veroquadnolindfd - Mathbox for Jiamin Zhao

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Theorem veroquadnolindfd 50800

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 21831	. . . . . . 7 ⊢ ℝ_fld ∈ Field
2		isfld 20902	. . . . . . 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 20896	. . . . 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 2762	. . . . 5 ⊢ (ℝ_fld freeLMod (1...6)) = (ℝ_fld freeLMod (1...6))
9	8	frlmlmod 21961	. . . 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 3475	. . . . 5 ⊢ ℝ_fld ∈ V
14		fzfi 14036	. . . . 5 ⊢ (1...6) ∈ Fin
15		rebase 21818	. . . . . 6 ⊢ ℝ = (Base‘ℝ_fld)
16	8, 15	frlmfibas 21974	. . . . 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 6887	. . . . . . . . . 10 ⊢ (𝑖 = 𝑢 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑢)))
21	20	fveq1d 6884	. . . . . . . . 9 ⊢ (𝑖 = 𝑢 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑗))
22		fveq2 6882	. . . . . . . . 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 2785	. . . . . . 7 ⊢ 𝑉 = (𝑢 ∈ (1...6), 𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
25	24	tposmpo 8264	. . . . . 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 50787	. . . . . 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 8073	. . . 4 ⊢ (𝜑 → tpos 𝑉:((1...6) × (1...6))⟶ℝ)
35		1nn 12269	. . . . . . . 8 ⊢ 1 ∈ ℕ
36		6nn 12355	. . . . . . . 8 ⊢ 6 ∈ ℕ
37		1re 11233	. . . . . . . . 9 ⊢ 1 ∈ ℝ
38		6re 12356	. . . . . . . . 9 ⊢ 6 ∈ ℝ
39		1lt6 12453	. . . . . . . . 9 ⊢ 1 < 6
40	37, 38, 39	ltleii 11358	. . . . . . . 8 ⊢ 1 ≤ 6
41		elfz1b 13648	. . . . . . . 8 ⊢ (1 ∈ (1...6) ↔ (1 ∈ ℕ ∧ 6 ∈ ℕ ∧ 1 ≤ 6))
42	35, 36, 40, 41	mpbir3an 1360	. . . . . . 7 ⊢ 1 ∈ (1...6)
43	42	ne0ii 4293	. . . . . 6 ⊢ (1...6) ≠ ∅
44		eldifsn 4751	. . . . . 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 11216	. . . . 5 ⊢ ℝ ∈ V
48	47	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
49		curf 8872	. . . 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 6693	. 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 8881	. . . 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 2864	. 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 50799	. 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 2762	. . 3 ⊢ (Base‘(ℝ_fld freeLMod (1...6))) = (Base‘(ℝ_fld freeLMod (1...6)))
60	8	frlmsca 21965	. . . 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 2762	. . 3 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6))) = ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))
63		eqid 2762	. . 3 ⊢ (0_g‘(ℝ_fld freeLMod (1...6))) = (0_g‘(ℝ_fld freeLMod (1...6)))
64		re0g 21824	. . 3 ⊢ 0 = (0_g‘ℝ_fld)
65	59, 61, 62, 63, 64, 59	nellindf 50785	. 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 2957 Vcvv 3453 ∖ cdif 3899 ∅c0 4282 {csn 4587 class class class wbr 5107 × cxp 5657 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 ∈ cmpo 7418 ∘_f cof 7679 tpos ctpos 8226 curry ccur 8266 ↑_m cmap 8829 Fincfn 8955 ℝcr 11124 0cc0 11125 1c1 11126 + caddc 11128 · cmul 11130 ≤ cle 11269 ℕcn 12258 2c2 12320 3c3 12321 4c4 12322 5c5 12323 6c6 12324 ...cfz 13561 ↑cexp 14125 Basecbs 17303 Scalarcsca 17347 ·_𝑠 cvsca 17348 0_gc0g 17526 Σ_g cgsu 17527 Ringcrg 20371 CRingccrg 20372 DivRingcdr 20889 Fieldcfield 20890 LModclmod 21043 ℝ_fldcrefld 21816 freeLMod cfrlm 21958 LIndF clindf 22016 veronesecveronese 50783

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-addf 11204 ax-mulf 11205

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8227 df-cur 8268 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-fz 13562 df-fzo 13710 df-seq 14066 df-exp 14126 df-hash 14395 df-struct 17241 df-sets 17258 df-slot 17276 df-ndx 17288 df-base 17304 df-ress 17325 df-plusg 17357 df-mulr 17358 df-starv 17359 df-sca 17360 df-vsca 17361 df-ip 17362 df-tset 17363 df-ple 17364 df-ds 17366 df-unif 17367 df-hom 17368 df-cco 17369 df-0g 17528 df-gsum 17529 df-prds 17534 df-pws 17536 df-mre 17672 df-mrc 17673 df-acs 17675 df-mgm 18732 df-sgrp 18821 df-mnd 18837 df-mhm 18890 df-submnd 18891 df-grp 19059 df-minusg 19060 df-sbg 19061 df-mulg 19190 df-subg 19245 df-ghm 19340 df-cntz 19443 df-cmn 19908 df-abl 19909 df-mgp 20273 df-rng 20287 df-ur 20320 df-ring 20373 df-cring 20374 df-oppr 20477 df-dvdsr 20497 df-unit 20498 df-invr 20528 df-dvr 20541 df-nzr 20672 df-subrng 20707 df-subrg 20731 df-drng 20891 df-field 20892 df-lmod 21045 df-lss 21115 df-lsp 21155 df-lmhm 21205 df-lbs 21258 df-sra 21356 df-rgmod 21357 df-cnfld 21585 df-refld 21817 df-dsmm 21944 df-frlm 21959 df-uvc 21995 df-lindf 22018 df-veronese 50784

This theorem is used by: veroquaddetzerod 50801

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