Theorem extvfval 34234

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 4594	. . . . . . 7 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	difeq2d 4066	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐼 ∖ {𝑎}) = (𝐼 ∖ {𝐴}))
3		extvfval.j	. . . . . 6 ⊢ 𝐽 = (𝐼 ∖ {𝐴})
4	2, 3	eqtr4di 2816	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐼 ∖ {𝑎}) = 𝐽)
5	4	fvoveq1d 7465	. . . 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 2816	. . 3 ⊢ (𝑎 = 𝐴 → (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) = 𝑀)
8		fveqeq2 6917	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑥‘𝑎) = 0 ↔ (𝑥‘𝐴) = 0))
9	4	reseq2d 5981	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑥 ↾ (𝐼 ∖ {𝑎})) = (𝑥 ↾ 𝐽))
10	9	fveq2d 6912	. . . . 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 5202	. . 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 5195	. 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 2763	. . 3 ⊢ (𝐼 ∖ {𝑎}) = (𝐼 ∖ {𝑎})
19		eqid 2763	. . 3 ⊢ (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) = (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅))
20	14, 15, 16, 17, 18, 19	extvval 34233	. 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 6922	. . . 4 ⊢ 𝑀 ∈ V
23	22	mptex 7252	. . 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 7041	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 2147  {crab 3416  Vcvv 3454   ∖ cdif 4048  ifcif 4485  {csn 4588   class class class wbr 5106   ↦ cmpt 5189   ↾ cres 5657  ‘cfv 6558  (class class class)co 7443   ↑_m cmap 8875   finSupp cfsupp 9381  0cc0 11227  ℕ₀cn0 12633  Basecbs 17416  0_gc0g 17639   mPoly cmpl 22244  extendVarscextv 34231

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-extv 34232

This theorem is used by:  extvfv  34235  extvfvalf  34239