Theorem extvfval 33697

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 4590	. . . . . . 7 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	difeq2d 4078	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐼 ∖ {𝑎}) = (𝐼 ∖ {𝐴}))
3		extvfval.j	. . . . . 6 ⊢ 𝐽 = (𝐼 ∖ {𝐴})
4	2, 3	eqtr4di 2789	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐼 ∖ {𝑎}) = 𝐽)
5	4	fvoveq1d 7380	. . . 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 2789	. . 3 ⊢ (𝑎 = 𝐴 → (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) = 𝑀)
8		fveqeq2 6843	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑥‘𝑎) = 0 ↔ (𝑥‘𝐴) = 0))
9	4	reseq2d 5938	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑥 ↾ (𝐼 ∖ {𝑎})) = (𝑥 ↾ 𝐽))
10	9	fveq2d 6838	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))) = (𝑓‘(𝑥 ↾ 𝐽)))
11	8, 10	ifbieq1d 4504	. . . 4 ⊢ (𝑎 = 𝐴 → if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 ) = if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))
12	11	mpteq2dv 5192	. . 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 5185	. 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 2736	. . 3 ⊢ (𝐼 ∖ {𝑎}) = (𝐼 ∖ {𝑎})
19		eqid 2736	. . 3 ⊢ (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) = (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅))
20	14, 15, 16, 17, 18, 19	extvval 33696	. 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 6848	. . . 4 ⊢ 𝑀 ∈ V
23	22	mptex 7169	. . 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 6965	1 ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ∖ cdif 3898  ifcif 4479  {csn 4580   class class class wbr 5098   ↦ cmpt 5179   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763   finSupp cfsupp 9264  0cc0 11026  ℕ₀cn0 12401  Basecbs 17136  0_gc0g 17359   mPoly cmpl 21862  extendVarscextv 33694

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-extv 33695

This theorem is referenced by:  extvfv  33698  extvfvalf  33702