extvfvcl - Mathbox for Thierry Arnoux

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Theorem extvfvcl 33587

Description: Closure for 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
extvfvvcl.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0}
extvfvvcl.3	⊢ 0 = (0_g‘𝑅)
extvfvvcl.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
extvfvvcl.r	⊢ (𝜑 → 𝑅 ∈ Ring)
extvfvvcl.b	⊢ 𝐵 = (Base‘𝑅)
extvfvvcl.j	⊢ 𝐽 = (𝐼 ∖ {𝐴})
extvfvvcl.m	⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))
extvfvvcl.1	⊢ (𝜑 → 𝐴 ∈ 𝐼)
extvfvvcl.f	⊢ (𝜑 → 𝐹 ∈ 𝑀)
extvfvcl.n	⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅))

**Assertion**
Ref	Expression
extvfvcl	⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁)

Distinct variable groups: 𝐴,ℎ ℎ,𝐼 ℎ,𝐽 Allowed substitution hints: 𝜑(ℎ) 𝐵(ℎ) 𝐷(ℎ) 𝑅(ℎ) 𝐹(ℎ) 𝑀(ℎ) 𝑁(ℎ) 𝑉(ℎ) 0 (ℎ)

Proof of Theorem extvfvcl Dummy variables 𝑥 𝑦 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		extvfvvcl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
2	1	fvexi 6842	. . . . 5 ⊢ 𝐵 ∈ V
3	2	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
4		extvfvvcl.d	. . . . . 6 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0}
5		ovex 7385	. . . . . 6 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
6	4, 5	rabex2 5281	. . . . 5 ⊢ 𝐷 ∈ V
7	6	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
8		fvexd 6843	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘(𝑥 ↾ 𝐽)) ∈ V)
9		extvfvvcl.3	. . . . . . . 8 ⊢ 0 = (0_g‘𝑅)
10	9	fvexi 6842	. . . . . . 7 ⊢ 0 ∈ V
11	10	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 0 ∈ V)
12	8, 11	ifcld 4521	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 ) ∈ V)
13		extvfvvcl.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
14		extvfvvcl.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
15		extvfvvcl.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐼)
16		extvfvvcl.j	. . . . . 6 ⊢ 𝐽 = (𝐼 ∖ {𝐴})
17		extvfvvcl.m	. . . . . 6 ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))
18		extvfvvcl.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑀)
19	4, 9, 13, 14, 15, 16, 17, 18	extvfv 33584	. . . . 5 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
20	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐼 ∈ 𝑉)
21	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑅 ∈ Ring)
22	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝐼)
23	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐹 ∈ 𝑀)
24		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
25	4, 9, 20, 21, 1, 16, 17, 22, 23, 24	extvfvvcl 33586	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑥) ∈ 𝐵)
26	12, 19, 25	fmpt2d 7063	. . . 4 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹):𝐷⟶𝐵)
27	3, 7, 26	elmapdd 8771	. . 3 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ (𝐵 ↑_m 𝐷))
28		eqid 2733	. . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
29	4	psrbasfsupp 33579	. . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
30		eqid 2733	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
31	28, 1, 29, 30, 13	psrbas 21872	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (𝐵 ↑_m 𝐷))
32	27, 31	eleqtrrd 2836	. 2 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)))
33	7	mptexd 7164	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) ∈ V)
34	10	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ V)
35	12	fmpttd 7054	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )):𝐷⟶V)
36	35	ffund 6660	. . . 4 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
37		fveq1 6827	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦‘𝐴) = (𝑥‘𝐴))
38	37	eqeq1d 2735	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑦‘𝐴) = 0 ↔ (𝑥‘𝐴) = 0))
39	38	cbvrabv 3406	. . . . . . . 8 ⊢ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} = {𝑥 ∈ 𝐷 ∣ (𝑥‘𝐴) = 0}
40	39	partfun2 32661	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ))
41	40	oveq1i 7362	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) supp 0 ) = (((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 )) supp 0 )
42	39, 7	rabexd 5280	. . . . . . . 8 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ∈ V)
43	42	mptexd 7164	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) ∈ V)
44	7	difexd 5271	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∈ V)
45	44	mptexd 7164	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) ∈ V)
