veroquadnolindfd - Mathbox for Jiamin Zhao

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Theorem veroquadnolindfd 50985

Description: A nonzero homogeneous quadratic equation satisfied by all six points gives a linear dependence among the columns of the Veronese matrix. (Contributed by Jiamin Zhao, 27-Aug-2026.)

**Hypotheses**
Ref	Expression
veroquad.a	⊢ 𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))
veroquad.f	⊢ (𝜑 → 𝐴:(1...6)⟶(ℝ ↑_m (1...3)))
veroquad.k	⊢ (𝜑 → 𝐾:(1...6)⟶ℝ)
veroquad.q	⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))) = 0)
veroquadnolindf.n	⊢ (𝜑 → 𝐾 ≠ ((1...6) × {0}))

**Assertion**
Ref	Expression
veroquadnolindfd	⊢ (𝜑 → ¬ curry tpos 𝑉 LIndF (ℝ_fld freeLMod (1...6)))

Distinct variable groups: 𝐴,𝑖,𝑗 𝑖,𝑉,𝑗 𝜑,𝑖 𝑖,𝐾 Allowed substitution hints: 𝜑(𝑗) 𝐾(𝑗)

Proof of Theorem veroquadnolindfd Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		refld 21925	. . . . . . 7 ⊢ ℝ_fld ∈ Field
2		isfld 20993	. . . . . . 7 ⊢ (ℝ_fld ∈ Field ↔ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing))
3	1, 2	mpbi 233	. . . . . 6 ⊢ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing)
4	3	simpli 489	. . . . 5 ⊢ ℝ_fld ∈ DivRing
5		drngring 20987	. . . . 5 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
6	4, 5	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
7		ovex 7453	. . . 4 ⊢ (1...6) ∈ V
8		eqid 2760	. . . . 5 ⊢ (ℝ_fld freeLMod (1...6)) = (ℝ_fld freeLMod (1...6))
9	8	frlmlmod 22055	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ (1...6) ∈ V) → (ℝ_fld freeLMod (1...6)) ∈ LMod)
10	6, 7, 9	mp2an 705	. . 3 ⊢ (ℝ_fld freeLMod (1...6)) ∈ LMod
11	10	a1i 11	. 2 ⊢ (𝜑 → (ℝ_fld freeLMod (1...6)) ∈ LMod)
12	7	a1i 11	. 2 ⊢ (𝜑 → (1...6) ∈ V)
13	1	elexi 3472	. . . . 5 ⊢ ℝ_fld ∈ V
14		fzfi 14115	. . . . 5 ⊢ (1...6) ∈ Fin
15		rebase 21912	. . . . . 6 ⊢ ℝ = (Base‘ℝ_fld)
16	8, 15	frlmfibas 22068	. . . . 5 ⊢ ((ℝ_fld ∈ V ∧ (1...6) ∈ Fin) → (ℝ ↑_m (1...6)) = (Base‘(ℝ_fld freeLMod (1...6))))
17	13, 14, 16	mp2an 705	. . . 4 ⊢ (ℝ ↑_m (1...6)) = (Base‘(ℝ_fld freeLMod (1...6)))
18	17	a1i 11	. . 3 ⊢ (𝜑 → (ℝ ↑_m (1...6)) = (Base‘(ℝ_fld freeLMod (1...6))))
19		veroquad.a	. . . . . . . 8 ⊢ 𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))
20		2fveq3 6890	. . . . . . . . . 10 ⊢ (𝑖 = 𝑢 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑢)))
21	20	fveq1d 6887	. . . . . . . . 9 ⊢ (𝑖 = 𝑢 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑗))
22		fveq2 6885	. . . . . . . . 9 ⊢ (𝑗 = 𝑣 → ((veronese‘(𝐴‘𝑢))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑣))
23	21, 22	cbvmpov 7515	. . . . . . . 8 ⊢ (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗)) = (𝑢 ∈ (1...6), 𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
