Theorem extvval 33789

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 33788	. . 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 486	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑖 = 𝐼)
4		difeq1 4073	. . . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 ∖ {𝑎}) = 𝐽)
8		simpr 488	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
9	7, 8	oveq12d 7410	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 ∖ {𝑎}) mPoly 𝑟) = (𝐽 mPoly 𝑅))
10	9	fveq2d 6867	. . . . . 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 7400	. . . . . . . . 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 484	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} = 𝐷)
18	4	reseq2d 5963	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑥 ↾ (𝑖 ∖ {𝑎})) = (𝑥 ↾ (𝐼 ∖ {𝑎})))
19	18	fveq2d 6867	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))) = (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))))
20	19	adantr 484	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))) = (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))))
21		fveq2 6863	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
22	21	adantl 485	. . . . . . . 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 4501	. . . . . 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 5186	. . . . 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 5186	. . . 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 5186	. . 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 485	. 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 3476	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
32		extvval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
33	32	elexd 3476	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
34	30	mptexd 7204	. 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 7544	1 ⊢ (𝜑 → (𝐼extendVars𝑅) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))))

Syntax hints:   → wi 4   ∧ wa 399   = 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   ∈ cmpo 7394   ↑_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:  extvfval  33790