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Theorem gsummoncoe1 21828

Description: A coefficient of the polynomial represented as a sum of scaled monomials is the coefficient of the corresponding scaled monomial. (Contributed by AV, 13-Oct-2019.)

**Hypotheses**
Ref	Expression
gsummonply1.p	⊢ 𝑃 = (Poly₁‘𝑅)
gsummonply1.b	⊢ 𝐵 = (Base‘𝑃)
gsummonply1.x	⊢ 𝑋 = (var₁‘𝑅)
gsummonply1.e	⊢ ↑ = (._g‘(mulGrp‘𝑃))
gsummonply1.r	⊢ (𝜑 → 𝑅 ∈ Ring)
gsummonply1.k	⊢ 𝐾 = (Base‘𝑅)
gsummonply1.m	⊢ ∗ = ( ·_𝑠 ‘𝑃)
gsummonply1.0	⊢ 0 = (0_g‘𝑅)
gsummonply1.a	⊢ (𝜑 → ∀𝑘 ∈ ℕ₀ 𝐴 ∈ 𝐾)
gsummonply1.f	⊢ (𝜑 → (𝑘 ∈ ℕ₀ ↦ 𝐴) finSupp 0 )
gsummonply1.l	⊢ (𝜑 → 𝐿 ∈ ℕ₀)

**Assertion**
Ref	Expression
gsummoncoe1	⊢ (𝜑 → ((coe₁‘(𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴)

Distinct variable groups: 𝐵,𝑘 𝑘,𝐾 𝜑,𝑘 ∗ ,𝑘 𝑘,𝐿 𝑃,𝑘 𝑅,𝑘 0 ,𝑘 ↑ ,𝑘 Allowed substitution hints: 𝐴(𝑘) 𝑋(𝑘)

