extvfvcl - Mathbox for Thierry Arnoux

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Theorem extvfvcl 33957

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 6902	. . . . 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 7456	. . . . . 6 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
6	4, 5	rabex2 5316	. . . . 5 ⊢ 𝐷 ∈ V
7	6	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
8		fvexd 6903	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘(𝑥 ↾ 𝐽)) ∈ V)
9		extvfvvcl.3	. . . . . . . 8 ⊢ 0 = (0_g‘𝑅)
10	9	fvexi 6902	. . . . . . 7 ⊢ 0 ∈ V
11	10	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 0 ∈ V)
12	8, 11	ifcld 4539	. . . . 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 33954	. . . . 5 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
20	13	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐼 ∈ 𝑉)
21	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑅 ∈ Ring)
22	15	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝐼)
23	18	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐹 ∈ 𝑀)
24		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
25	4, 9, 20, 21, 1, 16, 17, 22, 23, 24	extvfvvcl 33956	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑥) ∈ 𝐵)
26	12, 19, 25	fmpt2d 7127	. . . 4 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹):𝐷⟶𝐵)
27	3, 7, 26	elmapdd 8847	. . 3 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ (𝐵 ↑_m 𝐷))
28		eqid 2766	. . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
29	4	psrbasfsupp 33932	. . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
30		eqid 2766	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
31	28, 1, 29, 30, 13	psrbas 22121	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (𝐵 ↑_m 𝐷))
32	27, 31	eleqtrrd 2869	. 2 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)))
33	7	mptexd 7229	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) ∈ V)
34	10	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ V)
35	12	fmpttd 7117	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )):𝐷⟶V)
36	35	ffund 6717	. . . 4 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
37		fveq1 6887	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦‘𝐴) = (𝑥‘𝐴))
38	37	eqeq1d 2768	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑦‘𝐴) = 0 ↔ (𝑥‘𝐴) = 0))
39	38	cbvrabv 3429	. . . . . . . 8 ⊢ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} = {𝑥 ∈ 𝐷 ∣ (𝑥‘𝐴) = 0}
40	39	partfun2 33058	. . . . . . 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 7433	. . . . . 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 5315	. . . . . . . 8 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ∈ V)
43	42	mptexd 7229	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) ∈ V)
44	7	difexd 5307	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∈ V)
45	44	mptexd 7229	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) ∈ V)
46	43, 45, 34	suppun2 33066	. . . . . 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 2813	. . . . 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 2766	. . . . . . . . . 10 ⊢ (𝐽 mPoly 𝑅) = (𝐽 mPoly 𝑅)
49		eqid 2766	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0}
50	49	psrbasfsupp 33932	. . . . . . . . . 10 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡ℎ “ ℕ) ∈ Fin}
51	48, 1, 17, 50, 18	mplelf 22184	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0}⟶𝐵)
52		breq1 5117	. . . . . . . . . 10 ⊢ (ℎ = (𝑥 ↾ 𝐽) → (ℎ finSupp 0 ↔ (𝑥 ↾ 𝐽) finSupp 0))
53		nn0ex 12528	. . . . . . . . . . . 12 ⊢ ℕ₀ ∈ V
54	53	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → ℕ₀ ∈ V)
55	13	difexd 5307	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 ∖ {𝐴}) ∈ V)
56	16, 55	eqeltrid 2870	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ V)
57	56	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝐽 ∈ V)
58	13	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝐼 ∈ 𝑉)
59		ssrab2 4037	. . . . . . . . . . . . . . 15 ⊢ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ⊆ 𝐷
60		ssrab2 4037	. . . . . . . . . . . . . . . . 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 3978	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ⊆ (ℕ₀ ↑_m 𝐼))
63	59, 62	sstrid 3951	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ⊆ (ℕ₀ ↑_m 𝐼))
64	63	sselda 3940	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥 ∈ (ℕ₀ ↑_m 𝐼))
65	58, 54, 64	elmaprd 33062	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥:𝐼⟶ℕ₀)
66		difssd 4094	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐼 ∖ {𝐴}) ⊆ 𝐼)
67	16, 66	eqsstrid 3978	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
68	67	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝐽 ⊆ 𝐼)
69	65, 68	fssresd 6752	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → (𝑥 ↾ 𝐽):𝐽⟶ℕ₀)
70	54, 57, 69	elmapdd 8847	. . . . . . . . . 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 3940	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥 ∈ 𝐷)
73	29	psrbagfsupp 22106	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → 𝑥 finSupp 0)
74	72, 73	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥 finSupp 0)
75		c0ex 11218	. . . . . . . . . . . 12 ⊢ 0 ∈ V
76	75	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 0 ∈ V)
77	74, 76	fsuppres 9363	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → (𝑥 ↾ 𝐽) finSupp 0)
78	52, 70, 77	elrabd 3655	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → (𝑥 ↾ 𝐽) ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0})
79	51, 78	cofmpt 7135	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))) = (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))))
80	79	oveq1d 7438	. . . . . . 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 7229	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) ∈ V)
82		suppco 8211	. . . . . . . . 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 596	. . . . . . . 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 7117	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)):{𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}⟶(ℕ₀ ↑_m 𝐽))
85		simpr 490	. . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) = (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))
87		reseq1 5977	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑢 → (𝑥 ↾ 𝐽) = (𝑢 ↾ 𝐽))
