Theorem extvval 34145

Description: Value of the "variable extension" function. (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)
extvval.j	⊢ 𝐽 = (𝐼 ∖ {𝑎})
extvval.m	⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))

Assertion

Ref	Expression
extvval	⊢ (𝜑 → (𝐼extendVars𝑅) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))))

Distinct variable groups:   𝐼,𝑎,𝑓,ℎ,𝑥   𝑅,𝑎,𝑓,𝑥

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

Proof of Theorem extvval

Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-extv 34144	. . 3 ⊢ extendVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑎 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟))))))
2	1	a1i 11	. 2 ⊢ (𝜑 → extendVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑎 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟)))))))
3		simpl 488	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑖 = 𝐼)
4		difeq1 4067	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑖 ∖ {𝑎}) = (𝐼 ∖ {𝑎}))
5		extvval.j	. . . . . . . . . 10 ⊢ 𝐽 = (𝐼 ∖ {𝑎})
6	4, 5	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑖 ∖ {𝑎}) = 𝐽)
7	6	adantr 486	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 ∖ {𝑎}) = 𝐽)
8		simpr 490	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
9	7, 8	oveq12d 7430	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 ∖ {𝑎}) mPoly 𝑟) = (𝐽 mPoly 𝑅))
10	9	fveq2d 6881	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) = (Base‘(𝐽 mPoly 𝑅)))
11		extvval.m	. . . . . 6 ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))
12	10, 11	eqtr4di 2814	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) = 𝑀)
13		oveq2 7420	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
14	13	rabeqdv 3428	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
15		extvval.d	. . . . . . . 8 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
16	14, 15	eqtr4di 2814	. . . . . . 7 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} = 𝐷)
17	16	adantr 486	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} = 𝐷)
18	4	reseq2d 5970	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑥 ↾ (𝑖 ∖ {𝑎})) = (𝑥 ↾ (𝐼 ∖ {𝑎})))
19	18	fveq2d 6881	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))) = (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))))
20	19	adantr 486	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))) = (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))))
21		fveq2 6877	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
22	21	adantl 487	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (0g‘𝑟) = (0g‘𝑅))
23		extvval.1	. . . . . . . 8 ⊢ 0 = (0g‘𝑅)
24	22, 23	eqtr4di 2814	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (0g‘𝑟) = 0 )
25	20, 24	ifeq12d 4504	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟)) = if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 ))
26	17, 25	mpteq12dv 5192	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟))) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))
27	12, 26	mpteq12dv 5192	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟)))) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 ))))
28	3, 27	mpteq12dv 5192	. . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑎 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟))))) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))))
29	28	adantl 487	. 2 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → (𝑎 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟))))) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))))
30		extvval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
31	30	elexd 3474	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
32		extvval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
33	32	elexd 3474	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
34	30	mptexd 7222	. 2 ⊢ (𝜑 → (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))) ∈ V)
35	2, 29, 31, 33, 34	ovmpod 7564	1 ⊢ (𝜑 → (𝐼extendVars𝑅) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3413  Vcvv 3451   ∖ cdif 3896  ifcif 4482  {csn 4584   class class class wbr 5103   ↦ cmpt 5186   ↾ cres 5653  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414   ↑_m cmap 8831   finSupp cfsupp 9337  0cc0 11181  ℕ₀cn0 12587  Basecbs 17367  0_gc0g 17590   mPoly cmpl 22194  extendVarscextv 34143

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-extv 34144

This theorem is used by:  extvfval  34146