Theorem prjcrv0 40459

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 2740	. . 3 ⊢ ((0...𝑁) mHomP 𝐾) = ((0...𝑁) mHomP 𝐾)
2		eqid 2740	. . 3 ⊢ ((0...𝑁) eval 𝐾) = ((0...𝑁) eval 𝐾)
3		prjcrv0.p	. . 3 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)
4		eqid 2740	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
5		prjcrv0.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
6		prjcrv0.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ Field)
7		funmpt 6469	. . . . 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 2740	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝑌)
10		eqid 2740	. . . . . . 7 ⊢ {ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin}
11		ovexd 7304	. . . . . . 7 ⊢ (𝜑 → (0...𝑁) ∈ V)
12	1, 8, 9, 4, 10, 11, 6	mhpfval 21319	. . . . . 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 6453	. . . . 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 257	. . . 4 ⊢ (𝜑 → Fun ((0...𝑁) mHomP 𝐾))
15		prjcrv0.0	. . . . . 6 ⊢ 0 = (0g‘𝑌)
16	6	fldcrngd 40244	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ CRing)
17	16	crnggrpd 19787	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Grp)
18	8, 10, 4, 15, 11, 17	mpl0 21202	. . . . 5 ⊢ (𝜑 → 0 = ({ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝐾)}))
19		0nn0 12240	. . . . . . 7 ⊢ 0 ∈ ℕ0
20	19	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ0)
21	1, 4, 10, 11, 17, 20	mhp0cl 21326	. . . . 5 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝐾)}) ∈ (((0...𝑁) mHomP 𝐾)‘0))
22	18, 21	eqeltrd 2841	. . . 4 ⊢ (𝜑 → 0 ∈ (((0...𝑁) mHomP 𝐾)‘0))
23		elunirn2 7121	. . . 4 ⊢ ((Fun ((0...𝑁) mHomP 𝐾) ∧ 0 ∈ (((0...𝑁) mHomP 𝐾)‘0)) → 0 ∈ ∪ ran ((0...𝑁) mHomP 𝐾))
24	14, 22, 23	syl2anc 584	. . 3 ⊢ (𝜑 → 0 ∈ ∪ ran ((0...𝑁) mHomP 𝐾))
25	1, 2, 3, 4, 5, 6, 24	prjcrvval 40458	. 2 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = {𝑝 ∈ 𝑃 ∣ ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)}})
26		eqid 2740	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
27		ovexd 7304	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (0...𝑁) ∈ V)
28	16	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ∈ CRing)
29	2, 26, 8, 4, 15, 27, 28	evl0 40260	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (((0...𝑁) eval 𝐾)‘ 0 ) = (((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)}))
30	29	imaeq1d 5966	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = ((((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)}) “ 𝑝))
31	3	eleq2i 2832	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ (𝑁ℙ𝕣𝕠𝕛n𝐾))
32	31	biimpi 215	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → 𝑝 ∈ (𝑁ℙ𝕣𝕠𝕛n𝐾))
33	32	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ (𝑁ℙ𝕣𝕠𝕛n𝐾))
34	5	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑁 ∈ ℕ0)
35		isfld 19990	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
36	35	simplbi 498	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Field → 𝐾 ∈ DivRing)
37	6, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ DivRing)
38	37	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ∈ DivRing)
39		prjspnval 40444	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
40	34, 38, 39	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
41		eqid 2740	. . . . . . . . . . . . 13 ⊢ (𝐾 freeLMod (0...𝑁)) = (𝐾 freeLMod (0...𝑁))
42	41	frlmlvec 20958	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → (𝐾 freeLMod (0...𝑁)) ∈ LVec)
43	37, 11, 42	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 freeLMod (0...𝑁)) ∈ LVec)
44	43	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝐾 freeLMod (0...𝑁)) ∈ LVec)
45		eqid 2740	. . . . . . . . . . 11 ⊢ (0g‘(𝐾 freeLMod (0...𝑁))) = (0g‘(𝐾 freeLMod (0...𝑁)))
46		eqid 2740	. . . . . . . . . . 11 ⊢ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) = ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})
47		eqid 2740	. . . . . . . . . . 11 ⊢ (LSpan‘(𝐾 freeLMod (0...𝑁))) = (LSpan‘(𝐾 freeLMod (0...𝑁)))
48	45, 46, 47	prjspval2 40441	. . . . . . . . . 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 2780	. . . . . . . 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 2843	. . . . . . 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 4934	. . . . . . 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 2740	. . . . . . . . . . . . 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 20952	. . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} = (Base‘(𝐾 freeLMod (0...𝑁))))
57		ssrab2 4018	. . . . . . . . . . 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 723	. . . . . . . . 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 4066	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) → 𝑎 ∈ (Base‘(𝐾 freeLMod (0...𝑁))))
61	60	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})) → 𝑎 ∈ (Base‘(𝐾 freeLMod (0...𝑁))))
62	61	adantrr 714	. . . . . . . . 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 4583	. . . . . . . . . 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 623	. . . . . . . . 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 40247	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾 freeLMod (0...𝑁)) ∈ LMod)
67	66	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})) → (𝐾 freeLMod (0...𝑁)) ∈ LMod)
68		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐾 freeLMod (0...𝑁))) = (Base‘(𝐾 freeLMod (0...𝑁)))
69	68, 47	lspsnid 20245	. . . . . . . . . . . . 13 ⊢ (((𝐾 freeLMod (0...𝑁)) ∈ LMod ∧ 𝑎 ∈ (Base‘(𝐾 freeLMod (0...𝑁)))) → 𝑎 ∈ ((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}))
70	67, 61, 69	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})) → 𝑎 ∈ ((LSpan‘(𝐾 freeLMod (0...𝑁)))‘{𝑎}))
71		eldifn 4067	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) → ¬ 𝑎 ∈ {(0g‘(𝐾 freeLMod (0...𝑁)))})
72	71	adantl 482	. . . . . . . . . . . 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 2829	. . . . . . . . . . 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 246	. . . . . . . . . 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 455	. . . . . . . . 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 594	. . . . . . . 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 4133	. . . . . . 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 3222	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (((Base‘𝐾) ↑m (0...𝑁)) ∩ 𝑝) ≠ ∅)
81		xpima2 6085	. . . . 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 2780	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)})
84	83	rabeqcda 3428	. 2 ⊢ (𝜑 → {𝑝 ∈ 𝑃 ∣ ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)}} = 𝑃)
85	25, 84	eqtrd 2780	1 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110   ≠ wne 2945  ∃wrex 3067  {crab 3070  Vcvv 3431   ∖ cdif 3889   ∩ cin 3891   ⊆ wss 3892  ∅c0 4262  {csn 4567  ∪ cuni 4845  ∪ ciun 4930   class class class wbr 5079   ↦ cmpt 5162   × cxp 5587  ^◡ccnv 5588  ran crn 5590   “ cima 5592  Fun wfun 6425  ‘cfv 6431  (class class class)co 7269   supp csupp 7962   ↑_m cmap 8590  Fincfn 8708   finSupp cfsupp 9098  0cc0 10864  ℕcn 11965  ℕ₀cn0 12225  ...cfz 13230  Basecbs 16902   ↾_s cress 16931  0_gc0g 17140   Σ_g cgsu 17141  CRingccrg 19774  DivRingcdr 19981  Fieldcfield 19982  LModclmod 20113  LSpanclspn 20223  LVecclvec 20354  ℂ_fldccnfld 20587   freeLMod cfrlm 20943   mPoly cmpl 21099   eval cevl 21271   mHomP cmhp 21309  ℙ𝕣𝕠𝕛cprjsp 40429  ℙ𝕣𝕠𝕛_ncprjspn 40442  ℙ𝕣𝕠𝕛Crvcprjcrv 40455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-cnex 10920  ax-resscn 10921  ax-1cn 10922  ax-icn 10923  ax-addcl 10924  ax-addrcl 10925  ax-mulcl 10926  ax-mulrcl 10927  ax-mulcom 10928  ax-addass 10929  ax-mulass 10930  ax-distr 10931  ax-i2m1 10932  ax-1ne0 10933  ax-1rid 10934  ax-rnegex 10935  ax-rrecex 10936  ax-cnre 10937  ax-pre-lttri 10938  ax-pre-lttrn 10939  ax-pre-ltadd 10940  ax-pre-mulgt0 10941

