‘(1...𝑁)) ↔ (0...𝑁) ≼ (1...𝑁))
255		hashfz0 13079	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → (#‘(0...𝑁)) = (𝑁 + 1))
256	2, 255	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (#‘(0...𝑁)) = (𝑁 + 1))
257		hashfz1 12996	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → (#‘(1...𝑁)) = 𝑁)
258	2, 257	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (#‘(1...𝑁)) = 𝑁)
259	256, 258	breq12d 4596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((#‘(0...𝑁)) ≤ (#‘(1...𝑁)) ↔ (𝑁 + 1) ≤ 𝑁))
260	254, 259	syl5bbr 273	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((0...𝑁) ≼ (1...𝑁) ↔ (𝑁 + 1) ≤ 𝑁))
261	252, 260	syl5ib 233	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((0...𝑁) ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ∧ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)) → (𝑁 + 1) ≤ 𝑁))
262	251, 261	mpan2i 709	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0...𝑁) ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) → (𝑁 + 1) ≤ 𝑁))
263	248, 262	syl5 33	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))) → (𝑁 + 1) ≤ 𝑁))
264	263	expd 451	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) → (𝑁 + 1) ≤ 𝑁)))
265	247, 264	syl5 33	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) → (𝑁 + 1) ≤ 𝑁)))
266	265	com23 84	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁)) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
267	243, 266	syl5 33	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑣 ≼ ran ((ℂ_fld ↾_s 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)))
268	267	expdimp 452	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ≼ ran ((ℂ_fld ↾_s 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)))
269	241, 268	sylan2 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝒫 ran ((ℂ_fld ↾_s 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)))
270	269	adantrd 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝒫 ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))) → ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))))) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
271	270	rexlimdva 3013	. . . . . . . 8 ⊢ (𝜑 → (∃𝑣 ∈ 𝒫 ran ((ℂ_fld ↾_s ℚ) unitVec (1...𝑁))(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))))) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
272	236, 271	syld 46	. . . . . . 7 ⊢ (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
273	272	impd 446	. . . . . 6 ⊢ (𝜑 → ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁))) ∧ (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V) → (𝑁 + 1) ≤ 𝑁))
274	273	ancomsd 469	. . . . 5 ⊢ (𝜑 → (((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂ_fld ↾_s ℚ) freeLMod (1...𝑁)))) → (𝑁 + 1) ≤ 𝑁))
275	182, 274	sylbird 249	. . . 4 ⊢ (𝜑 → (¬ ∃𝑤 ∈ ((ℚ ↑_𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 → (𝑁 + 1) ≤ 𝑁))
276	9, 275	mt3d 139	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ ((ℚ ↑_𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)
277		eldifsn 4260	. . . . 5 ⊢ (𝑤 ∈ ((ℚ ↑_𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ↔ (𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁)) ∧ 𝑤 ≠ ((0...𝑁) × {0})))
278	44	anim1i 590	. . . . 5 ⊢ ((𝑤 ∈ (ℚ ↑_𝑚 (0...𝑁)) ∧ 𝑤 ≠ ((0...𝑁) × {0})) → (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})))
279	277, 278	sylbi 206	. . . 4 ⊢ (𝑤 ∈ ((ℚ ↑_𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) → (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})))
280	97	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ℚ ⊆ ℂ)
281	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → 𝑁 ∈ ℕ₀)
282	280, 281, 56	elplyd 23762	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) ∈ (Poly‘ℚ))
283	282	adantrr 749	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) ∈ (Poly‘ℚ))
284		uzdisj 12282	. . . . . . . . . . . . . . . . . 18 ⊢ ((0...((𝑁 + 1) − 1)) ∩ (ℤ_≥‘(𝑁 + 1))) = ∅
285	2	nn0cnd 11230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ ℂ)
