Theorem veronesematrowd 51259

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 6910	. . . . . . 7 ⊢ (𝑖 = 𝑢 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑢)))
3	2	fveq1d 6907	. . . . . 6 ⊢ (𝑖 = 𝑢 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑗))
4		fveq2 6905	. . . . . 6 ⊢ (𝑗 = 𝑣 → ((veronese‘(𝐴‘𝑢))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑣))
5	3, 4	cbvmpov 7535	. . . . 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 2786	. . . 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 7106	. . . . . 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 51250	. . . . . 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 3209	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (1...6)∀𝑣 ∈ (1...6)((veronese‘(𝐴‘𝑢))‘𝑣) ∈ ℝ)
15		1nn 12370	. . . . . . 7 ⊢ 1 ∈ ℕ
16		6nn 12456	. . . . . . 7 ⊢ 6 ∈ ℕ
17		1re 11332	. . . . . . . 8 ⊢ 1 ∈ ℝ
18		6re 12457	. . . . . . . 8 ⊢ 6 ∈ ℝ
19		1lt6 12554	. . . . . . . 8 ⊢ 1 < 6
20	17, 18, 19	ltleii 11457	. . . . . . 7 ⊢ 1 ≤ 6
21		elfz1b 13751	. . . . . . 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 8311	. . 3 ⊢ (𝜑 → curry 𝑉 = (𝑢 ∈ (1...6) ↦ (𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))))
26		ovex 7473	. . . . . . . 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 2763	. . . . . . . 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 6701	. . . . . . 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 7105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (1...6)) → (𝐴‘𝑢) ∈ (ℝ ↑m (1...3)))
30	29	veronesevald 51249	. . . . . . . 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 6651	. . . . . . 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 6963	. . . . . 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 2769	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (1...6)) → (𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣)) = (veronese‘(𝐴‘𝑢)))
36	35	mpteq2dva 5201	. . 3 ⊢ (𝜑 → (𝑢 ∈ (1...6) ↦ (𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))) = (𝑢 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑢))))
37	25, 36	eqtrd 2798	. 2 ⊢ (𝜑 → curry 𝑉 = (𝑢 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑢))))
38		2fveq3 6910	. . 3 ⊢ (𝑢 = 𝑖 → (veronese‘(𝐴‘𝑢)) = (veronese‘(𝐴‘𝑖)))
39	38	cbvmptv 5212	. 2 ⊢ (𝑢 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑢))) = (𝑖 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑖)))
40	37, 39	eqtrdi 2814	1 ⊢ (𝜑 → curry 𝑉 = (𝑖 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑖))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∅c0 4281  ifcif 4485   class class class wbr 5106   ↦ cmpt 5189   Fn wfn 6550  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ∈ cmpo 7442  curry ccur 8307   ↑_m cmap 8872  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  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

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-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-op 4839  df-uni 4871  df-iun 4956  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-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-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-cur 8309  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-n0 12631  df-z 12718  df-uz 12990  df-fz 13664  df-seq 14169  df-exp 14229  df-veronese 51247

This theorem is used by:  veronesematrowexpd  51260  veroquadgsumlem  51261  veroquadmodzerod  51262