Theorem extvfval 34050

Description: The "variable extension" function evaluated for 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))

Assertion

Ref	Expression
extvfval	⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))))

Distinct variable groups:   𝑓,𝐼,ℎ,𝑥   𝑅,𝑓,𝑥   𝐴,𝑓,𝑥   𝑓,𝑀

Allowed substitution hints:   𝜑(𝑥, 𝑓, ℎ)   𝐴(ℎ)   𝐷(𝑥, 𝑓, ℎ)   𝑅(ℎ)   𝐽(𝑥, 𝑓, ℎ)   𝑀(𝑥, ℎ)   𝑉(𝑥, 𝑓, ℎ)   𝑊(𝑥, 𝑓, ℎ)   0 (𝑥, 𝑓, ℎ)

Proof of Theorem extvfval

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4597	. . . . . . 7 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	difeq2d 4077	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐼 ∖ {𝑎}) = (𝐼 ∖ {𝐴}))
3		extvfval.j	. . . . . 6 ⊢ 𝐽 = (𝐼 ∖ {𝐴})
4	2, 3	eqtr4di 2815	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐼 ∖ {𝑎}) = 𝐽)
5	4	fvoveq1d 7439	. . . 4 ⊢ (𝑎 = 𝐴 → (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) = (Base‘(𝐽 mPoly 𝑅)))
6		extvfval.m	. . . 4 ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))
7	5, 6	eqtr4di 2815	. . 3 ⊢ (𝑎 = 𝐴 → (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) = 𝑀)
8		fveqeq2 6891	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑥‘𝑎) = 0 ↔ (𝑥‘𝐴) = 0))
9	4	reseq2d 5976	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑥 ↾ (𝐼 ∖ {𝑎})) = (𝑥 ↾ 𝐽))
10	9	fveq2d 6886	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))) = (𝑓‘(𝑥 ↾ 𝐽)))
11	8, 10	ifbieq1d 4510	. . . 4 ⊢ (𝑎 = 𝐴 → if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 ) = if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))
12	11	mpteq2dv 5203	. . 3 ⊢ (𝑎 = 𝐴 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 )))
13	7, 12	mpteq12dv 5196	. 2 ⊢ (𝑎 = 𝐴 → (𝑓 ∈ (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 ))) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))))
14		extvval.d	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
15		extvval.1	. . 3 ⊢ 0 = (0g‘𝑅)
16		extvval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
17		extvval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
18		eqid 2762	. . 3 ⊢ (𝐼 ∖ {𝑎}) = (𝐼 ∖ {𝑎})
19		eqid 2762	. . 3 ⊢ (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) = (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅))
20	14, 15, 16, 17, 18, 19	extvval 34049	. 2 ⊢ (𝜑 → (𝐼extendVars𝑅) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))))
21		extvfval.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐼)
22	6	fvexi 6896	. . . 4 ⊢ 𝑀 ∈ V
23	22	mptex 7226	. . 3 ⊢ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))) ∈ V
24	23	a1i 11	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))) ∈ V)
25	13, 20, 21, 24	fvmptd4 7015	1 ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453   ∖ cdif 3899  ifcif 4485  {csn 4587   class class class wbr 5107   ↦ cmpt 5190   ↾ cres 5661  ‘cfv 6537  (class class class)co 7417   ↑_m cmap 8830   finSupp cfsupp 9335  0cc0 11128  ℕ₀cn0 12532  Basecbs 17307  0_gc0g 17530   mPoly cmpl 22127  extendVarscextv 34047

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-extv 34048

This theorem is used by:  extvfv  34051  extvfvalf  34055