286		pncan1 10333	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁)
287	285, 286	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
288	287	oveq2d 6565	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
289	288	ineq1d 3775	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((0...((𝑁 + 1) − 1)) ∩ (ℤ_≥‘(𝑁 + 1))) = ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1))))
290	284, 289	syl5eqr 2658	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∅ = ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1))))
291	290	eqcomd 2616	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1))) = ∅)
292	134	fconst 6004	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℤ_≥‘(𝑁 + 1)) × {0}):(ℤ_≥‘(𝑁 + 1))⟶{0}
293		snssi 4280	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ ℚ → {0} ⊆ ℚ)
294	101, 102, 293	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ {0} ⊆ ℚ
295	294, 97	sstri 3577	. . . . . . . . . . . . . . . . . 18 ⊢ {0} ⊆ ℂ
296		fss 5969	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℤ_≥‘(𝑁 + 1)) × {0}):(ℤ_≥‘(𝑁 + 1))⟶{0} ∧ {0} ⊆ ℂ) → ((ℤ_≥‘(𝑁 + 1)) × {0}):(ℤ_≥‘(𝑁 + 1))⟶ℂ)
297	292, 295, 296	mp2an 704	. . . . . . . . . . . . . . . . 17 ⊢ ((ℤ_≥‘(𝑁 + 1)) × {0}):(ℤ_≥‘(𝑁 + 1))⟶ℂ
298		fun 5979	. . . . . . . . . . . . . . . . 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))
299	297, 298	mpanl2 713	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤:(0...𝑁)⟶ℚ ∧ ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1))) = ∅) → (𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
300	291, 299	sylan2 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝜑) → (𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
301	300	ancoms 468	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
302		nn0uz 11598	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ₀ = (ℤ_≥‘0)
303	6, 302	syl6eleq 2698	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘0))
304		uzsplit 12281	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘0) → (ℤ_≥‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))))
305	303, 304	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℤ_≥‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))))
306	302, 305	syl5eq 2656	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ₀ = ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))))
307	288	uneq1d 3728	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
308	306, 307	eqtr2d 2645	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))) = ℕ₀)
309		ssequn1 3745	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ ⊆ ℂ ↔ (ℚ ∪ ℂ) = ℂ)
310	97, 309	mpbi 219	. . . . . . . . . . . . . . . . 17 ⊢ (ℚ ∪ ℂ) = ℂ
311	310	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℚ ∪ ℂ) = ℂ)
312	308, 311	feq23d 5953	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1)))⟶(ℚ ∪ ℂ) ↔ (𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})):ℕ₀⟶ℂ))
313	312	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1)))⟶(ℚ ∪ ℂ) ↔ (𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})):ℕ₀⟶ℂ))
314	301, 313	mpbid 221	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})):ℕ₀⟶ℂ)
315		ffn 5958	. . . . . . . . . . . . . . . 16 ⊢ (𝑤:(0...𝑁)⟶ℚ → 𝑤 Fn (0...𝑁))
316		fnimadisj 5925	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1))) = ∅) → (𝑤 “ (ℤ_≥‘(𝑁 + 1))) = ∅)
317	315, 291, 316	syl2anr 494	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑤 “ (ℤ_≥‘(𝑁 + 1))) = ∅)
318	2	nn0zd 11356	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℤ)
319	318	peano2zd 11361	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
320		uzid 11578	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 + 1) ∈ ℤ → (𝑁 + 1) ∈ (ℤ_≥‘(𝑁 + 1)))
321		ne0i 3880	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘(𝑁 + 1)) → (ℤ_≥‘(𝑁 + 1)) ≠ ∅)
322	319, 320, 321	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℤ_≥‘(𝑁 + 1)) ≠ ∅)