46	43, 45, 34	suppun2 32669	. . . . . 6 ⊢ (𝜑 → (((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 )) supp 0 ) = (((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) supp 0 ) ∪ ((𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) supp 0 )))
47	41, 46	eqtrid 2780	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) supp 0 ) = (((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) supp 0 ) ∪ ((𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) supp 0 )))
48		eqid 2733	. . . . . . . . . 10 ⊢ (𝐽 mPoly 𝑅) = (𝐽 mPoly 𝑅)
49		eqid 2733	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0}
50	49	psrbasfsupp 33579	. . . . . . . . . 10 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡ℎ “ ℕ) ∈ Fin}
51	48, 1, 17, 50, 18	mplelf 21936	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0}⟶𝐵)
52		breq1 5096	. . . . . . . . . 10 ⊢ (ℎ = (𝑥 ↾ 𝐽) → (ℎ finSupp 0 ↔ (𝑥 ↾ 𝐽) finSupp 0))
53		nn0ex 12394	. . . . . . . . . . . 12 ⊢ ℕ₀ ∈ V
54	53	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → ℕ₀ ∈ V)
55	13	difexd 5271	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 ∖ {𝐴}) ∈ V)
56	16, 55	eqeltrid 2837	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ V)
57	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝐽 ∈ V)
58	13	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝐼 ∈ 𝑉)
59		ssrab2 4029	. . . . . . . . . . . . . . 15 ⊢ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ⊆ 𝐷
60		ssrab2 4029	. . . . . . . . . . . . . . . . 17 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ₀ ↑_m 𝐼)
61	60	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ₀ ↑_m 𝐼))
62	4, 61	eqsstrid 3969	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ⊆ (ℕ₀ ↑_m 𝐼))
63	59, 62	sstrid 3942	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ⊆ (ℕ₀ ↑_m 𝐼))
64	63	sselda 3930	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥 ∈ (ℕ₀ ↑_m 𝐼))
65	58, 54, 64	elmaprd 32665	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥:𝐼⟶ℕ₀)
66		difssd 4086	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐼 ∖ {𝐴}) ⊆ 𝐼)
67	16, 66	eqsstrid 3969	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
68	67	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝐽 ⊆ 𝐼)
69	65, 68	fssresd 6695	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → (𝑥 ↾ 𝐽):𝐽⟶ℕ₀)
70	54, 57, 69	elmapdd 8771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → (𝑥 ↾ 𝐽) ∈ (ℕ₀ ↑_m 𝐽))
71	59	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ⊆ 𝐷)
72	71	sselda 3930	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥 ∈ 𝐷)
73	29	psrbagfsupp 21858	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → 𝑥 finSupp 0)
74	72, 73	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥 finSupp 0)
75		c0ex 11113	. . . . . . . . . . . 12 ⊢ 0 ∈ V
76	75	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 0 ∈ V)
77	74, 76	fsuppres 9284	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → (𝑥 ↾ 𝐽) finSupp 0)
78	52, 70, 77	elrabd 3645	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → (𝑥 ↾ 𝐽) ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0})
79	51, 78	cofmpt 7071	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))) = (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))))
80	79	oveq1d 7367	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))) supp 0 ) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) supp 0 ))
81	42	mptexd 7164	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) ∈ V)
82		suppco 8142	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝑀 ∧ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) ∈ V) → ((𝐹 ∘ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))) supp 0 ) = (^◡(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) “ (𝐹 supp 0 )))
83	18, 81, 82	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ∘ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))) supp 0 ) = (^◡(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) “ (𝐹 supp 0 )))
84	70	fmpttd 7054	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)):{𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}⟶(ℕ₀ ↑_m 𝐽))
85		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣))
86		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) = (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))
87		reseq1 5926	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑢 → (𝑥 ↾ 𝐽) = (𝑢 ↾ 𝐽))
88		simpllr 775	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0})
