Theorem extvfv 33584

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 6827	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 ↾ 𝐽)) = (𝐹‘(𝑥 ↾ 𝐽)))
2	1	ifeq1d 4494	. . 3 ⊢ (𝑓 = 𝐹 → if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ) = if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 ))
3	2	mpteq2dv 5187	. 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 33583	. 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 7385	. . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V
14	4, 13	rabex2 5281	. . . 4 ⊢ 𝐷 ∈ V
15	14	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
16	15	mptexd 7164	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) ∈ V)
17	3, 11, 12, 16	fvmptd4 6959	1 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {crab 3396  Vcvv 3437   ∖ cdif 3895  ifcif 4474  {csn 4575   class class class wbr 5093   ↦ cmpt 5174   ↾ cres 5621  ‘cfv 6486  (class class class)co 7352   ↑_m cmap 8756   finSupp cfsupp 9252  0cc0 11013  ℕ₀cn0 12388  Basecbs 17122  0_gc0g 17345   mPoly cmpl 21845  extendVarscextv 33580

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-extv 33581

This theorem is referenced by:  extvfvv  33585  extvfvcl  33587  esplyind  33613