323		inidm 3784	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℤ_≥‘(𝑁 + 1)) ∩ (ℤ_≥‘(𝑁 + 1))) = (ℤ_≥‘(𝑁 + 1))
324	323	neeq1i 2846	. . . . . . . . . . . . . . . . . 18 ⊢ (((ℤ_≥‘(𝑁 + 1)) ∩ (ℤ_≥‘(𝑁 + 1))) ≠ ∅ ↔ (ℤ_≥‘(𝑁 + 1)) ≠ ∅)
325	322, 324	sylibr 223	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((ℤ_≥‘(𝑁 + 1)) ∩ (ℤ_≥‘(𝑁 + 1))) ≠ ∅)
326		xpima2 5497	. . . . . . . . . . . . . . . . 17 ⊢ (((ℤ_≥‘(𝑁 + 1)) ∩ (ℤ_≥‘(𝑁 + 1))) ≠ ∅ → (((ℤ_≥‘(𝑁 + 1)) × {0}) “ (ℤ_≥‘(𝑁 + 1))) = {0})
327	325, 326	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((ℤ_≥‘(𝑁 + 1)) × {0}) “ (ℤ_≥‘(𝑁 + 1))) = {0})
328	327	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (((ℤ_≥‘(𝑁 + 1)) × {0}) “ (ℤ_≥‘(𝑁 + 1))) = {0})
329	317, 328	uneq12d 3730	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ((𝑤 “ (ℤ_≥‘(𝑁 + 1))) ∪ (((ℤ_≥‘(𝑁 + 1)) × {0}) “ (ℤ_≥‘(𝑁 + 1)))) = (∅ ∪ {0}))
330		imaundir 5465	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})) “ (ℤ_≥‘(𝑁 + 1))) = ((𝑤 “ (ℤ_≥‘(𝑁 + 1))) ∪ (((ℤ_≥‘(𝑁 + 1)) × {0}) “ (ℤ_≥‘(𝑁 + 1))))
331		uncom 3719	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ {0}) = ({0} ∪ ∅)
332		un0 3919	. . . . . . . . . . . . . . 15 ⊢ ({0} ∪ ∅) = {0}
333	331, 332	eqtr2i 2633	. . . . . . . . . . . . . 14 ⊢ {0} = (∅ ∪ {0})
334	329, 330, 333	3eqtr4g 2669	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})) “ (ℤ_≥‘(𝑁 + 1))) = {0})
335	291, 315	anim12ci 589	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1))) = ∅))
336		fnconstg 6006	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 ∈ V → ((ℤ_≥‘(𝑁 + 1)) × {0}) Fn (ℤ_≥‘(𝑁 + 1)))
337	134, 336	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ_≥‘(𝑁 + 1)) × {0}) Fn (ℤ_≥‘(𝑁 + 1))
338		fvun1 6179	. . . . . . . . . . . . . . . . . . . 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) = (𝑤‘𝑘))
339	337, 338	mp3an2 1404	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 Fn (0...𝑁) ∧ (((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1))) = ∅ ∧ 𝑘 ∈ (0...𝑁))) → ((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤‘𝑘))
340	339	anassrs 678	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1))) = ∅) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤‘𝑘))
341	335, 340	sylan 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤‘𝑘))
342	341	eqcomd 2616	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤‘𝑘) = ((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0}))‘𝑘))
343	342	oveq1d 6564	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤‘𝑘) · (𝑦↑𝑘)) = (((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦↑𝑘)))
344	343	sumeq2dv 14281	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)) = Σ𝑘 ∈ (0...𝑁)(((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦↑𝑘)))
345	344	mpteq2dv 4673	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦↑𝑘))))
346	282, 281, 314, 334, 345	coeeq 23787	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))) = (𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})))
347	346	reseq1d 5316	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))) ↾ (0...𝑁)) = ((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)))
348		res0 5321	. . . . . . . . . . . . . 14 ⊢ (𝑤 ↾ ∅) = ∅
349	290	reseq2d 5317	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑤 ↾ ∅) = (𝑤 ↾ ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1)))))
350		res0 5321	. . . . . . . . . . . . . . 15 ⊢ (((ℤ_≥‘(𝑁 + 1)) × {0}) ↾ ∅) = ∅
351	290	reseq2d 5317	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((ℤ_≥‘(𝑁 + 1)) × {0}) ↾ ∅) = (((ℤ_≥‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1)))))
352	350, 351	syl5eqr 2658	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∅ = (((ℤ_≥‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1)))))
353	348, 349, 352	3eqtr3a 2668	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑤 ↾ ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1)))) = (((ℤ_≥‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ_≥‘(𝑁 + 1)))))
