veroquadnolindfd - Mathbox for Jiamin Zhao

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Theorem veroquadnolindfd 50883

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 21864	. . . . . . 7 ⊢ ℝ_fld ∈ Field
2		isfld 20932	. . . . . . 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 20926	. . . . 5 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
6	4, 5	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
7		ovex 7449	. . . 4 ⊢ (1...6) ∈ V
8		eqid 2760	. . . . 5 ⊢ (ℝ_fld freeLMod (1...6)) = (ℝ_fld freeLMod (1...6))
9	8	frlmlmod 21994	. . . 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 14061	. . . . 5 ⊢ (1...6) ∈ Fin
15		rebase 21851	. . . . . 6 ⊢ ℝ = (Base‘ℝ_fld)
16	8, 15	frlmfibas 22007	. . . . 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 6886	. . . . . . . . . 10 ⊢ (𝑖 = 𝑢 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑢)))
21	20	fveq1d 6883	. . . . . . . . 9 ⊢ (𝑖 = 𝑢 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑗))
22		fveq2 6881	. . . . . . . . 9 ⊢ (𝑗 = 𝑣 → ((veronese‘(𝐴‘𝑢))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑣))
23	21, 22	cbvmpov 7511	. . . . . . . 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 8266	. . . . . 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 7081	. . . . . 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 50870	. . . . . 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 8075	. . . 4 ⊢ (𝜑 → tpos 𝑉:((1...6) × (1...6))⟶ℝ)
35		1nn 12293	. . . . . . . 8 ⊢ 1 ∈ ℕ
36		6nn 12379	. . . . . . . 8 ⊢ 6 ∈ ℕ
37		1re 11257	. . . . . . . . 9 ⊢ 1 ∈ ℝ
38		6re 12380	. . . . . . . . 9 ⊢ 6 ∈ ℝ
39		1lt6 12477	. . . . . . . . 9 ⊢ 1 < 6
40	37, 38, 39	ltleii 11382	. . . . . . . 8 ⊢ 1 ≤ 6
41		elfz1b 13673	. . . . . . . 8 ⊢ (1 ∈ (1...6) ↔ (1 ∈ ℕ ∧ 6 ∈ ℕ ∧ 1 ≤ 6))
42	35, 36, 40, 41	mpbir3an 1360	. . . . . . 7 ⊢ 1 ∈ (1...6)
43	42	ne0ii 4290	. . . . . 6 ⊢ (1...6) ≠ ∅
44		eldifsn 4748	. . . . . 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 11240	. . . . 5 ⊢ ℝ ∈ V
48	47	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
49		curf 8876	. . . 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 6692	. 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 8885	. . . 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 50882	. 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 21998	. . . 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 21857	. . 3 ⊢ 0 = (0_g‘ℝ_fld)
65	59, 61, 62, 63, 64, 59	nellindf 50868	. 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 1570 ∈ wcel 2145 ≠ wne 2955 Vcvv 3450 ∖ cdif 3896 ∅c0 4279 {csn 4584 class class class wbr 5103 × cxp 5653 ⟶wf 6531 ‘cfv 6535 (class class class)co 7416 ∈ cmpo 7418 ∘_f cof 7682 tpos ctpos 8228 curry ccur 8268 ↑_m cmap 8833 Fincfn 8959 ℝcr 11148 0cc0 11149 1c1 11150 + caddc 11152 · cmul 11154 ≤ cle 11293 ℕcn 12282 2c2 12344 3c3 12345 4c4 12346 5c5 12347 6c6 12348 ...cfz 13586 ↑cexp 14150 Basecbs 17326 Scalarcsca 17370 ·_𝑠 cvsca 17371 0_gc0g 17549 Σ_g cgsu 17550 Ringcrg 20398 CRingccrg 20399 DivRingcdr 20919 Fieldcfield 20920 LModclmod 21074 ℝ_fldcrefld 21849 freeLMod cfrlm 21991 LIndF clindf 22049 veronesecveronese 50866

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-addf 11228 ax-mulf 11229

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8164 df-tpos 8229 df-cur 8270 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9339 df-sup 9419 df-oi 9489 df-card 9969 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-fz 13587 df-fzo 13735 df-seq 14091 df-exp 14151 df-hash 14420 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-sca 17383 df-vsca 17384 df-ip 17385 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-hom 17391 df-cco 17392 df-0g 17551 df-gsum 17552 df-prds 17557 df-pws 17559 df-mre 17695 df-mrc 17696 df-acs 17698 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-mhm 18917 df-submnd 18918 df-grp 19086 df-minusg 19087 df-sbg 19088 df-mulg 19217 df-subg 19272 df-ghm 19367 df-cntz 19470 df-cmn 19935 df-abl 19936 df-mgp 20300 df-rng 20314 df-ur 20347 df-ring 20400 df-cring 20401 df-oppr 20506 df-dvdsr 20526 df-unit 20527 df-invr 20557 df-dvr 20570 df-nzr 20702 df-subrng 20737 df-subrg 20761 df-drng 20921 df-field 20922 df-lmod 21076 df-lss 21146 df-lsp 21186 df-lmhm 21236 df-lbs 21289 df-sra 21387 df-rgmod 21388 df-cnfld 21618 df-refld 21850 df-dsmm 21977 df-frlm 21992 df-uvc 22028 df-lindf 22051 df-veronese 50867

This theorem is used by: veroquaddetzerod 50884

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