Theorem prjcrv0 40507

Description: The "curve" (zero set) corresponding to the zero polynomial contains all coordinates. (Contributed by SN, 23-Nov-2024.)

Hypotheses

Ref	Expression
prjcrv0.y	⊢ 𝑌 = ((0...𝑁) mPoly 𝐾)
prjcrv0.0	⊢ 0 = (0g‘𝑌)
prjcrv0.p	⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)
prjcrv0.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
prjcrv0.k	⊢ (𝜑 → 𝐾 ∈ Field)

Assertion

Ref	Expression
prjcrv0	⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = 𝑃)

Proof of Theorem prjcrv0

Dummy variables 𝑎 𝑘 𝑝 ℎ 𝑓 𝑛 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ ((0...𝑁) mHomP 𝐾) = ((0...𝑁) mHomP 𝐾)
2		eqid 2736	. . 3 ⊢ ((0...𝑁) eval 𝐾) = ((0...𝑁) eval 𝐾)
3		prjcrv0.p	. . 3 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)
4		eqid 2736	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
5		prjcrv0.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
6		prjcrv0.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ Field)
7		funmpt 6501	. . . . 5 ⊢ Fun (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘𝑌) ∣ (𝑓 supp (0g‘𝐾)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})
8		prjcrv0.y	. . . . . . 7 ⊢ 𝑌 = ((0...𝑁) mPoly 𝐾)
9		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝑌)
10		eqid 2736	. . . . . . 7 ⊢ {ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin}
11		ovexd 7342	. . . . . . 7 ⊢ (𝜑 → (0...𝑁) ∈ V)
12	1, 8, 9, 4, 10, 11, 6	mhpfval 21374	. . . . . 6 ⊢ (𝜑 → ((0...𝑁) mHomP 𝐾) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘𝑌) ∣ (𝑓 supp (0g‘𝐾)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
13	12	funeqd 6485	. . . . 5 ⊢ (𝜑 → (Fun ((0...𝑁) mHomP 𝐾) ↔ Fun (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘𝑌) ∣ (𝑓 supp (0g‘𝐾)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})))
14	7, 13	mpbiri 258	. . . 4 ⊢ (𝜑 → Fun ((0...𝑁) mHomP 𝐾))
15		prjcrv0.0	. . . . . 6 ⊢ 0 = (0g‘𝑌)
16	6	fldcrngd 40292	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ CRing)
17	16	crnggrpd 19842	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Grp)
18	8, 10, 4, 15, 11, 17	mpl0 21257	. . . . 5 ⊢ (𝜑 → 0 = ({ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝐾)}))
19		0nn0 12294	. . . . . . 7 ⊢ 0 ∈ ℕ0
20	19	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ0)
21	1, 4, 10, 11, 17, 20	mhp0cl 21381	. . . . 5 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝐾)}) ∈ (((0...𝑁) mHomP 𝐾)‘0))
22	18, 21	eqeltrd 2837	. . . 4 ⊢ (𝜑 → 0 ∈ (((0...𝑁) mHomP 𝐾)‘0))
23		elunirn2 7158	. . . 4 ⊢ ((Fun ((0...𝑁) mHomP 𝐾) ∧ 0 ∈ (((0...𝑁) mHomP 𝐾)‘0)) → 0 ∈ ∪ ran ((0...𝑁) mHomP 𝐾))
24	14, 22, 23	syl2anc 585	. . 3 ⊢ (𝜑 → 0 ∈ ∪ ran ((0...𝑁) mHomP 𝐾))
25	1, 2, 3, 4, 5, 6, 24	prjcrvval 40506	. 2 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = {𝑝 ∈ 𝑃 ∣ ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)}})
26		eqid 2736	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
27		ovexd 7342	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (0...𝑁) ∈ V)
28	16	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ∈ CRing)
29	2, 26, 8, 4, 15, 27, 28	evl0 40308	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (((0...𝑁) eval 𝐾)‘ 0 ) = (((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)}))
30	29	imaeq1d 5978	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = ((((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)}) “ 𝑝))
31	3	eleq2i 2828	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ (𝑁ℙ𝕣𝕠𝕛n𝐾))
32	31	biimpi 215	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → 𝑝 ∈ (𝑁ℙ𝕣𝕠𝕛n𝐾))
33	32	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ (𝑁ℙ𝕣𝕠𝕛n𝐾))
34	5	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑁 ∈ ℕ0)
35		isfld 20045	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
36	35	simplbi 499	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Field → 𝐾 ∈ DivRing)
37	6, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ DivRing)
38	37	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ∈ DivRing)
39		prjspnval 40492	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
40	34, 38, 39	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
41		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝐾 freeLMod (0...𝑁)) = (𝐾 freeLMod (0...𝑁))
42	41	frlmlvec 21013	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → (𝐾 freeLMod (0...𝑁)) ∈ LVec)
43	37, 11, 42	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 freeLMod (0...𝑁)) ∈ LVec)
44	43	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝐾 freeLMod (0...𝑁)) ∈ LVec)
45		eqid 2736	. . . . . . . . . . 11 ⊢ (0g‘(𝐾 freeLMod (0...𝑁))) = (0g‘(𝐾 freeLMod (0...𝑁)))
46		eqid 2736	. . . . . . . . . . 11 ⊢ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) = ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})
47		eqid 2736	. . . . . . . . . . 11 ⊢ (LSpan‘(𝐾 freeLMod (0...𝑁))) = (LSpan‘(𝐾 freeLMod (0...𝑁)))
48	45, 46, 47	prjspval2 40489	. . . . . . . . . 10 ⊢ ((𝐾 freeLMod (0...𝑁)) ∈ LVec → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = ∪ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}){(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})
49	44, 48	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = ∪ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}){(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})
50	40, 49	eqtrd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = ∪ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}){(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})
51	33, 50	eleqtrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ ∪ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}){(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})