354		fss 5969	. . . . . . . . . . . . . . 15 ⊢ ((((ℤ_≥‘(𝑁 + 1)) × {0}):(ℤ_≥‘(𝑁 + 1))⟶{0} ∧ {0} ⊆ ℚ) → ((ℤ_≥‘(𝑁 + 1)) × {0}):(ℤ_≥‘(𝑁 + 1))⟶ℚ)
355	292, 294, 354	mp2an 704	. . . . . . . . . . . . . 14 ⊢ ((ℤ_≥‘(𝑁 + 1)) × {0}):(ℤ_≥‘(𝑁 + 1))⟶ℚ
356		fresaunres1 5990	. . . . . . . . . . . . . 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)
357	355, 356	mp3an2 1404	. . . . . . . . . . . . 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)
358	353, 357	sylan2 490	. . . . . . . . . . . 12 ⊢ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝜑) → ((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
359	358	ancoms 468	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ_≥‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
360	347, 359	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))) ↾ (0...𝑁)) = 𝑤)
361		fveq2 6103	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) = 0_𝑝 → (coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))) = (coeff‘0_𝑝))
362	361	reseq1d 5316	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) = 0_𝑝 → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))) ↾ (0...𝑁)) = ((coeff‘0_𝑝) ↾ (0...𝑁)))
363		eqtr2 2630	. . . . . . . . . . . 12 ⊢ ((((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))) ↾ (0...𝑁)) = 𝑤 ∧ ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))) ↾ (0...𝑁)) = ((coeff‘0_𝑝) ↾ (0...𝑁))) → 𝑤 = ((coeff‘0_𝑝) ↾ (0...𝑁)))
364		coe0 23816	. . . . . . . . . . . . . 14 ⊢ (coeff‘0_𝑝) = (ℕ₀ × {0})
365	364	reseq1i 5313	. . . . . . . . . . . . 13 ⊢ ((coeff‘0_𝑝) ↾ (0...𝑁)) = ((ℕ₀ × {0}) ↾ (0...𝑁))
366		elfznn0 12302	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℕ₀)
367	366	ssriv 3572	. . . . . . . . . . . . . 14 ⊢ (0...𝑁) ⊆ ℕ₀
368		xpssres 5354	. . . . . . . . . . . . . 14 ⊢ ((0...𝑁) ⊆ ℕ₀ → ((ℕ₀ × {0}) ↾ (0...𝑁)) = ((0...𝑁) × {0}))
369	367, 368	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((ℕ₀ × {0}) ↾ (0...𝑁)) = ((0...𝑁) × {0})
370	365, 369	eqtri 2632	. . . . . . . . . . . 12 ⊢ ((coeff‘0_𝑝) ↾ (0...𝑁)) = ((0...𝑁) × {0})
371	363, 370	syl6eq 2660	. . . . . . . . . . 11 ⊢ ((((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))) ↾ (0...𝑁)) = 𝑤 ∧ ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))) ↾ (0...𝑁)) = ((coeff‘0_𝑝) ↾ (0...𝑁))) → 𝑤 = ((0...𝑁) × {0}))
372	371	ex 449	. . . . . . . . . 10 ⊢ (((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))) ↾ (0...𝑁)) = 𝑤 → (((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))) ↾ (0...𝑁)) = ((coeff‘0_𝑝) ↾ (0...𝑁)) → 𝑤 = ((0...𝑁) × {0})))
373	360, 362, 372	syl2im 39	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) = 0_𝑝 → 𝑤 = ((0...𝑁) × {0})))
374	373	necon3d 2803	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → (𝑤 ≠ ((0...𝑁) × {0}) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) ≠ 0_𝑝))
375	374	impr 647	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) ≠ 0_𝑝)
376		eldifsn 4260	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) ∈ ((Poly‘ℚ) ∖ {0_𝑝}) ↔ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) ∈ (Poly‘ℚ) ∧ (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) ≠ 0_𝑝))
377	283, 375, 376	sylanbrc 695	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) ∈ ((Poly‘ℚ) ∖ {0_𝑝}))
378	377	adantrr 749	. . . . 5 ⊢ ((𝜑 ∧ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})) ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) ∈ ((Poly‘ℚ) ∖ {0_𝑝}))
379		oveq1 6556	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑦↑𝑘) = (𝐴↑𝑘))
380	379	oveq2d 6565	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → ((𝑤‘𝑘) · (𝑦↑𝑘)) = ((𝑤‘𝑘) · (𝐴↑𝑘)))
381	380	sumeq2sdv 14282	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝐴↑𝑘)))
382		eqid 2610	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))
383		sumex 14266	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝐴↑𝑘)) ∈ V
384	381, 382, 383	fvmpt 6191	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝐴↑𝑘)))