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  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 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-isom 6440  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-of 7525  df-ofr 7526  df-om 7702  df-1st 7818  df-2nd 7819  df-supp 7963  df-tpos 8027  df-frecs 8082  df-wrecs 8113  df-recs 8187  df-rdg 8226  df-1o 8282  df-er 8473  df-ec 8475  df-qs 8479  df-map 8592  df-pm 8593  df-ixp 8661  df-en 8709  df-dom 8710  df-sdom 8711  df-fin 8712  df-fsupp 9099  df-sup 9171  df-oi 9239  df-card 9690  df-pnf 11004  df-mnf 11005  df-xr 11006  df-ltxr 11007  df-le 11008  df-sub 11199  df-neg 11200  df-nn 11966  df-2 12028  df-3 12029  df-4 12030  df-5 12031  df-6 12032  df-7 12033  df-8 12034  df-9 12035  df-n0 12226  df-z 12312  df-dec 12429  df-uz 12574  df-fz 13231  df-fzo 13374  df-seq 13712  df-hash 14035  df-struct 16838  df-sets 16855  df-slot 16873  df-ndx 16885  df-base 16903  df-ress 16932  df-plusg 16965  df-mulr 16966  df-sca 16968  df-vsca 16969  df-ip 16970  df-tset 16971  df-ple 16972  df-ds 16974  df-hom 16976  df-cco 16977  df-0g 17142  df-gsum 17143  df-prds 17148  df-pws 17150  df-mre 17285  df-mrc 17286  df-acs 17288  df-mgm 18316  df-sgrp 18365  df-mnd 18376  df-mhm 18420  df-submnd 18421  df-grp 18570  df-minusg 18571  df-sbg 18572  df-mulg 18691  df-subg 18742  df-ghm 18822  df-cntz 18913  df-cmn 19378  df-abl 19379  df-mgp 19711  df-ur 19728  df-srg 19732  df-ring 19775  df-cring 19776  df-oppr 19852  df-dvdsr 19873  df-unit 19874  df-invr 19904  df-rnghom 19949  df-drng 19983  df-field 19984  df-subrg 20012  df-lmod 20115  df-lss 20184  df-lsp 20224  df-lvec 20355  df-sra 20424  df-rgmod 20425  df-dsmm 20929  df-frlm 20944  df-assa 21050  df-asp 21051  df-ascl 21052  df-psr 21102  df-mvr 21103  df-mpl 21104  df-evls 21272  df-evl 21273  df-mhp 21313  df-prjsp 40430  df-prjspn 40443  df-prjcrv 40456

This theorem is referenced by: (None)