Proof of Theorem gsummoncoe1 Dummy variables 𝑛 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsummonply1.f	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ₀ ↦ 𝐴) finSupp 0 )
2		gsummonply1.a	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ₀ 𝐴 ∈ 𝐾)
3	2	r19.21bi 3249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ 𝐾)
4	3	fmpttd 7115	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ₀ ↦ 𝐴):ℕ₀⟶𝐾)
5		gsummonply1.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑅)
6	5	fvexi 6906	. . . . . . 7 ⊢ 𝐾 ∈ V
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ V)
8		nn0ex 12478	. . . . . 6 ⊢ ℕ₀ ∈ V
9		elmapg 8833	. . . . . 6 ⊢ ((𝐾 ∈ V ∧ ℕ₀ ∈ V) → ((𝑘 ∈ ℕ₀ ↦ 𝐴) ∈ (𝐾 ↑_m ℕ₀) ↔ (𝑘 ∈ ℕ₀ ↦ 𝐴):ℕ₀⟶𝐾))
10	7, 8, 9	sylancl 587	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ ℕ₀ ↦ 𝐴) ∈ (𝐾 ↑_m ℕ₀) ↔ (𝑘 ∈ ℕ₀ ↦ 𝐴):ℕ₀⟶𝐾))
11	4, 10	mpbird 257	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ₀ ↦ 𝐴) ∈ (𝐾 ↑_m ℕ₀))
12		gsummonply1.0	. . . . 5 ⊢ 0 = (0_g‘𝑅)
13	12	fvexi 6906	. . . 4 ⊢ 0 ∈ V
14		fsuppmapnn0ub 13960	. . . 4 ⊢ (((𝑘 ∈ ℕ₀ ↦ 𝐴) ∈ (𝐾 ↑_m ℕ₀) ∧ 0 ∈ V) → ((𝑘 ∈ ℕ₀ ↦ 𝐴) finSupp 0 → ∃𝑠 ∈ ℕ₀ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = 0 )))
15	11, 13, 14	sylancl 587	. . 3 ⊢ (𝜑 → ((𝑘 ∈ ℕ₀ ↦ 𝐴) finSupp 0 → ∃𝑠 ∈ ℕ₀ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = 0 )))
16	1, 15	mpd 15	. 2 ⊢ (𝜑 → ∃𝑠 ∈ ℕ₀ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = 0 ))
17		simpr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → 𝑥 ∈ ℕ₀)
18	2	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → ∀𝑘 ∈ ℕ₀ 𝐴 ∈ 𝐾)
19		rspcsbela 4436	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ₀ ∧ ∀𝑘 ∈ ℕ₀ 𝐴 ∈ 𝐾) → ⦋𝑥 / 𝑘⦌𝐴 ∈ 𝐾)
20	17, 18, 19	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → ⦋𝑥 / 𝑘⦌𝐴 ∈ 𝐾)
21		eqid 2733	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ ↦ 𝐴) = (𝑘 ∈ ℕ₀ ↦ 𝐴)
22	21	fvmpts 7002	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ₀ ∧ ⦋𝑥 / 𝑘⦌𝐴 ∈ 𝐾) → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐴)
23	17, 20, 22	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐴)
24	23	eqeq1d 2735	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → (((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = 0 ↔ ⦋𝑥 / 𝑘⦌𝐴 = 0 ))
25	24	imbi2d 341	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → ((𝑠 < 𝑥 → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = 0 ) ↔ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )))
26	25	biimpd 228	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → ((𝑠 < 𝑥 → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = 0 ) → (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )))
27	26	ralimdva 3168	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = 0 ) → ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )))
28		gsummonply1.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑃)
29		eqid 2733	. . . . . . . . 9 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
30		gsummonply1.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Ring)
31		gsummonply1.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly₁‘𝑅)
32	31	ply1ring 21770	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
33		ringcmn 20099	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
34	30, 32, 33	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ CMnd)
35	34	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑃 ∈ CMnd)
36	30	3ad2ant1 1134	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝐾) → 𝑅 ∈ Ring)
37		simp3 1139	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝐾) → 𝐴 ∈ 𝐾)
38		simp2 1138	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝐾) → 𝑘 ∈ ℕ₀)
39		gsummonply1.x	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = (var₁‘𝑅)
40		gsummonply1.m	. . . . . . . . . . . . . . 15 ⊢ ∗ = ( ·_𝑠 ‘𝑃)
41		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
42		gsummonply1.e	. . . . . . . . . . . . . . 15 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
43	5, 31, 39, 40, 41, 42, 28	ply1tmcl 21794	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ₀) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
44	36, 37, 38, 43	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝐾) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
45	44	3expia 1122	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴 ∈ 𝐾 → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵))
46	45	ralimdva 3168	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑘 ∈ ℕ₀ 𝐴 ∈ 𝐾 → ∀𝑘 ∈ ℕ₀ (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵))
47	2, 46	mpd 15	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ ℕ₀ (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
48	47	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ ℕ₀ (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
49		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑠 ∈ ℕ₀)
50		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑠 < 𝑥
51		nfcsb1v 3919	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴
52	51	nfeq1 2919	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴 = 0
53	50, 52	nfim 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )
54		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 )
55		breq2 5153	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑘))
56		csbeq1 3897	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝑘 / 𝑘⦌𝐴)
57	56	eqeq1d 2735	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (⦋𝑥 / 𝑘⦌𝐴 = 0 ↔ ⦋𝑘 / 𝑘⦌𝐴 = 0 ))
58	55, 57	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → ((𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 )))
59	53, 54, 58	cbvralw 3304	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ ∀𝑘 ∈ ℕ₀ (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ))
60		csbid 3907	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑘 / 𝑘⦌𝐴 = 𝐴
61	60	eqeq1i 2738	. . . . . . . . . . . . . 14 ⊢ (⦋𝑘 / 𝑘⦌𝐴 = 0 ↔ 𝐴 = 0 )
62		oveq1 7416	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = ( 0 ∗ (𝑘 ↑ 𝑋)))
63	31	ply1sca 21775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
64	30, 63	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
65	64	fveq2d 6896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0_g‘𝑅) = (0_g‘(Scalar‘𝑃)))
66	12, 65	eqtrid 2785	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 = (0_g‘(Scalar‘𝑃)))
67	66	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 0 = (0_g‘(Scalar‘𝑃)))
68	67	oveq1d 7424	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ( 0 ∗ (𝑘 ↑ 𝑋)) = ((0_g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)))
69	31	ply1lmod 21774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
70	30, 69	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ LMod)
71	70	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑃 ∈ LMod)
72		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘𝑃) = (Base‘𝑃)
73	41, 72	mgpbas 19993	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
