Theorem extvfvv 33699

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)
extvfvv.1	⊢ (𝜑 → 𝑋 ∈ 𝐷)

Assertion

Ref	Expression
extvfvv	⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) = if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ))

Distinct variable group:   ℎ,𝐼

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

Proof of Theorem extvfvv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6833	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥‘𝐴) = (𝑋‘𝐴))
2	1	eqeq1d 2738	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝐴) = 0 ↔ (𝑋‘𝐴) = 0))
3		reseq1 5932	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ↾ 𝐽) = (𝑋 ↾ 𝐽))
4	3	fveq2d 6838	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 ↾ 𝐽)) = (𝐹‘(𝑋 ↾ 𝐽)))
5	2, 4	ifbieq1d 4504	. 2 ⊢ (𝑥 = 𝑋 → if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 ) = if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ))
6		extvval.d	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
7		extvval.1	. . 3 ⊢ 0 = (0g‘𝑅)
8		extvval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
9		extvval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
10		extvfval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐼)
11		extvfval.j	. . 3 ⊢ 𝐽 = (𝐼 ∖ {𝐴})
12		extvfval.m	. . 3 ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))
13		extvfv.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑀)
14	6, 7, 8, 9, 10, 11, 12, 13	extvfv 33698	. 2 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
15		extvfvv.1	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
16		fvexd 6849	. . 3 ⊢ (𝜑 → (𝐹‘(𝑋 ↾ 𝐽)) ∈ V)
17	7	fvexi 6848	. . . 4 ⊢ 0 ∈ V
18	17	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
19	16, 18	ifcld 4526	. 2 ⊢ (𝜑 → if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ) ∈ V)
20	5, 14, 15, 19	fvmptd4 6965	1 ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) = if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ∖ cdif 3898  ifcif 4479  {csn 4580   class class class wbr 5098   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763   finSupp cfsupp 9264  0cc0 11026  ℕ₀cn0 12401  Basecbs 17136  0_gc0g 17359   mPoly cmpl 21862  extendVarscextv 33694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-extv 33695

This theorem is referenced by:  extvfvvcl  33700  evlextv  33707  esplyind  33731