extvfvcl - Mathbox for Thierry Arnoux

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Theorem extvfvcl 33727

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 6848	. . . . 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 7396	. . . . . 6 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
6	4, 5	rabex2 5276	. . . . 5 ⊢ 𝐷 ∈ V
7	6	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
8		fvexd 6849	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘(𝑥 ↾ 𝐽)) ∈ V)
9		extvfvvcl.3	. . . . . . . 8 ⊢ 0 = (0_g‘𝑅)
10	9	fvexi 6848	. . . . . . 7 ⊢ 0 ∈ V
11	10	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 0 ∈ V)
12	8, 11	ifcld 4508	. . . . 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 33724	. . . . 5 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
20	13	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐼 ∈ 𝑉)
21	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑅 ∈ Ring)
22	15	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝐼)
23	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐹 ∈ 𝑀)
24		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
25	4, 9, 20, 21, 1, 16, 17, 22, 23, 24	extvfvvcl 33726	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑥) ∈ 𝐵)
26	12, 19, 25	fmpt2d 7073	. . . 4 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹):𝐷⟶𝐵)
27	3, 7, 26	elmapdd 8785	. . 3 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ (𝐵 ↑_m 𝐷))
28		eqid 2740	. . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
29	4	psrbasfsupp 33702	. . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
30		eqid 2740	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
31	28, 1, 29, 30, 13	psrbas 21916	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (𝐵 ↑_m 𝐷))
32	27, 31	eleqtrrd 2843	. 2 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)))
33	7	mptexd 7175	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) ∈ V)
34	10	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ V)
35	12	fmpttd 7063	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )):𝐷⟶V)
36	35	ffund 6666	. . . 4 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
37		fveq1 6833	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦‘𝐴) = (𝑥‘𝐴))
38	37	eqeq1d 2742	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑦‘𝐴) = 0 ↔ (𝑥‘𝐴) = 0))
39	38	cbvrabv 3402	. . . . . . . 8 ⊢ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} = {𝑥 ∈ 𝐷 ∣ (𝑥‘𝐴) = 0}
40	39	partfun2 32775	. . . . . . 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 7373	. . . . . 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 5275	. . . . . . . 8 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ∈ V)
43	42	mptexd 7175	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) ∈ V)
44	7	difexd 5266	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ∈ V)
45	44	mptexd 7175	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) ∈ V)
46	43, 45, 34	suppun2 32783	. . . . . 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 2787	. . . . 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 2740	. . . . . . . . . 10 ⊢ (𝐽 mPoly 𝑅) = (𝐽 mPoly 𝑅)
49		eqid 2740	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0}
50	49	psrbasfsupp 33702	. . . . . . . . . 10 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡ℎ “ ℕ) ∈ Fin}
51	48, 1, 17, 50, 18	mplelf 21979	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0}⟶𝐵)
52		breq1 5082	. . . . . . . . . 10 ⊢ (ℎ = (𝑥 ↾ 𝐽) → (ℎ finSupp 0 ↔ (𝑥 ↾ 𝐽) finSupp 0))
53		nn0ex 12441	. . . . . . . . . . . 12 ⊢ ℕ₀ ∈ V
54	53	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → ℕ₀ ∈ V)
55	13	difexd 5266	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 ∖ {𝐴}) ∈ V)
56	16, 55	eqeltrid 2844	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ V)
57	56	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝐽 ∈ V)
58	13	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝐼 ∈ 𝑉)
59		ssrab2 4018	. . . . . . . . . . . . . . 15 ⊢ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ⊆ 𝐷
60		ssrab2 4018	. . . . . . . . . . . . . . . . 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 3960	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ⊆ (ℕ₀ ↑_m 𝐼))
63	59, 62	sstrid 3933	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ⊆ (ℕ₀ ↑_m 𝐼))
64	63	sselda 3922	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥 ∈ (ℕ₀ ↑_m 𝐼))
65	58, 54, 64	elmaprd 32779	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥:𝐼⟶ℕ₀)
66		difssd 4074	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐼 ∖ {𝐴}) ⊆ 𝐼)
67	16, 66	eqsstrid 3960	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
68	67	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝐽 ⊆ 𝐼)
69	65, 68	fssresd 6701	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → (𝑥 ↾ 𝐽):𝐽⟶ℕ₀)
70	54, 57, 69	elmapdd 8785	. . . . . . . . . 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 3922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥 ∈ 𝐷)
73	29	psrbagfsupp 21901	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → 𝑥 finSupp 0)
74	72, 73	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 𝑥 finSupp 0)
75		c0ex 11136	. . . . . . . . . . . 12 ⊢ 0 ∈ V
76	75	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → 0 ∈ V)
77	74, 76	fsuppres 9303	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → (𝑥 ↾ 𝐽) finSupp 0)
78	52, 70, 77	elrabd 3638	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) → (𝑥 ↾ 𝐽) ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0})
79	51, 78	cofmpt 7081	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))) = (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))))
80	79	oveq1d 7378	. . . . . . 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 7175	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) ∈ V)
82		suppco 8153	. . . . . . . . 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 590	. . . . . . . 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 7063	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)):{𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}⟶(ℕ₀ ↑_m 𝐽))
85		simpr 485	. . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) = (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))
87		reseq1 5932	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑢 → (𝑥 ↾ 𝐽) = (𝑢 ↾ 𝐽))
88		simpllr 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)) → 𝑢 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0})