89	88	resexd 5981	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → (𝑢 ↾ 𝐽) ∈ V)
90	86, 87, 88, 89	fvmptd3 6958	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = (𝑢 ↾ 𝐽))
91		reseq1 5926	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑣 → (𝑥 ↾ 𝐽) = (𝑣 ↾ 𝐽))
92		simplr 768	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0})
93	92	resexd 5981	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → (𝑣 ↾ 𝐽) ∈ V)
94	86, 91, 92, 93	fvmptd3 6958	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣) = (𝑣 ↾ 𝐽))
95	85, 90, 94	3eqtr3d 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → (𝑢 ↾ 𝐽) = (𝑣 ↾ 𝐽))
96	16	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝐽 = (𝐼 ∖ {𝐴}))
97	96	reseq2d 5932	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → (𝑢 ↾ 𝐽) = (𝑢 ↾ (𝐼 ∖ {𝐴})))
98	96	reseq2d 5932	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → (𝑣 ↾ 𝐽) = (𝑣 ↾ (𝐼 ∖ {𝐴})))
99	95, 97, 98	3eqtr3d 2776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → (𝑢 ↾ (𝐼 ∖ {𝐴})) = (𝑣 ↾ (𝐼 ∖ {𝐴})))
100		fveq1 6827	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑢 → (𝑦‘𝐴) = (𝑢‘𝐴))
101	100	eqeq1d 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑢 → ((𝑦‘𝐴) = 0 ↔ (𝑢‘𝐴) = 0))
102	101, 88	elrabrd 32480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → (𝑢‘𝐴) = 0)
103		fveq1 6827	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑣 → (𝑦‘𝐴) = (𝑣‘𝐴))
104	103	eqeq1d 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑣 → ((𝑦‘𝐴) = 0 ↔ (𝑣‘𝐴) = 0))
105	104, 92	elrabrd 32480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → (𝑣‘𝐴) = 0)
106	102, 105	eqtr4d 2771	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → (𝑢‘𝐴) = (𝑣‘𝐴))
107	106	opeq2d 4831	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → ⟨𝐴, (𝑢‘𝐴)⟩ = ⟨𝐴, (𝑣‘𝐴)⟩)
108	107	sneqd 4587	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → {⟨𝐴, (𝑢‘𝐴)⟩} = {⟨𝐴, (𝑣‘𝐴)⟩})
109	99, 108	uneq12d 4118	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → ((𝑢 ↾ (𝐼 ∖ {𝐴})) ∪ {⟨𝐴, (𝑢‘𝐴)⟩}) = ((𝑣 ↾ (𝐼 ∖ {𝐴})) ∪ {⟨𝐴, (𝑣‘𝐴)⟩}))
110	13	ad3antrrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝐼 ∈ 𝑉)
111	53	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → ℕ₀ ∈ V)
112	62	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝐷 ⊆ (ℕ₀ ↑_m 𝐼))
113	59, 88	sselid 3928	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑢 ∈ 𝐷)
114	112, 113	sseldd 3931	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑢 ∈ (ℕ₀ ↑_m 𝐼))
115	110, 111, 114	elmaprd 32665	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑢:𝐼⟶ℕ₀)
116	115	ffnd 6657	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑢 Fn 𝐼)
117	15	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝐴 ∈ 𝐼)
118		fnsnsplit 7124	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 Fn 𝐼 ∧ 𝐴 ∈ 𝐼) → 𝑢 = ((𝑢 ↾ (𝐼 ∖ {𝐴})) ∪ {⟨𝐴, (𝑢‘𝐴)⟩}))
119	116, 117, 118	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑢 = ((𝑢 ↾ (𝐼 ∖ {𝐴})) ∪ {⟨𝐴, (𝑢‘𝐴)⟩}))
120	59, 92	sselid 3928	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑣 ∈ 𝐷)
121	112, 120	sseldd 3931	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑣 ∈ (ℕ₀ ↑_m 𝐼))
122	110, 111, 121	elmaprd 32665	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑣:𝐼⟶ℕ₀)
123	122	ffnd 6657	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑣 Fn 𝐼)
124		fnsnsplit 7124	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 Fn 𝐼 ∧ 𝐴 ∈ 𝐼) → 𝑣 = ((𝑣 ↾ (𝐼 ∖ {𝐴})) ∪ {⟨𝐴, (𝑣‘𝐴)⟩}))
125	123, 117, 124	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑣 = ((𝑣 ↾ (𝐼 ∖ {𝐴})) ∪ {⟨𝐴, (𝑣‘𝐴)⟩}))
126	109, 119, 125	3eqtr4d 2778	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣)) → 𝑢 = 𝑣)
127	126	ex 412	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → (((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣) → 𝑢 = 𝑣))
128	127	anasss 466	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ∧ 𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0})) → (((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣) → 𝑢 = 𝑣))
129	128	ralrimivva 3176	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}∀𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} (((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣) → 𝑢 = 𝑣))
