veroquadnolindfd - Mathbox for Jiamin Zhao

	Mathbox for Jiamin Zhao		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > veroquadnolindfd		Structured version Visualization version GIF version

Theorem veroquadnolindfd 51263

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 21952	. . . . . . 7 ⊢ ℝ_fld ∈ Field
2		isfld 21020	. . . . . . 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 21014	. . . . 5 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
6	4, 5	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
7		ovex 7473	. . . 4 ⊢ (1...6) ∈ V
8		eqid 2763	. . . . 5 ⊢ (ℝ_fld freeLMod (1...6)) = (ℝ_fld freeLMod (1...6))
9	8	frlmlmod 22082	. . . 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 3476	. . . . 5 ⊢ ℝ_fld ∈ V
14		fzfi 14139	. . . . 5 ⊢ (1...6) ∈ Fin
15		rebase 21939	. . . . . 6 ⊢ ℝ = (Base‘ℝ_fld)
16	8, 15	frlmfibas 22095	. . . . 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 6910	. . . . . . . . . 10 ⊢ (𝑖 = 𝑢 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑢)))
21	20	fveq1d 6907	. . . . . . . . 9 ⊢ (𝑖 = 𝑢 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑗))
22		fveq2 6905	. . . . . . . . 9 ⊢ (𝑗 = 𝑣 → ((veronese‘(𝐴‘𝑢))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑣))
23	21, 22	cbvmpov 7535	. . . . . . . 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 2786	. . . . . . 7 ⊢ 𝑉 = (𝑢 ∈ (1...6), 𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
25	24	tposmpo 8305	. . . . . 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 7106	. . . . . 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 51250	. . . . . 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 8106	. . . 4 ⊢ (𝜑 → tpos 𝑉:((1...6) × (1...6))⟶ℝ)
35		1nn 12370	. . . . . . . 8 ⊢ 1 ∈ ℕ
36		6nn 12456	. . . . . . . 8 ⊢ 6 ∈ ℕ
37		1re 11332	. . . . . . . . 9 ⊢ 1 ∈ ℝ
38		6re 12457	. . . . . . . . 9 ⊢ 6 ∈ ℝ
39		1lt6 12554	. . . . . . . . 9 ⊢ 1 < 6
40	37, 38, 39	ltleii 11457	. . . . . . . 8 ⊢ 1 ≤ 6
41		elfz1b 13751	. . . . . . . 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 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 11315	. . . . 5 ⊢ ℝ ∈ V
48	47	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
49		curf 8915	. . . 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 1399	. . 3 ⊢ (𝜑 → curry tpos 𝑉:(1...6)⟶(ℝ ↑_m (1...6)))
51	18, 50	feq3dd 6715	. 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 8924	. . . 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 2866	. 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 51262	. 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 2763	. . 3 ⊢ (Base‘(ℝ_fld freeLMod (1...6))) = (Base‘(ℝ_fld freeLMod (1...6)))
60	8	frlmsca 22086	. . . 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 2763	. . 3 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6))) = ( ·_𝑠 ‘(ℝ_fld freeLMod (1...6)))
63		eqid 2763	. . 3 ⊢ (0_g‘(ℝ_fld freeLMod (1...6))) = (0_g‘(ℝ_fld freeLMod (1...6)))
64		re0g 21945	. . 3 ⊢ 0 = (0_g‘ℝ_fld)
65	59, 61, 62, 63, 64, 59	nellindf 51248	. 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 1413	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 2147 ≠ wne 2959 Vcvv 3454 ∖ cdif 4048 ∅c0 4281 {csn 4588 class class class wbr 5106 × cxp 5653 ⟶wf 6551 ‘cfv 6555 (class class class)co 7440 ∈ cmpo 7442 ∘_f cof 7712 tpos ctpos 8267 curry ccur 8307 ↑_m cmap 8872 Fincfn 8998 ℝcr 11223 0cc0 11224 1c1 11225 + caddc 11227 · cmul 11229 ≤ cle 11368 ℕcn 12359 2c2 12421 3c3 12422 4c4 12423 5c5 12424 6c6 12425 ...cfz 13663 ↑cexp 14228 Basecbs 17413 Scalarcsca 17457 ·_𝑠 cvsca 17458 0_gc0g 17636 Σ_g cgsu 17637 Ringcrg 20486 CRingccrg 20487 DivRingcdr 21007 Fieldcfield 21008 LModclmod 21162 ℝ_fldcrefld 21937 freeLMod cfrlm 22079 LIndF clindf 22137 veronesecveronese 51246

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-addf 11303 ax-mulf 11304

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 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 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-tpos 8268 df-cur 8309 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-sup 9458 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-fz 13664 df-fzo 13813 df-seq 14169 df-exp 14229 df-hash 14499 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-starv 17469 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-unif 17477 df-hom 17478 df-cco 17479 df-0g 17638 df-gsum 17639 df-prds 17644 df-pws 17646 df-mre 17782 df-mrc 17783 df-acs 17785 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-mhm 19004 df-submnd 19005 df-grp 19173 df-minusg 19174 df-sbg 19175 df-mulg 19304 df-subg 19359 df-ghm 19454 df-cntz 19558 df-cmn 20023 df-abl 20024 df-mgp 20388 df-rng 20402 df-ur 20435 df-ring 20488 df-cring 20489 df-oppr 20594 df-dvdsr 20614 df-unit 20615 df-invr 20645 df-dvr 20658 df-nzr 20790 df-subrng 20825 df-subrg 20849 df-drng 21009 df-field 21010 df-lmod 21164 df-lss 21234 df-lsp 21274 df-lmhm 21324 df-lbs 21377 df-sra 21475 df-rgmod 21476 df-cnfld 21706 df-refld 21938 df-dsmm 22065 df-frlm 22080 df-uvc 22116 df-lindf 22139 df-veronese 51247

This theorem is used by: veroquaddetzerod 51264

W3C validator