89	88	resexd 5987	. . . . . . . . . . . . . . . . . . 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 6966	. . . . . . . . . . . . . . . . . 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 5932	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑣 → (𝑥 ↾ 𝐽) = (𝑣 ↾ 𝐽))
92		simplr 774	. . . . . . . . . . . . . . . . . . 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 5987	. . . . . . . . . . . . . . . . . . 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 6966	. . . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . . 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 5938	. . . . . . . . . . . . . . . . 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 5938	. . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . 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 6833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑢 → (𝑦‘𝐴) = (𝑢‘𝐴))
101	100	eqeq1d 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑢 → ((𝑦‘𝐴) = 0 ↔ (𝑢‘𝐴) = 0))
102	101, 88	elrabrd 32593	. . . . . . . . . . . . . . . . . . 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 6833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑣 → (𝑦‘𝐴) = (𝑣‘𝐴))
104	103	eqeq1d 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑣 → ((𝑦‘𝐴) = 0 ↔ (𝑣‘𝐴) = 0))
105	104, 92	elrabrd 32593	. . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . 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 4818	. . . . . . . . . . . . . . . . 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 4574	. . . . . . . . . . . . . . . 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 4106	. . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . . . . 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 32779	. . . . . . . . . . . . . . . . 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 6663	. . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . 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 7135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 Fn 𝐼 ∧ 𝐴 ∈ 𝐼) → 𝑢 = ((𝑢 ↾ (𝐼 ∖ {𝐴})) ∪ {⟨𝐴, (𝑢‘𝐴)⟩}))
119	116, 117, 118	syl2anc 590	. . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . . . . 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 32779	. . . . . . . . . . . . . . . . 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 6663	. . . . . . . . . . . . . . . 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 7135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 Fn 𝐼 ∧ 𝐴 ∈ 𝐼) → 𝑣 = ((𝑣 ↾ (𝐼 ∖ {𝐴})) ∪ {⟨𝐴, (𝑣‘𝐴)⟩}))
125	123, 117, 124	syl2anc 590	. . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . . 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 467	. . . . . . . . . . . 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 3183	. . . . . . . . . . 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 7205	. . . . . . . . . . 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 589	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)):{𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}–1-1→(ℕ₀ ↑_m 𝐽))
132		df-f1 6497	. . . . . . . . . . 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 498	. . . . . . . . . 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 21976	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 finSupp 0 )
136	135	fsuppimpd 9279	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
137		imafi 9222	. . . . . . . . 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 590	. . . . . . . 8 ⊢ (𝜑 → (^◡(𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽)) “ (𝐹 supp 0 )) ∈ Fin)
139	83, 138	eqeltrd 2840	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝑥 ↾ 𝐽))) supp 0 ) ∈ Fin)
140	80, 139	eqeltrrd 2841	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0} ↦ (𝐹‘(𝑥 ↾ 𝐽))) supp 0 ) ∈ Fin)
141		fconstmpt 5687	. . . . . . . . . 10 ⊢ ((𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) × { 0 }) = (𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 )
142	141	oveq1i 7373	. . . . . . . . 9 ⊢ (((𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) × { 0 }) supp 0 ) = ((𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) supp 0 )
143		fczsupp0 8140	. . . . . . . . 9 ⊢ (((𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) × { 0 }) supp 0 ) = ∅
144	142, 143	eqtr3i 2765	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐷 ∖ {𝑦 ∈ 𝐷 ∣ (𝑦‘𝐴) = 0}) ↦ 0 ) supp 0 ) = ∅
145		0fi 8986	. . . . . . . 8 ⊢ ∅ ∈ Fin
146	144, 145	eqeltri 2836	. . . . . . 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 9103	. . . . 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 2840	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) supp 0 ) ∈ Fin)
150	33, 34, 36, 149	isfsuppd 9276	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )) finSupp 0 )
151	19, 150	eqbrtrd 5101	. 2 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) finSupp 0 )
152		eqid 2740	. . 3 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
153		extvfvcl.n	. . 3 ⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅))
154	152, 28, 30, 9, 153	mplelbas 21972	. 2 ⊢ ((((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁 ↔ ((((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (((𝐼extendVars𝑅)‘𝐴)‘𝐹) finSupp 0 ))
155	32, 151, 154	sylanbrc 589	1 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3054 {crab 3392 Vcvv 3432 ∖ cdif 3887 ∪ cun 3888 ⊆ wss 3890 ∅c0 4268 ifcif 4461 {csn 4562 ⟨cop 4568 class class class wbr 5079 ↦ cmpt 5160 × cxp 5623 ^◡ccnv 5624 ↾ cres 5627 “ cima 5628 ∘ ccom 5629 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 –1-1→wf1 6489 ‘cfv 6492 (class class class)co 7363 supp csupp 8107 ↑_m cmap 8770 Fincfn 8890 finSupp cfsupp 9271 0cc0 11036 ℕ₀cn0 12435 Basecbs 17177 0_gc0g 17400 Ringcrg 20212 mPwSer cmps 21886 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-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 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-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 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-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 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-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 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-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-tset 17237 df-0g 17402 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-grp 18910 df-ring 20214 df-psr 21891 df-mpl 21893 df-extv 33721

This theorem is referenced by: extvfvalf 33728 evlextv 33733 esplyindfv 33767

W3C validator