Theorem extvval 33722

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 33721	. . 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 483	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑖 = 𝐼)
4		difeq1 4057	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑖 ∖ {𝑎}) = (𝐼 ∖ {𝑎}))
5		extvval.j	. . . . . . . . . 10 ⊢ 𝐽 = (𝐼 ∖ {𝑎})
6	4, 5	eqtr4di 2793	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑖 ∖ {𝑎}) = 𝐽)
7	6	adantr 481	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 ∖ {𝑎}) = 𝐽)
8		simpr 485	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
9	7, 8	oveq12d 7381	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 ∖ {𝑎}) mPoly 𝑟) = (𝐽 mPoly 𝑅))
10	9	fveq2d 6838	. . . . . 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 2793	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) = 𝑀)
13		oveq2 7371	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
14	13	rabeqdv 3407	. . . . . . . 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 2793	. . . . . . 7 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} = 𝐷)
17	16	adantr 481	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} = 𝐷)
18	4	reseq2d 5938	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑥 ↾ (𝑖 ∖ {𝑎})) = (𝑥 ↾ (𝐼 ∖ {𝑎})))
19	18	fveq2d 6838	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))) = (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))))
20	19	adantr 481	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))) = (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))))
21		fveq2 6834	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
22	21	adantl 482	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (0g‘𝑟) = (0g‘𝑅))
23		extvval.1	. . . . . . . 8 ⊢ 0 = (0g‘𝑅)
24	22, 23	eqtr4di 2793	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (0g‘𝑟) = 0 )
25	20, 24	ifeq12d 4483	. . . . . 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 5166	. . . . 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 5166	. . . 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 5166	. . 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 482	. 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 3456	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
32		extvval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
33	32	elexd 3456	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
34	30	mptexd 7175	. 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 7515	1 ⊢ (𝜑 → (𝐼extendVars𝑅) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3392  Vcvv 3432   ∖ cdif 3887  ifcif 4461  {csn 4562   class class class wbr 5079   ↦ cmpt 5160   ↾ cres 5627  ‘cfv 6492  (class class class)co 7363   ∈ cmpo 7365   ↑_m cmap 8770   finSupp cfsupp 9271  0cc0 11036  ℕ₀cn0 12435  Basecbs 17177  0_gc0g 17400   mPoly cmpl 21888  extendVarscextv 33720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7366  df-oprab 7367  df-mpo 7368  df-extv 33721

This theorem is referenced by:  extvfval  33723