Theorem extvfv 33901

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 6882	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 ↾ 𝐽)) = (𝐹‘(𝑥 ↾ 𝐽)))
2	1	ifeq1d 4508	. . 3 ⊢ (𝑓 = 𝐹 → if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ) = if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 ))
3	2	mpteq2dv 5206	. 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 33900	. 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 7445	. . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V
14	4, 13	rabex2 5313	. . . 4 ⊢ 𝐷 ∈ V
15	14	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
16	15	mptexd 7224	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) ∈ V)
17	3, 11, 12, 16	fvmptd4 7016	1 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   ∖ cdif 3903  ifcif 4488  {csn 4590   class class class wbr 5110   ↦ cmpt 5193   ↾ cres 5665  ‘cfv 6538  (class class class)co 7412   ↑_m cmap 8825   finSupp cfsupp 9322  0cc0 11101  ℕ₀cn0 12505  Basecbs 17270  0_gc0g 17493   mPoly cmpl 22037  extendVarscextv 33897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7415  df-oprab 7416  df-mpo 7417  df-extv 33898

This theorem is referenced by:  extvfvv  33902  extvfvcl  33904  esplyind  33943