Theorem extvfvv 34212

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 6884	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥‘𝐴) = (𝑋‘𝐴))
2	1	eqeq1d 2762	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝐴) = 0 ↔ (𝑋‘𝐴) = 0))
3		reseq1 5963	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ↾ 𝐽) = (𝑋 ↾ 𝐽))
4	3	fveq2d 6889	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 ↾ 𝐽)) = (𝐹‘(𝑋 ↾ 𝐽)))
5	2, 4	ifbieq1d 4506	. 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 34211	. 2 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
15		extvfvv.1	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
16		fvexd 6900	. . 3 ⊢ (𝜑 → (𝐹‘(𝑋 ↾ 𝐽)) ∈ V)
17	7	fvexi 6899	. . . 4 ⊢ 0 ∈ V
18	17	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
19	16, 18	ifcld 4528	. 2 ⊢ (𝜑 → if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ) ∈ V)
20	5, 14, 15, 19	fvmptd4 7018	1 ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) = if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {crab 3412  Vcvv 3450   ∖ cdif 4045  ifcif 4481  {csn 4584   class class class wbr 5102   ↾ cres 5652  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8850   finSupp cfsupp 9356  0cc0 11203  ℕ₀cn0 12609  Basecbs 17392  0_gc0g 17615   mPoly cmpl 22219  extendVarscextv 34207

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-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  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-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-extv 34208

This theorem is used by:  extvfvvcl  34213  evlextv  34220  esplyind  34253