52		eliun 4935	. . . . . . 7 ⊢ (𝑝 ∈ ∪ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}){(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})} ↔ ∃𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})𝑝 ∈ {(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})
53	51, 52	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ∃𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})𝑝 ∈ {(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})
54		eqid 2736	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} = {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)}
55	41, 26, 4, 54	frlmbas 21007	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Field ∧ (0...𝑁) ∈ V) → {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} = (Base‘(𝐾 freeLMod (0...𝑁))))
56	6, 11, 55	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} = (Base‘(𝐾 freeLMod (0...𝑁))))
57		ssrab2 4019	. . . . . . . . . . 11 ⊢ {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} ⊆ ((Base‘𝐾) ↑m (0...𝑁))
58	56, 57	eqsstrrdi 3981	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(𝐾 freeLMod (0...𝑁))) ⊆ ((Base‘𝐾) ↑m (0...𝑁)))
59	58	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑝 ∈ {(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})) → (Base‘(𝐾 freeLMod (0...𝑁))) ⊆ ((Base‘𝐾) ↑m (0...𝑁)))
60		eldifi 4067	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) → 𝑎 ∈ (Base‘(𝐾 freeLMod (0...𝑁))))
61	60	adantl 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})) → 𝑎 ∈ (Base‘(𝐾 freeLMod (0...𝑁))))
62	61	adantrr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑝 ∈ {(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})) → 𝑎 ∈ (Base‘(𝐾 freeLMod (0...𝑁))))
63	59, 62	sseldd 3927	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑝 ∈ {(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})) → 𝑎 ∈ ((Base‘𝐾) ↑m (0...𝑁)))
64		velsn 4581	. . . . . . . . . 10 ⊢ (𝑝 ∈ {(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})} ↔ 𝑝 = (((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}))
65	64	anbi2i 624	. . . . . . . . 9 ⊢ ((𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑝 ∈ {(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})}) ↔ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑝 = (((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})))
66	43	lveclmodd 40295	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾 freeLMod (0...𝑁)) ∈ LMod)
67	66	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})) → (𝐾 freeLMod (0...𝑁)) ∈ LMod)
68		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐾 freeLMod (0...𝑁))) = (Base‘(𝐾 freeLMod (0...𝑁)))
69	68, 47	lspsnid 20300	. . . . . . . . . . . . 13 ⊢ (((𝐾 freeLMod (0...𝑁)) ∈ LMod ∧ 𝑎 ∈ (Base‘(𝐾 freeLMod (0...𝑁)))) → 𝑎 ∈ ((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}))
70	67, 61, 69	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})) → 𝑎 ∈ ((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}))
71		eldifn 4068	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) → ¬ 𝑎 ∈ {(0g‘(𝐾 freeLMod (0...𝑁)))})
72	71	adantl 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})) → ¬ 𝑎 ∈ {(0g‘(𝐾 freeLMod (0...𝑁)))})
73	70, 72	eldifd 3903	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})) → 𝑎 ∈ (((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}))
74		eleq2 2825	. . . . . . . . . . 11 ⊢ (𝑝 = (((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) → (𝑎 ∈ 𝑝 ↔ 𝑎 ∈ (((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})))
75	73, 74	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})) → (𝑝 = (((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) → 𝑎 ∈ 𝑝))
76	75	impr 456	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑝 = (((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}))) → 𝑎 ∈ 𝑝)
77	65, 76	sylan2b 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑝 ∈ {(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})) → 𝑎 ∈ 𝑝)
78	63, 77	elind 4134	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑝 ∈ {(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})) → 𝑎 ∈ (((Base‘𝐾) ↑m (0...𝑁)) ∩ 𝑝))
79	78	ne0d 4275	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑝 ∈ {(((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})})) → (((Base‘𝐾) ↑m (0...𝑁)) ∩ 𝑝) ≠ ∅)
80	53, 79	rexlimddv 3155	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (((Base‘𝐾) ↑m (0...𝑁)) ∩ 𝑝) ≠ ∅)
81		xpima2 6102	. . . . 5 ⊢ ((((Base‘𝐾) ↑m (0...𝑁)) ∩ 𝑝) ≠ ∅ → ((((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)}) “ 𝑝) = {(0g‘𝐾)})
82	80, 81	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)}) “ 𝑝) = {(0g‘𝐾)})
83	30, 82	eqtrd 2776	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)})
84	83	rabeqcda 3436	. 2 ⊢ (𝜑 → {𝑝 ∈ 𝑃 ∣ ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)}} = 𝑃)
85	25, 84	eqtrd 2776	1 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104   ≠ wne 2941  ∃wrex 3071  {crab 3284  Vcvv 3437   ∖ cdif 3889   ∩ cin 3891   ⊆ wss 3892  ∅c0 4262  {csn 4565  ∪ cuni 4844  ∪ ciun 4931   class class class wbr 5081   ↦ cmpt 5164   × cxp 5598  ^◡ccnv 5599  ran crn 5601   “ cima 5603  Fun wfun 6452  ‘cfv 6458  (class class class)co 7307   supp csupp 8008   ↑_m cmap 8646  Fincfn 8764   finSupp cfsupp 9172  0cc0 10917  ℕcn 12019  ℕ₀cn0 12279  ...cfz 13285  Basecbs 16957   ↾_s cress 16986  0_gc0g 17195   Σ_g cgsu 17196  CRingccrg 19829  DivRingcdr 20036  Fieldcfield 20037  LModclmod 20168  LSpanclspn 20278  LVecclvec 20409  ℂ_fldccnfld 20642   freeLMod cfrlm 20998   mPoly cmpl 21154   eval cevl 21326   mHomP cmhp 21364  ℙ𝕣𝕠𝕛cprjsp 40477  ℙ𝕣𝕠𝕛_ncprjspn 40490  ℙ𝕣𝕠𝕛Crvcprjcrv 40503

