Theorem extvfv 33954

Description: The "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)

Hypotheses

Ref	Expression
extvval.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
extvval.1	⊢ 0 = (0g‘𝑅)
extvval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
extvval.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)
extvfval.a	⊢ (𝜑 → 𝐴 ∈ 𝐼)
extvfval.j	⊢ 𝐽 = (𝐼 ∖ {𝐴})
extvfval.m	⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))
extvfv.1	⊢ (𝜑 → 𝐹 ∈ 𝑀)

Assertion

Ref	Expression
extvfv	⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))

Distinct variable groups:   ℎ,𝐼,𝑥   𝑥,𝑅   𝑥,𝐴   𝑥,𝐷   𝑥,𝐹

Allowed substitution hints:   𝜑(𝑥, ℎ)   𝐴(ℎ)   𝐷(ℎ)   𝑅(ℎ)   𝐹(ℎ)   𝐽(𝑥, ℎ)   𝑀(𝑥, ℎ)   𝑉(𝑥, ℎ)   𝑊(𝑥, ℎ)   0 (𝑥, ℎ)

Proof of Theorem extvfv

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6887	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 ↾ 𝐽)) = (𝐹‘(𝑥 ↾ 𝐽)))
2	1	ifeq1d 4512	. . 3 ⊢ (𝑓 = 𝐹 → if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ) = if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 ))
3	2	mpteq2dv 5210	. 2 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 )) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
4		extvval.d	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
5		extvval.1	. . 3 ⊢ 0 = (0g‘𝑅)
6		extvval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
7		extvval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
8		extvfval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐼)
9		extvfval.j	. . 3 ⊢ 𝐽 = (𝐼 ∖ {𝐴})
10		extvfval.m	. . 3 ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))
11	4, 5, 6, 7, 8, 9, 10	extvfval 33953	. 2 ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))))
12		extvfv.1	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑀)
13		ovex 7456	. . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V
14	4, 13	rabex2 5316	. . . 4 ⊢ 𝐷 ∈ V
15	14	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
16	15	mptexd 7229	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) ∈ V)
17	3, 11, 12, 16	fvmptd4 7021	1 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {crab 3419  Vcvv 3458   ∖ cdif 3905  ifcif 4492  {csn 4594   class class class wbr 5114   ↦ cmpt 5197   ↾ cres 5668  ‘cfv 6543  (class class class)co 7423   ↑_m cmap 8833   finSupp cfsupp 9331  0cc0 11118  ℕ₀cn0 12522  Basecbs 17294  0_gc0g 17517   mPoly cmpl 22093  extendVarscextv 33950

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-extv 33951

This theorem is used by:  extvfvv  33955  extvfvcl  33957  esplyind  33996