385	1, 384	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝐴↑𝑘)))
386	385	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝐴↑𝑘)))
387	112	adantll 746	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤‘𝑘) ∈ ℂ)
388		aacllem.2	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
389	388	adantlr 747	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
390	115, 389	mulcld 9939	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝐶 · 𝑋) ∈ ℂ)
391	390	adantllr 751	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝐶 · 𝑋) ∈ ℂ)
392	54, 387, 391	fsummulc2 14358	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤‘𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)) = Σ𝑛 ∈ (1...𝑁)((𝑤‘𝑘) · (𝐶 · 𝑋)))
393		aacllem.4	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑𝑘) = Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋))
394	393	oveq2d 6565	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤‘𝑘) · (𝐴↑𝑘)) = ((𝑤‘𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)))
395	394	adantlr 747	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤‘𝑘) · (𝐴↑𝑘)) = ((𝑤‘𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)))
396	387	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑤‘𝑘) ∈ ℂ)
397	115	adantllr 751	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℂ)
398		simpll 786	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → 𝜑)
399	398, 388	sylan 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
400	396, 397, 399	mulassd 9942	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((𝑤‘𝑘) · 𝐶) · 𝑋) = ((𝑤‘𝑘) · (𝐶 · 𝑋)))
401	400	sumeq2dv 14281	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁)((𝑤‘𝑘) · (𝐶 · 𝑋)))
402	392, 395, 401	3eqtr4d 2654	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤‘𝑘) · (𝐴↑𝑘)) = Σ𝑛 ∈ (1...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋))
403	402	sumeq2dv 14281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝐴↑𝑘)) = Σ𝑘 ∈ (0...𝑁)Σ𝑛 ∈ (1...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋))
404	112	ad2ant2lr 780	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → (𝑤‘𝑘) ∈ ℂ)
405	115	anasss 677	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝐶 ∈ ℂ)
406	405	adantlr 747	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝐶 ∈ ℂ)
407	404, 406	mulcld 9939	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → ((𝑤‘𝑘) · 𝐶) ∈ ℂ)
408	388	ad2ant2rl 781	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝑋 ∈ ℂ)
409	407, 408	mulcld 9939	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → (((𝑤‘𝑘) · 𝐶) · 𝑋) ∈ ℂ)
410	45, 72, 409	fsumcom 14349	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)Σ𝑛 ∈ (1...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋))
411	403, 410	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝐴↑𝑘)) = Σ𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋))
412	411	adantrr 749	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝐴↑𝑘)) = Σ𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋))
413		nfv 1830	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝜑
414		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛 𝑤:(0...𝑁)⟶ℚ
415		nfra1 2925	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0
416	414, 415	nfan 1816	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)
417	413, 416	nfan 1816	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0))
418		rspa 2914	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)
419	418	oveq1d 6564	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0 ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
420	419	adantll 746	. . . . . . . . . . . . . 14 ⊢ (((𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
421	420	adantll 746	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
422	388	adantlr 747	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
423	96, 422, 118	fsummulc1 14359	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) · 𝑋) = Σ𝑘 ∈ (0...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋))
