Theorem veronesematrowd 50879

Description: Currying the Veronese matrix gives the indexed family of Veronese images of the points 𝐴‘𝑖. (Contributed by Jiamin Zhao, 19-Aug-2026.)

Hypotheses

Ref	Expression
veronesemat.a	⊢ 𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))
veronesemat.f	⊢ (𝜑 → 𝐴:(1...6)⟶(ℝ ↑m (1...3)))

Assertion

Ref	Expression
veronesematrowd	⊢ (𝜑 → curry 𝑉 = (𝑖 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑖))))

Distinct variable groups:   𝐴,𝑖,𝑗   𝑖,𝑉,𝑗

Allowed substitution hints:   𝜑(𝑖, 𝑗)

Proof of Theorem veronesematrowd

Dummy variables 𝑘 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		veronesemat.a	. . . . 5 ⊢ 𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))
2		2fveq3 6886	. . . . . . 7 ⊢ (𝑖 = 𝑢 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑢)))
3	2	fveq1d 6883	. . . . . 6 ⊢ (𝑖 = 𝑢 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑗))
4		fveq2 6881	. . . . . 6 ⊢ (𝑗 = 𝑣 → ((veronese‘(𝐴‘𝑢))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑣))
5	3, 4	cbvmpov 7511	. . . . 5 ⊢ (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗)) = (𝑢 ∈ (1...6), 𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
6	1, 5	eqtri 2783	. . . 4 ⊢ 𝑉 = (𝑢 ∈ (1...6), 𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))
7		veronesemat.f	. . . . . . . 8 ⊢ (𝜑 → 𝐴:(1...6)⟶(ℝ ↑m (1...3)))
8	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...6) ∧ 𝑣 ∈ (1...6))) → 𝐴:(1...6)⟶(ℝ ↑m (1...3)))
9		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...6) ∧ 𝑣 ∈ (1...6))) → 𝑢 ∈ (1...6))
10	8, 9	ffvelcdmd 7081	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...6) ∧ 𝑣 ∈ (1...6))) → (𝐴‘𝑢) ∈ (ℝ ↑m (1...3)))
11		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...6) ∧ 𝑣 ∈ (1...6))) → 𝑣 ∈ (1...6))
12		veronesefvcl 50870	. . . . . 6 ⊢ (((𝐴‘𝑢) ∈ (ℝ ↑m (1...3)) ∧ 𝑣 ∈ (1...6)) → ((veronese‘(𝐴‘𝑢))‘𝑣) ∈ ℝ)
13	10, 11, 12	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...6) ∧ 𝑣 ∈ (1...6))) → ((veronese‘(𝐴‘𝑢))‘𝑣) ∈ ℝ)
14	13	ralrimivva 3205	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (1...6)∀𝑣 ∈ (1...6)((veronese‘(𝐴‘𝑢))‘𝑣) ∈ ℝ)
15		1nn 12293	. . . . . . 7 ⊢ 1 ∈ ℕ
16		6nn 12379	. . . . . . 7 ⊢ 6 ∈ ℕ
17		1re 11257	. . . . . . . 8 ⊢ 1 ∈ ℝ
18		6re 12380	. . . . . . . 8 ⊢ 6 ∈ ℝ
19		1lt6 12477	. . . . . . . 8 ⊢ 1 < 6
20	17, 18, 19	ltleii 11382	. . . . . . 7 ⊢ 1 ≤ 6
21		elfz1b 13673	. . . . . . 7 ⊢ (1 ∈ (1...6) ↔ (1 ∈ ℕ ∧ 6 ∈ ℕ ∧ 1 ≤ 6))
22	15, 16, 20, 21	mpbir3an 1360	. . . . . 6 ⊢ 1 ∈ (1...6)
23	22	ne0ii 4290	. . . . 5 ⊢ (1...6) ≠ ∅
24	23	a1i 11	. . . 4 ⊢ (𝜑 → (1...6) ≠ ∅)
25	6, 14, 24	mpocurryd 8272	. . 3 ⊢ (𝜑 → curry 𝑉 = (𝑢 ∈ (1...6) ↦ (𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))))
26		ovex 7449	. . . . . . . 8 ⊢ (((if(𝑘 = 1, (((𝐴‘𝑢)‘1)↑2), 0) + if(𝑘 = 2, (((𝐴‘𝑢)‘2)↑2), 0)) + if(𝑘 = 3, (((𝐴‘𝑢)‘3)↑2), 0)) + ((if(𝑘 = 4, (((𝐴‘𝑢)‘1) · ((𝐴‘𝑢)‘2)), 0) + if(𝑘 = 5, (((𝐴‘𝑢)‘2) · ((𝐴‘𝑢)‘3)), 0)) + if(𝑘 = 6, (((𝐴‘𝑢)‘3) · ((𝐴‘𝑢)‘1)), 0))) ∈ V
27		eqid 2760	. . . . . . . 8 ⊢ (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, (((𝐴‘𝑢)‘1)↑2), 0) + if(𝑘 = 2, (((𝐴‘𝑢)‘2)↑2), 0)) + if(𝑘 = 3, (((𝐴‘𝑢)‘3)↑2), 0)) + ((if(𝑘 = 4, (((𝐴‘𝑢)‘1) · ((𝐴‘𝑢)‘2)), 0) + if(𝑘 = 5, (((𝐴‘𝑢)‘2) · ((𝐴‘𝑢)‘3)), 0)) + if(𝑘 = 6, (((𝐴‘𝑢)‘3) · ((𝐴‘𝑢)‘1)), 0)))) = (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, (((𝐴‘𝑢)‘1)↑2), 0) + if(𝑘 = 2, (((𝐴‘𝑢)‘2)↑2), 0)) + if(𝑘 = 3, (((𝐴‘𝑢)‘3)↑2), 0)) + ((if(𝑘 = 4, (((𝐴‘𝑢)‘1) · ((𝐴‘𝑢)‘2)), 0) + if(𝑘 = 5, (((𝐴‘𝑢)‘2) · ((𝐴‘𝑢)‘3)), 0)) + if(𝑘 = 6, (((𝐴‘𝑢)‘3) · ((𝐴‘𝑢)‘1)), 0))))
