Theorem extvfv 33677

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 6839	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 ↾ 𝐽)) = (𝐹‘(𝑥 ↾ 𝐽)))
2	1	ifeq1d 4486	. . 3 ⊢ (𝑓 = 𝐹 → if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ) = if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 ))
3	2	mpteq2dv 5179	. 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 33676	. 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 7400	. . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V
14	4, 13	rabex2 5282	. . . 4 ⊢ 𝐷 ∈ V
15	14	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
16	15	mptexd 7179	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) ∈ V)
17	3, 11, 12, 16	fvmptd4 6972	1 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3389  Vcvv 3429   ∖ cdif 3886  ifcif 4466  {csn 4567   class class class wbr 5085   ↦ cmpt 5166   ↾ cres 5633  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773   finSupp cfsupp 9274  0cc0 11038  ℕ₀cn0 12437  Basecbs 17179  0_gc0g 17402   mPoly cmpl 21886  extendVarscextv 33673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-extv 33674

This theorem is referenced by:  extvfvv  33678  extvfvcl  33680  esplyind  33719