424	423	adantlrr 753	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) · 𝑋) = Σ𝑘 ∈ (0...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋))
425	388	mul02d 10113	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (0 · 𝑋) = 0)
426	425	adantlr 747	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (0 · 𝑋) = 0)
427	421, 424, 426	3eqtr3d 2652	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋) = 0)
428	427	ex 449	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → (𝑛 ∈ (1...𝑁) → Σ𝑘 ∈ (0...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋) = 0))
429	417, 428	ralrimi 2940	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋) = 0)
430	429	sumeq2d 14280	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → Σ𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)(((𝑤‘𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁)0)
431	412, 430	eqtrd 2644	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝐴↑𝑘)) = Σ𝑛 ∈ (1...𝑁)0)
432	24	olci 405	. . . . . . . . 9 ⊢ ((1...𝑁) ⊆ (ℤ_≥‘𝐵) ∨ (1...𝑁) ∈ Fin)
433		sumz 14300	. . . . . . . . 9 ⊢ (((1...𝑁) ⊆ (ℤ_≥‘𝐵) ∨ (1...𝑁) ∈ Fin) → Σ𝑛 ∈ (1...𝑁)0 = 0)
434	432, 433	ax-mp 5	. . . . . . . 8 ⊢ Σ𝑛 ∈ (1...𝑁)0 = 0
435	431, 434	syl6eq 2660	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝐴↑𝑘)) = 0)
436	386, 435	eqtrd 2644	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))‘𝐴) = 0)
437	436	adantrlr 755	. . . . 5 ⊢ ((𝜑 ∧ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})) ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))‘𝐴) = 0)
438		fveq1 6102	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) → (𝑥‘𝐴) = ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))‘𝐴))
439	438	eqeq1d 2612	. . . . . 6 ⊢ (𝑥 = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) → ((𝑥‘𝐴) = 0 ↔ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))‘𝐴) = 0))
440	439	rspcev 3282	. . . . 5 ⊢ (((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘))) ∈ ((Poly‘ℚ) ∖ {0_𝑝}) ∧ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · (𝑦↑𝑘)))‘𝐴) = 0) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0_𝑝})(𝑥‘𝐴) = 0)
441	378, 437, 440	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})) ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0_𝑝})(𝑥‘𝐴) = 0)
442	279, 441	sylanr1 682	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ ((ℚ ↑_𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ∧ ∀𝑛 ∈ (1...𝑁)Σ𝑘 ∈ (0...𝑁)((𝑤‘𝑘) · 𝐶) = 0)) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0_𝑝})(𝑥‘𝐴) = 0)
443	276, 442	rexlimddv 3017	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0_𝑝})(𝑥‘𝐴) = 0)
444		elqaa 23881	. 2 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0_𝑝})(𝑥‘𝐴) = 0))
445	1, 443, 444	sylanbrc 695	1 ⊢ (𝜑 → 𝐴 ∈ 𝔸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ↑_𝑚 cmap 7744 ≈ cen 7838 ≼ cdom 7839 Fincfn 7841 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ℚcq 11664 ...cfz 12197 ↑cexp 12722 #chash 12979 Σcsu 14264 Basecbs 15695 ↾_s cress 15696 ._rcmulr 15769 Scalarcsca 15771 ·_𝑠 cvsca 15772 0_gc0g 15923 Σ_g cgsu 15924 Moorecmre 16065 mrClscmrc 16066 mrIndcmri 16067 ACScacs 16068 Ringcrg 18370 DivRingcdr 18570 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 LBasisclbs 18895 LVecclvec 18923 NzRingcnzr 19078 ℂ_fldccnfld 19567 freeLMod cfrlm 19909 unitVec cuvc 19940 LIndF clindf 19962 LIndSclinds 19963 0_𝑝c0p 23242 Polycply 23744 coeffccoe 23746 𝔸caa 23873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-mri 16071 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lmhm 18843 df-lbs 18896 df-lvec 18924 df-sra 18993 df-rgmod 18994 df-nzr 19079 df-cnfld 19568 df-dsmm 19895 df-frlm 19910 df-uvc 19941 df-lindf 19964 df-linds 19965 df-0p 23243 df-ply 23748 df-coe 23750 df-dgr 23751 df-aa 23874

This theorem is referenced by: (None)

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