24	19, 23	eqtri 2783	. . . . . . 7 ⊢ 𝑉 = (𝑢 ∈ (1...6), 𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
25	24	tposmpo 8280	. . . . . 6 ⊢ tpos 𝑉 = (𝑣 ∈ (1...6), 𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
26	25	a1i 11	. . . . 5 ⊢ (𝜑 → tpos 𝑉 = (𝑣 ∈ (1...6), 𝑢 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣)))
27		veroquad.f	. . . . . . . 8 ⊢ (𝜑 → 𝐴:(1...6)⟶(ℝ ↑_m (1...3)))
28	27	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ (1...6) ∧ 𝑢 ∈ (1...6))) → 𝐴:(1...6)⟶(ℝ ↑_m (1...3)))
29		simprr 785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ (1...6) ∧ 𝑢 ∈ (1...6))) → 𝑢 ∈ (1...6))
30	28, 29	ffvelcdmd 7085	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ (1...6) ∧ 𝑢 ∈ (1...6))) → (𝐴‘𝑢) ∈ (ℝ ↑_m (1...3)))
31		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ (1...6) ∧ 𝑢 ∈ (1...6))) → 𝑣 ∈ (1...6))
32		veronesefvcl 50972	. . . . . 6 ⊢ (((𝐴‘𝑢) ∈ (ℝ ↑_m (1...3)) ∧ 𝑣 ∈ (1...6)) → ((veronese‘(𝐴‘𝑢))‘𝑣) ∈ ℝ)
33	30, 31, 32	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ (1...6) ∧ 𝑢 ∈ (1...6))) → ((veronese‘(𝐴‘𝑢))‘𝑣) ∈ ℝ)
34	26, 33	fmpod 8084	. . . 4 ⊢ (𝜑 → tpos 𝑉:((1...6) × (1...6))⟶ℝ)
35		1nn 12346	. . . . . . . 8 ⊢ 1 ∈ ℕ
36		6nn 12432	. . . . . . . 8 ⊢ 6 ∈ ℕ
37		1re 11308	. . . . . . . . 9 ⊢ 1 ∈ ℝ
38		6re 12433	. . . . . . . . 9 ⊢ 6 ∈ ℝ
39		1lt6 12530	. . . . . . . . 9 ⊢ 1 < 6
40	37, 38, 39	ltleii 11433	. . . . . . . 8 ⊢ 1 ≤ 6
41		elfz1b 13727	. . . . . . . 8 ⊢ (1 ∈ (1...6) ↔ (1 ∈ ℕ ∧ 6 ∈ ℕ ∧ 1 ≤ 6))
42	35, 36, 40, 41	mpbir3an 1360	. . . . . . 7 ⊢ 1 ∈ (1...6)
43	42	ne0ii 4289	. . . . . 6 ⊢ (1...6) ≠ ∅
44		eldifsn 4747	. . . . . 6 ⊢ ((1...6) ∈ (V ∖ {∅}) ↔ ((1...6) ∈ V ∧ (1...6) ≠ ∅))
45	7, 43, 44	mpbir2an 724	. . . . 5 ⊢ (1...6) ∈ (V ∖ {∅})
46	45	a1i 11	. . . 4 ⊢ (𝜑 → (1...6) ∈ (V ∖ {∅}))
47		reex 11291	. . . . 5 ⊢ ℝ ∈ V
48	47	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
49		curf 8890	. . . 4 ⊢ ((tpos 𝑉:((1...6) × (1...6))⟶ℝ ∧ (1...6) ∈ (V ∖ {∅}) ∧ ℝ ∈ V) → curry tpos 𝑉:(1...6)⟶(ℝ ↑_m (1...6)))
50	34, 46, 48, 49	syl3anc 1398	. . 3 ⊢ (𝜑 → curry tpos 𝑉:(1...6)⟶(ℝ ↑_m (1...6)))
51	18, 50	feq3dd 6696	. 2 ⊢ (𝜑 → curry tpos 𝑉:(1...6)⟶(Base‘(ℝ_fld freeLMod (1...6))))
52		veroquad.k	. . . 4 ⊢ (𝜑 → 𝐾:(1...6)⟶ℝ)
53	47, 7	elmap 8899	. . . 4 ⊢ (𝐾 ∈ (ℝ ↑_m (1...6)) ↔ 𝐾:(1...6)⟶ℝ)
54	52, 53	sylibr 237	. . 3 ⊢ (𝜑 → 𝐾 ∈ (ℝ ↑_m (1...6)))
55	54, 18	eleqtrd 2862	. 2 ⊢ (𝜑 → 𝐾 ∈ (Base‘(ℝ_fld freeLMod (1...6))))
56		veroquadnolindf.n	. 2 ⊢ (𝜑 → 𝐾 ≠ ((1...6) × {0}))
57		veroquad.q	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))) = 0)
58	19, 27, 52, 57	veroquadmodzerod 50984	. 2 ⊢ (𝜑 → ((ℝ_fld freeLMod (1...6)) Σ_g (𝐾 ∘_f ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))curry tpos 𝑉)) = (0_g‘(ℝ_fld freeLMod (1...6))))