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-cnex 10973  ax-resscn 10974  ax-1cn 10975  ax-icn 10976  ax-addcl 10977  ax-addrcl 10978  ax-mulcl 10979  ax-mulrcl 10980  ax-mulcom 10981  ax-addass 10982  ax-mulass 10983  ax-distr 10984  ax-i2m1 10985  ax-1ne0 10986  ax-1rid 10987  ax-rnegex 10988  ax-rrecex 10989  ax-cnre 10990  ax-pre-lttri 10991  ax-pre-lttrn 10992  ax-pre-ltadd 10993  ax-pre-mulgt0 10994

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-tp 4570  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-iin 4934  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-se 5556  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-isom 6467  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-of 7565  df-ofr 7566  df-om 7745  df-1st 7863  df-2nd 7864  df-supp 8009  df-tpos 8073  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-er 8529  df-ec 8531  df-qs 8535  df-map 8648  df-pm 8649  df-ixp 8717  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-fsupp 9173  df-sup 9245  df-oi 9313  df-card 9741  df-pnf 11057  df-mnf 11058  df-xr 11059  df-ltxr 11060  df-le 11061  df-sub 11253  df-neg 11254  df-nn 12020  df-2 12082  df-3 12083  df-4 12084  df-5 12085  df-6 12086  df-7 12087  df-8 12088  df-9 12089  df-n0 12280  df-z 12366  df-dec 12484  df-uz 12629  df-fz 13286  df-fzo 13429  df-seq 13768  df-hash 14091  df-struct 16893  df-sets 16910  df-slot 16928  df-ndx 16940  df-base 16958  df-ress 16987  df-plusg 17020  df-mulr 17021  df-sca 17023  df-vsca 17024  df-ip 17025  df-tset 17026  df-ple 17027  df-ds 17029  df-hom 17031  df-cco 17032  df-0g 17197  df-gsum 17198  df-prds 17203  df-pws 17205  df-mre 17340  df-mrc 17341  df-acs 17343  df-mgm 18371  df-sgrp 18420  df-mnd 18431  df-mhm 18475  df-submnd 18476  df-grp 18625  df-minusg 18626  df-sbg 18627  df-mulg 18746  df-subg 18797  df-ghm 18877  df-cntz 18968  df-cmn 19433  df-abl 19434  df-mgp 19766  df-ur 19783  df-srg 19787  df-ring 19830  df-cring 19831  df-oppr 19907  df-dvdsr 19928  df-unit 19929  df-invr 19959  df-rnghom 20004  df-drng 20038  df-field 20039  df-subrg 20067  df-lmod 20170  df-lss 20239  df-lsp 20279  df-lvec 20410  df-sra 20479  df-rgmod 20480  df-dsmm 20984  df-frlm 20999  df-assa 21105  df-asp 21106  df-ascl 21107  df-psr 21157  df-mvr 21158  df-mpl 21159  df-evls 21327  df-evl 21328  df-mhp 21368  df-prjsp 40478  df-prjspn 40491  df-prjcrv 40504

This theorem is referenced by: (None)