Theorem extvfv 34235

Description: The "variable extension" function evaluated for converting a given polynomial 𝐹 by 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))
extvfv.1	⊢ (𝜑 → 𝐹 ∈ 𝑀)

Assertion

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

Distinct variable groups:   ℎ,𝐼,𝑥   𝑥,𝑅   𝑥,𝐴   𝑥,𝐷   𝑥,𝐹

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

Proof of Theorem extvfv

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6907	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 ↾ 𝐽)) = (𝐹‘(𝑥 ↾ 𝐽)))
2	1	ifeq1d 4505	. . 3 ⊢ (𝑓 = 𝐹 → if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ) = if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 ))
3	2	mpteq2dv 5202	. 2 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 )) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
4		extvval.d	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
5		extvval.1	. . 3 ⊢ 0 = (0g‘𝑅)
6		extvval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
7		extvval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
8		extvfval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐼)
9		extvfval.j	. . 3 ⊢ 𝐽 = (𝐼 ∖ {𝐴})
10		extvfval.m	. . 3 ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))
11	4, 5, 6, 7, 8, 9, 10	extvfval 34234	. 2 ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))))
12		extvfv.1	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑀)
13		ovex 7476	. . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V
14	4, 13	rabex2 5306	. . . 4 ⊢ 𝐷 ∈ V
15	14	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
16	15	mptexd 7253	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) ∈ V)
17	3, 11, 12, 16	fvmptd4 7041	1 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (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-pw 4562  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:  extvfvv  34236  extvfvcl  34238  esplyind  34277