Theorem veronesematrowd 50981

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 6890	. . . . . . 7 ⊢ (𝑖 = 𝑢 → (veronese‘(𝐴‘𝑖)) = (veronese‘(𝐴‘𝑢)))
3	2	fveq1d 6887	. . . . . 6 ⊢ (𝑖 = 𝑢 → ((veronese‘(𝐴‘𝑖))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑗))
4		fveq2 6885	. . . . . 6 ⊢ (𝑗 = 𝑣 → ((veronese‘(𝐴‘𝑢))‘𝑗) = ((veronese‘(𝐴‘𝑢))‘𝑣))
5	3, 4	cbvmpov 7515	. . . . 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 7085	. . . . . 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 50972	. . . . . 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 12346	. . . . . . 7 ⊢ 1 ∈ ℕ
16		6nn 12432	. . . . . . 7 ⊢ 6 ∈ ℕ
17		1re 11308	. . . . . . . 8 ⊢ 1 ∈ ℝ
18		6re 12433	. . . . . . . 8 ⊢ 6 ∈ ℝ
19		1lt6 12530	. . . . . . . 8 ⊢ 1 < 6
20	17, 18, 19	ltleii 11433	. . . . . . 7 ⊢ 1 ≤ 6
21		elfz1b 13727	. . . . . . 7 ⊢ (1 ∈ (1...6) ↔ (1 ∈ ℕ ∧ 6 ∈ ℕ ∧ 1 ≤ 6))
22	15, 16, 20, 21	mpbir3an 1360	. . . . . 6 ⊢ 1 ∈ (1...6)
23	22	ne0ii 4289	. . . . 5 ⊢ (1...6) ≠ ∅
24	23	a1i 11	. . . 4 ⊢ (𝜑 → (1...6) ≠ ∅)
25	6, 14, 24	mpocurryd 8286	. . 3 ⊢ (𝜑 → curry 𝑉 = (𝑢 ∈ (1...6) ↦ (𝑣 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑢))‘𝑣))))
26		ovex 7453	. . . . . . . 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 6682	. . . . . . 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 7084	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (1...6)) → (𝐴‘𝑢) ∈ (ℝ ↑m (1...3)))
30	29	veronesevald 50971	. . . . . . . 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 6632	. . . . . . 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 6943	. . . . . 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 5197	. . 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 6890	. . 3 ⊢ (𝑢 = 𝑖 → (veronese‘(𝐴‘𝑢)) = (veronese‘(𝐴‘𝑖)))
39	38	cbvmptv 5208	. 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 1569   ∈ wcel 2144   ≠ wne 2955  ∅c0 4278  ifcif 4481   class class class wbr 5102   ↦ cmpt 5185   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  curry ccur 8282   ↑_m cmap 8847  ℝcr 11199  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205   ≤ cle 11344  ℕcn 12335  2c2 12397  3c3 12398  4c4 12399  5c5 12400  6c6 12401  ...cfz 13639  ↑cexp 14204  veronesecveronese 50968

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-cur 8284  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640  df-seq 14145  df-exp 14205  df-veronese 50969

This theorem is used by:  veronesematrowexpd  50982  veroquadgsumlem  50983  veroquadmodzerod  50984