Theorem extvfvv 34148

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 6876	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥‘𝐴) = (𝑋‘𝐴))
2	1	eqeq1d 2763	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝐴) = 0 ↔ (𝑋‘𝐴) = 0))
3		reseq1 5964	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ↾ 𝐽) = (𝑋 ↾ 𝐽))
4	3	fveq2d 6881	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 ↾ 𝐽)) = (𝐹‘(𝑋 ↾ 𝐽)))
5	2, 4	ifbieq1d 4507	. 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 34147	. 2 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
15		extvfvv.1	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
16		fvexd 6892	. . 3 ⊢ (𝜑 → (𝐹‘(𝑋 ↾ 𝐽)) ∈ V)
17	7	fvexi 6891	. . . 4 ⊢ 0 ∈ V
18	17	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
19	16, 18	ifcld 4529	. 2 ⊢ (𝜑 → if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ) ∈ V)
20	5, 14, 15, 19	fvmptd4 7010	1 ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) = if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {crab 3413  Vcvv 3451   ∖ cdif 3896  ifcif 4482  {csn 4584   class class class wbr 5103   ↾ cres 5653  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831   finSupp cfsupp 9337  0cc0 11181  ℕ₀cn0 12587  Basecbs 17367  0_gc0g 17590   mPoly cmpl 22194  extendVarscextv 34143

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-extv 34144

This theorem is used by:  extvfvvcl  34149  evlextv  34156  esplyind  34189