Theorem extvfval 33790

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 4591	. . . . . . 7 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	difeq2d 4080	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐼 ∖ {𝑎}) = (𝐼 ∖ {𝐴}))
3		extvfval.j	. . . . . 6 ⊢ 𝐽 = (𝐼 ∖ {𝐴})
4	2, 3	eqtr4di 2814	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐼 ∖ {𝑎}) = 𝐽)
5	4	fvoveq1d 7414	. . . 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 2814	. . 3 ⊢ (𝑎 = 𝐴 → (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) = 𝑀)
8		fveqeq2 6872	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑥‘𝑎) = 0 ↔ (𝑥‘𝐴) = 0))
9	4	reseq2d 5963	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑥 ↾ (𝐼 ∖ {𝑎})) = (𝑥 ↾ 𝐽))
10	9	fveq2d 6867	. . . . 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 5193	. . 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 5186	. 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 2761	. . 3 ⊢ (𝐼 ∖ {𝑎}) = (𝐼 ∖ {𝑎})
19		eqid 2761	. . 3 ⊢ (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅)) = (Base‘((𝐼 ∖ {𝑎}) mPoly 𝑅))
20	14, 15, 16, 17, 18, 19	extvval 33789	. 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 6877	. . . 4 ⊢ 𝑀 ∈ V
23	22	mptex 7203	. . 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 6996	1 ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453   ∖ cdif 3901  ifcif 4479  {csn 4581   class class class wbr 5099   ↦ cmpt 5180   ↾ cres 5647  ‘cfv 6517  (class class class)co 7392   ↑_m cmap 8803   finSupp cfsupp 9304  0cc0 11070  ℕ₀cn0 12478  Basecbs 17228  0_gc0g 17451   mPoly cmpl 21938  extendVarscextv 33787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-extv 33788

This theorem is referenced by:  extvfv  33791  extvfvalf  33795