Theorem extvfvv 33841

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 6870	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥‘𝐴) = (𝑋‘𝐴))
2	1	eqeq1d 2767	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝐴) = 0 ↔ (𝑋‘𝐴) = 0))
3		reseq1 5963	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ↾ 𝐽) = (𝑋 ↾ 𝐽))
4	3	fveq2d 6875	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 ↾ 𝐽)) = (𝐹‘(𝑋 ↾ 𝐽)))
5	2, 4	ifbieq1d 4508	. 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 33840	. 2 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
15		extvfvv.1	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
16		fvexd 6886	. . 3 ⊢ (𝜑 → (𝐹‘(𝑋 ↾ 𝐽)) ∈ V)
17	7	fvexi 6885	. . . 4 ⊢ 0 ∈ V
18	17	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
19	16, 18	ifcld 4530	. 2 ⊢ (𝜑 → if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ) ∈ V)
20	5, 14, 15, 19	fvmptd4 7004	1 ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) = if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  {crab 3417  Vcvv 3457   ∖ cdif 3904  ifcif 4483  {csn 4585   class class class wbr 5105   ↾ cres 5654  ‘cfv 6525  (class class class)co 7400   ↑_m cmap 8812   finSupp cfsupp 9309  0cc0 11088  ℕ₀cn0 12495  Basecbs 17259  0_gc0g 17482   mPoly cmpl 22016  extendVarscextv 33836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-extv 33837

This theorem is referenced by:  extvfvvcl  33842  evlextv  33849  esplyind  33882