Theorem extvfval 33583

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 4585	. . . . . . 7 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	difeq2d 4075	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐼 ∖ {𝑎}) = (𝐼 ∖ {𝐴}))
3		extvfval.j	. . . . . 6 ⊢ 𝐽 = (𝐼 ∖ {𝐴})
4	2, 3	eqtr4di 2786	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐼 ∖ {𝑎}) = 𝐽)
5	4	fvoveq1d 7374	. . . 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 2786	. . 3 ⊢ (𝑎 = 𝐴 → (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) = 𝑀)
8		fveqeq2 6837	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑥‘𝑎) = 0 ↔ (𝑥‘𝐴) = 0))
9	4	reseq2d 5932	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑥 ↾ (𝐼 ∖ {𝑎})) = (𝑥 ↾ 𝐽))
10	9	fveq2d 6832	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))) = (𝑓‘(𝑥 ↾ 𝐽)))
11	8, 10	ifbieq1d 4499	. . . 4 ⊢ (𝑎 = 𝐴 → if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 ) = if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))
12	11	mpteq2dv 5187	. . 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 5180	. 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 2733	. . 3 ⊢ (𝐼 ∖ {𝑎}) = (𝐼 ∖ {𝑎})
19		eqid 2733	. . 3 ⊢ (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) = (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅))
20	14, 15, 16, 17, 18, 19	extvval 33582	. 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 6842	. . . 4 ⊢ 𝑀 ∈ V
23	22	mptex 7163	. . 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 6959	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 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-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:  extvfv  33584  extvfvalf  33588