59		eqid 2760	. . 3 ⊢ (Base‘(ℝ_fld freeLMod (1...6))) = (Base‘(ℝ_fld freeLMod (1...6)))
60	8	frlmsca 22059	. . . 4 ⊢ ((ℝ_fld ∈ V ∧ (1...6) ∈ V) → ℝ_fld = (Scalar‘(ℝ_fld freeLMod (1...6))))
61	13, 7, 60	mp2an 705	. . 3 ⊢ ℝ_fld = (Scalar‘(ℝ_fld freeLMod (1...6)))
62		eqid 2760	. . 3 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6))) = ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))
63		eqid 2760	. . 3 ⊢ (0_g‘(ℝ_fld freeLMod (1...6))) = (0_g‘(ℝ_fld freeLMod (1...6)))
64		re0g 21918	. . 3 ⊢ 0 = (0_g‘ℝ_fld)
65	59, 61, 62, 63, 64, 59	nellindf 50970	. 2 ⊢ ((((ℝ_fld freeLMod (1...6)) ∈ LMod ∧ (1...6) ∈ V ∧ curry tpos 𝑉:(1...6)⟶(Base‘(ℝ_fld freeLMod (1...6)))) ∧ (𝐾 ∈ (Base‘(ℝ_fld freeLMod (1...6))) ∧ 𝐾 ≠ ((1...6) × {0}) ∧ ((ℝ_fld freeLMod (1...6)) Σ_g (𝐾 ∘_f ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))curry tpos 𝑉)) = (0_g‘(ℝ_fld freeLMod (1...6))))) → ¬ curry tpos 𝑉 LIndF (ℝ_fld freeLMod (1...6)))
66	11, 12, 51, 55, 56, 58, 65	syl33anc 1412	1 ⊢ (𝜑 → ¬ curry tpos 𝑉 LIndF (ℝ_fld freeLMod (1...6)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 Vcvv 3450 ∖ cdif 4045 ∅c0 4278 {csn 4583 class class class wbr 5102 × cxp 5648 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 ∘_f cof 7691 tpos ctpos 8242 curry ccur 8282 ↑_m cmap 8847 Fincfn 8973 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 ≤ cle 11344 ℕcn 12335 2c2 12397 3c3 12398 4c4 12399 5c5 12400 6c6 12401 ...cfz 13639 ↑cexp 14204 Basecbs 17387 Scalarcsca 17431 ·_𝑠 cvsca 17432 0_gc0g 17610 Σ_g cgsu 17611 Ringcrg 20459 CRingccrg 20460 DivRingcdr 20980 Fieldcfield 20981 LModclmod 21135 ℝ_fldcrefld 21910 freeLMod cfrlm 22052 LIndF clindf 22110 veronesecveronese 50968

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-addf 11279 ax-mulf 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-tpos 8243 df-cur 8284 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-sup 9434 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-fz 13640 df-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-0g 17612 df-gsum 17613 df-prds 17618 df-pws 17620 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-submnd 18979 df-grp 19147 df-minusg 19148 df-sbg 19149 df-mulg 19278 df-subg 19333 df-ghm 19428 df-cntz 19531 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-dvr 20631 df-nzr 20763 df-subrng 20798 df-subrg 20822 df-drng 20982 df-field 20983 df-lmod 21137 df-lss 21207 df-lsp 21247 df-lmhm 21297 df-lbs 21350 df-sra 21448 df-rgmod 21449 df-cnfld 21679 df-refld 21911 df-dsmm 22038 df-frlm 22053 df-uvc 22089 df-lindf 22112 df-veronese 50969

This theorem is used by: veroquaddetzerod 50986

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