28	26, 27	fnmpti 6678	. . . . . . 7 ⊢ (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, (((𝐴‘𝑢)‘1)↑2), 0) + if(𝑘 = 2, (((𝐴‘𝑢)‘2)↑2), 0)) + if(𝑘 = 3, (((𝐴‘𝑢)‘3)↑2), 0)) + ((if(𝑘 = 4, (((𝐴‘𝑢)‘1) · ((𝐴‘𝑢)‘2)), 0) + if(𝑘 = 5, (((𝐴‘𝑢)‘2) · ((𝐴‘𝑢)‘3)), 0)) + if(𝑘 = 6, (((𝐴‘𝑢)‘3) · ((𝐴‘𝑢)‘1)), 0)))) Fn (1...6)
29	7	ffvelcdmda 7080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (1...6)) → (𝐴‘𝑢) ∈ (ℝ ↑m (1...3)))
30	29	veronesevald 50869	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (1...6)) → (veronese‘(𝐴‘𝑢)) = (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, (((𝐴‘𝑢)‘1)↑2), 0) + if(𝑘 = 2, (((𝐴‘𝑢)‘2)↑2), 0)) + if(𝑘 = 3, (((𝐴‘𝑢)‘3)↑2), 0)) + ((if(𝑘 = 4, (((𝐴‘𝑢)‘1) · ((𝐴‘𝑢)‘2)), 0) + if(𝑘 = 5, (((𝐴‘𝑢)‘2) · ((𝐴‘𝑢)‘3)), 0)) + if(𝑘 = 6, (((𝐴‘𝑢)‘3) · ((𝐴‘𝑢)‘1)), 0)))))
31	30	fneq1d 6628	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (1...6)) → ((veronese‘(𝐴‘𝑢)) Fn (1...6) ↔ (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, (((𝐴‘𝑢)‘1)↑2), 0) + if(𝑘 = 2, (((𝐴‘𝑢)‘2)↑2), 0)) + if(𝑘 = 3, (((𝐴‘𝑢)‘3)↑2), 0)) + ((if(𝑘 = 4, (((𝐴‘𝑢)‘1) · ((𝐴‘𝑢)‘2)), 0) + if(𝑘 = 5, (((𝐴‘𝑢)‘2) · ((𝐴‘𝑢)‘3)), 0)) + if(𝑘 = 6, (((𝐴‘𝑢)‘3) · ((𝐴‘𝑢)‘1)), 0)))) Fn (1...6)))
32	28, 31	mpbiri 261	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (1...6)) → (veronese‘(𝐴‘𝑢)) Fn (1...6))
33		dffn5 6939	. . . . . 6 ⊢ ((veronese‘(𝐴‘𝑢)) Fn (1...6) ↔ (veronese‘(𝐴‘𝑢)) = (𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣)))
34	32, 33	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (1...6)) → (veronese‘(𝐴‘𝑢)) = (𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣)))
35	34	eqcomd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (1...6)) → (𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣)) = (veronese‘(𝐴‘𝑢)))
36	35	mpteq2dva 5198	. . 3 ⊢ (𝜑 → (𝑢 ∈ (1...6) ↦ (𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))) = (𝑢 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑢))))
37	25, 36	eqtrd 2795	. 2 ⊢ (𝜑 → curry 𝑉 = (𝑢 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑢))))
38		2fveq3 6886	. . 3 ⊢ (𝑢 = 𝑖 → (veronese‘(𝐴‘𝑢)) = (veronese‘(𝐴‘𝑖)))
39	38	cbvmptv 5209	. 2 ⊢ (𝑢 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑢))) = (𝑖 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑖)))
40	37, 39	eqtrdi 2811	1 ⊢ (𝜑 → curry 𝑉 = (𝑖 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑖))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∅c0 4279  ifcif 4482   class class class wbr 5103   ↦ cmpt 5186   Fn wfn 6530  ⟶wf 6531  ‘cfv 6535  (class class class)co 7416   ∈ cmpo 7418  curry ccur 8268   ↑_m cmap 8833  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  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

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-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-op 4591  df-uni 4868  df-iun 4953  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-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-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-cur 8270  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-er 8703  df-map 8835  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-nn 12283  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-n0 12554  df-z 12641  df-uz 12913  df-fz 13587  df-seq 14091  df-exp 14151  df-veronese 50867

This theorem is used by:  veronesematrowexpd  50880  veroquadgsumlem  50881  veroquadmodzerod  50882