88		simpllr 788	. . . . . . . . . . . . . . . . . . 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 6032	. . . . . . . . . . . . . . . . . . 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 7020	. . . . . . . . . . . . . . . . . 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 5977	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑣 → (𝑥 ↾ 𝐽) = (𝑣 ↾ 𝐽))
92		simplr 781	. . . . . . . . . . . . . . . . . . 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 6032	. . . . . . . . . . . . . . . . . . 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 7020	. . . . . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . . . . 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 5983	. . . . . . . . . . . . . . . . 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 5983	. . . . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . . . 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 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑢 → (𝑦‘𝐴) = (𝑢‘𝐴))
101	100	eqeq1d 2768	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑢 → ((𝑦‘𝐴) = 0 ↔ (𝑢‘𝐴) = 0))
102	101, 88	elrabrd 3656	. . . . . . . . . . . . . . . . . . 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 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑣 → (𝑦‘𝐴) = (𝑣‘𝐴))
104	103	eqeq1d 2768	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑣 → ((𝑦‘𝐴) = 0 ↔ (𝑣‘𝐴) = 0))
105	104, 92	elrabrd 3656	. . . . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . . . . 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 4850	. . . . . . . . . . . . . . . . 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 4606	. . . . . . . . . . . . . . . 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 4126	. . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . . 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 33062	. . . . . . . . . . . . . . . . 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 6713	. . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . 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 7189	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 Fn 𝐼 ∧ 𝐴 ∈ 𝐼) → 𝑢 = ((𝑢 ↾ (𝐼 ∖ {𝐴})) ∪ {⟨𝐴, (𝑢‘𝐴)⟩}))
119	116, 117, 118	syl2anc 596	. . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . . 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 33062	. . . . . . . . . . . . . . . . 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 6713	. . . . . . . . . . . . . . . 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 7189	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 Fn 𝐼 ∧ 𝐴 ∈ 𝐼) → 𝑣 = ((𝑣 ↾ (𝐼 ∖ {𝐴})) ∪ {⟨𝐴, (𝑣‘𝐴)⟩}))
125	123, 117, 124	syl2anc 596	. . . . . . . . . . . . . . 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 2811	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 472	. . . . . . . . . . . 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 3211	. . . . . . . . . . 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 7259	. . . . . . . . . . 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 595	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)):{𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}–1-1→(ℕ₀ ↑_m 𝐽))
132		df-f1 6548	. . . . . . . . . . 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 503	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ (𝜑 → Fun ^◡(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)))
135	48, 17, 9, 18	mplelsfi 22181	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 finSupp 0 )
136	135	fsuppimpd 9339	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
137		imafi 9285	. . . . . . . . 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 596	. . . . . . . 8 ⊢ (𝜑 → (^◡(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) “ (𝐹 supp 0 )) ∈ Fin)
139	83, 138	eqeltrd 2866	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))) supp 0 ) ∈ Fin)
140	80, 139	eqeltrrd 2867	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) supp 0 ) ∈ Fin)
141		fconstmpt 5728	. . . . . . . . . 10 ⊢ ((𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) × { 0 }) = (𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 )
142	141	oveq1i 7433	. . . . . . . . 9 ⊢ (((𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) × { 0 }) supp 0 ) = ((𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) supp 0 )
143		fczsupp0 8198	. . . . . . . . 9 ⊢ (((𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) × { 0 }) supp 0 ) = ∅
144	142, 143	eqtr3i 2791	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) supp 0 ) = ∅
145		0fi 9049	. . . . . . . 8 ⊢ ∅ ∈ Fin
146	144, 145	eqeltri 2862	. . . . . . 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 9166	. . . . 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 2866	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) supp 0 ) ∈ Fin)
150	33, 34, 36, 149	isfsuppd 9336	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) finSupp 0 )
151	19, 150	eqbrtrd 5138	. 2 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) finSupp 0 )
152		eqid 2766	. . 3 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
153		extvfvcl.n	. . 3 ⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅))
154	152, 28, 30, 9, 153	mplelbas 22177	. 2 ⊢ ((((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁 ↔ ((((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (((𝐼extendVars𝑅)‘𝐴)‘𝐹) finSupp 0 ))
155	32, 151, 154	sylanbrc 595	1 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3082 {crab 3419 Vcvv 3458 ∖ cdif 3905 ∪ cun 3906 ⊆ wss 3908 ∅c0 4289 ifcif 4492 {csn 4594 ⟨cop 4600 class class class wbr 5114 ↦ cmpt 5197 × cxp 5664 ^◡ccnv 5665 ↾ cres 5668 “ cima 5669 ∘ ccom 5670 Fun wfun 6537 Fn wfn 6538 ⟶wf 6539 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7423 supp csupp 8165 ↑_m cmap 8833 Fincfn 8952 finSupp cfsupp 9331 0cc0 11118 ℕ₀cn0 12522 Basecbs 17294 0_gc0g 17517 Ringcrg 20346 mPwSer cmps 22091 mPoly cmpl 22093 extendVarscextv 33950

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-uz 12881 df-fz 13554 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-sca 17351 df-vsca 17352 df-tset 17354 df-0g 17519 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-grp 19034 df-ring 20348 df-psr 22096 df-mpl 22098 df-extv 33951

This theorem is used by: extvfvalf 33958 evlextv 33963 esplyindfv 33997

W3C validator