Theorem veronesematrowd 50796

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 6887	. . . . . . 7 ⊢ (𝑖 = 𝑢 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑢)))
3	2	fveq1d 6884	. . . . . 6 ⊢ (𝑖 = 𝑢 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑗))
4		fveq2 6882	. . . . . 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 2785	. . . 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 50787	. . . . . 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 3207	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (1...6)∀𝑣 ∈ (1...6)((veronese‘(𝐴‘𝑢))‘𝑣) ∈ ℝ)
15		1nn 12269	. . . . . . 7 ⊢ 1 ∈ ℕ
16		6nn 12355	. . . . . . 7 ⊢ 6 ∈ ℕ
17		1re 11233	. . . . . . . 8 ⊢ 1 ∈ ℝ
18		6re 12356	. . . . . . . 8 ⊢ 6 ∈ ℝ
19		1lt6 12453	. . . . . . . 8 ⊢ 1 < 6
20	17, 18, 19	ltleii 11358	. . . . . . 7 ⊢ 1 ≤ 6
21		elfz1b 13648	. . . . . . 7 ⊢ (1 ∈ (1...6) ↔ (1 ∈ ℕ ∧ 6 ∈ ℕ ∧ 1 ≤ 6))
22	15, 16, 20, 21	mpbir3an 1360	. . . . . 6 ⊢ 1 ∈ (1...6)
23	22	ne0ii 4293	. . . . 5 ⊢ (1...6) ≠ ∅
24	23	a1i 11	. . . 4 ⊢ (𝜑 → (1...6) ≠ ∅)
25	6, 14, 24	mpocurryd 8270	. . 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 2762	. . . . . . . 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 6679	. . . . . . 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 50786	. . . . . . . 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 6629	. . . . . . 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 6940	. . . . . 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 2768	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (1...6)) → (𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣)) = (veronese‘(𝐴‘𝑢)))
36	35	mpteq2dva 5202	. . 3 ⊢ (𝜑 → (𝑢 ∈ (1...6) ↦ (𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))) = (𝑢 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑢))))
37	25, 36	eqtrd 2797	. 2 ⊢ (𝜑 → curry 𝑉 = (𝑢 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑢))))
38		2fveq3 6887	. . 3 ⊢ (𝑢 = 𝑖 → (veronese‘(𝐴‘𝑢)) = (veronese‘(𝐴‘𝑖)))
39	38	cbvmptv 5213	. 2 ⊢ (𝑢 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑢))) = (𝑖 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑖)))
40	37, 39	eqtrdi 2813	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 2957  ∅c0 4282  ifcif 4485   class class class wbr 5107   ↦ cmpt 5190   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418  curry ccur 8266   ↑_m cmap 8829  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  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

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-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-op 4594  df-uni 4871  df-iun 4956  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-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-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-cur 8268  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-5 12331  df-6 12332  df-n0 12530  df-z 12617  df-uz 12889  df-fz 13562  df-seq 14066  df-exp 14126  df-veronese 50784

This theorem is used by:  veronesematrowexpd  50797  veroquadgsumlem  50798  veroquadmodzerod  50799