130		dff13 7194	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)):{𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}–1-1→(ℕ₀ ↑_m 𝐽) ↔ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)):{𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}⟶(ℕ₀ ↑_m 𝐽) ∧ ∀𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}∀𝑣 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} (((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑢) = ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))‘𝑣) → 𝑢 = 𝑣)))
131	84, 129, 130	sylanbrc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)):{𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}–1-1→(ℕ₀ ↑_m 𝐽))
132		df-f1 6491	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)):{𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}–1-1→(ℕ₀ ↑_m 𝐽) ↔ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)):{𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}⟶(ℕ₀ ↑_m 𝐽) ∧ Fun ^◡(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))))
133	132	simprbi 496	. . . . . . . . . 10 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)):{𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}–1-1→(ℕ₀ ↑_m 𝐽) → Fun ^◡(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)))
134	131, 133	syl 17	. . . . . . . . 9 ⊢ (𝜑 → Fun ^◡(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)))
135	48, 17, 9, 18	mplelsfi 21933	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 finSupp 0 )
136	135	fsuppimpd 9260	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
137		imafi 9206	. . . . . . . . 9 ⊢ ((Fun ^◡(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) ∧ (𝐹 supp 0 ) ∈ Fin) → (^◡(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) “ (𝐹 supp 0 )) ∈ Fin)
138	134, 136, 137	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (^◡(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) “ (𝐹 supp 0 )) ∈ Fin)
139	83, 138	eqeltrd 2833	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))) supp 0 ) ∈ Fin)
140	80, 139	eqeltrrd 2834	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) supp 0 ) ∈ Fin)
141		fconstmpt 5681	. . . . . . . . . 10 ⊢ ((𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) × { 0 }) = (𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 )
142	141	oveq1i 7362	. . . . . . . . 9 ⊢ (((𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) × { 0 }) supp 0 ) = ((𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) supp 0 )
143		fczsupp0 8129	. . . . . . . . 9 ⊢ (((𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) × { 0 }) supp 0 ) = ∅
144	142, 143	eqtr3i 2758	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) supp 0 ) = ∅
145		0fi 8971	. . . . . . . 8 ⊢ ∅ ∈ Fin
146	144, 145	eqeltri 2829	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) supp 0 ) ∈ Fin
147	146	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) supp 0 ) ∈ Fin)
148	140, 147	unfid 9088	. . . . 5 ⊢ (𝜑 → (((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) supp 0 ) ∪ ((𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) supp 0 )) ∈ Fin)
149	47, 148	eqeltrd 2833	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) supp 0 ) ∈ Fin)
150	33, 34, 36, 149	isfsuppd 9257	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) finSupp 0 )
151	19, 150	eqbrtrd 5115	. 2 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) finSupp 0 )
152		eqid 2733	. . 3 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
153		extvfvcl.n	. . 3 ⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅))
154	152, 28, 30, 9, 153	mplelbas 21929	. 2 ⊢ ((((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁 ↔ ((((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (((𝐼extendVars𝑅)‘𝐴)‘𝐹) finSupp 0 ))
155	32, 151, 154	sylanbrc 583	1 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3048 {crab 3396 Vcvv 3437 ∖ cdif 3895 ∪ cun 3896 ⊆ wss 3898 ∅c0 4282 ifcif 4474 {csn 4575 ⟨cop 4581 class class class wbr 5093 ↦ cmpt 5174 × cxp 5617 ^◡ccnv 5618 ↾ cres 5621 “ cima 5622 ∘ ccom 5623 Fun wfun 6480 Fn wfn 6481 ⟶wf 6482 –1-1→wf1 6483 ‘cfv 6486 (class class class)co 7352 supp csupp 8096 ↑_m cmap 8756 Fincfn 8875 finSupp cfsupp 9252 0cc0 11013 ℕ₀cn0 12388 Basecbs 17122 0_gc0g 17345 Ringcrg 20153 mPwSer cmps 21843 mPoly cmpl 21845 extendVarscextv 33580

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-uz 12739 df-fz 13410 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-sca 17179 df-vsca 17180 df-tset 17182 df-0g 17347 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-grp 18851 df-ring 20155 df-psr 21848 df-mpl 21850 df-extv 33581

This theorem is referenced by: extvfvalf 33588

W3C validator