74	41	ringmgp 20062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
75	30, 32, 74	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
76	75	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (mulGrp‘𝑃) ∈ Mnd)
77		simpr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
78	39, 31, 72	vr1cl 21741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
79	30, 78	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑃))
80	79	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑋 ∈ (Base‘𝑃))
81	73, 42, 76, 77, 80	mulgnn0cld 18975	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑃))
82		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
83		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘(Scalar‘𝑃)) = (0_g‘(Scalar‘𝑃))
84	72, 82, 40, 83, 29	lmod0vs 20505	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ LMod ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑃)) → ((0_g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃))
85	71, 81, 84	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((0_g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃))
86	68, 85	eqtrd 2773	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ( 0 ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃))
87	62, 86	sylan9eqr 2795	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝐴 = 0 ) → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃))
88	87	ex 414	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃)))
89	61, 88	biimtrid 241	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (⦋𝑘 / 𝑘⦌𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃)))
90	89	imim2d 57	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃))))
91	90	ralimdva 3168	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (∀𝑘 ∈ ℕ₀ (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ) → ∀𝑘 ∈ ℕ₀ (𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃))))
92	59, 91	biimtrid 241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) → ∀𝑘 ∈ ℕ₀ (𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃))))
93	92	imp 408	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ ℕ₀ (𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃)))
94	28, 29, 35, 48, 49, 93	gsummptnn0fz 19854	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))) = (𝑃 Σ_g (𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))
95	94	fveq2d 6896	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (coe₁‘(𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ (𝐴 ∗ (𝑘 ↑ 𝑋))))) = (coe₁‘(𝑃 Σ_g (𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋))))))
96	95	fveq1d 6894	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ((coe₁‘(𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ((coe₁‘(𝑃 Σ_g (𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿))
97	30	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑅 ∈ Ring)
98		gsummonply1.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℕ₀)
99	98	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝐿 ∈ ℕ₀)
100		elfznn0 13594	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑠) → 𝑘 ∈ ℕ₀)
101		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝜑)
102	3	adantlr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ 𝐾)
103	101, 77, 102	3jca 1129	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝜑 ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝐾))
104	100, 103	sylan2 594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑠)) → (𝜑 ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝐾))
105	104, 44	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑠)) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
106	105	ralrimiva 3147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → ∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
107	106	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
108		fzfid 13938	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (0...𝑠) ∈ Fin)
109	31, 28, 97, 99, 107, 108	coe1fzgsumd 21826	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ((coe₁‘(𝑃 Σ_g (𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ ((coe₁‘(𝐴 ∗ (𝑘 ↑ 𝑋)))‘𝐿))))
110		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑠 ∈ ℕ₀)
111		nfcv 2904	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ₀
112	111, 53	nfralw 3309	. . . . . . . . . 10 ⊢ Ⅎ𝑘∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )
113	110, 112	nfan 1903	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ))
114	30	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝑅 ∈ Ring)
115	3	expcom 415	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → (𝜑 → 𝐴 ∈ 𝐾))
116	115, 100	syl11 33	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾))
117	116	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾))
118	117	imp 408	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ 𝐾)
119	100	adantl 483	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝑘 ∈ ℕ₀)
120	12, 5, 31, 39, 40, 41, 42	coe1tm 21795	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ₀) → (coe₁‘(𝐴 ∗ (𝑘 ↑ 𝑋))) = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 𝑘, 𝐴, 0 )))
121	114, 118, 119, 120	syl3anc 1372	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → (coe₁‘(𝐴 ∗ (𝑘 ↑ 𝑋))) = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 𝑘, 𝐴, 0 )))
122		eqeq1 2737	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐿 → (𝑛 = 𝑘 ↔ 𝐿 = 𝑘))
123	122	ifbid 4552	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐿 → if(𝑛 = 𝑘, 𝐴, 0 ) = if(𝐿 = 𝑘, 𝐴, 0 ))
124	123	adantl 483	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) ∧ 𝑛 = 𝐿) → if(𝑛 = 𝑘, 𝐴, 0 ) = if(𝐿 = 𝑘, 𝐴, 0 ))
125	98	ad3antrrr 729	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐿 ∈ ℕ₀)
126	5, 12	ring0cl 20084	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
127	30, 126	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ 𝐾)
128	127	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 0 ∈ 𝐾)
129	118, 128	ifcld 4575	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → if(𝐿 = 𝑘, 𝐴, 0 ) ∈ 𝐾)
130	121, 124, 125, 129	fvmptd 7006	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → ((coe₁‘(𝐴 ∗ (𝑘 ↑ 𝑋)))‘𝐿) = if(𝐿 = 𝑘, 𝐴, 0 ))
131	113, 130	mpteq2da 5247	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑘 ∈ (0...𝑠) ↦ ((coe₁‘(𝐴 ∗ (𝑘 ↑ 𝑋)))‘𝐿)) = (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 )))
132	131	oveq2d 7425	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ ((coe₁‘(𝐴 ∗ (𝑘 ↑ 𝑋)))‘𝐿))) = (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))))
133		breq2 5153	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (𝑠 < 𝑥 ↔ 𝑠 < 𝐿))
134		csbeq1 3897	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐿 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝐿 / 𝑘⦌𝐴)
135	134	eqeq1d 2735	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (⦋𝑥 / 𝑘⦌𝐴 = 0 ↔ ⦋𝐿 / 𝑘⦌𝐴 = 0 ))
136	133, 135	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐿 → ((𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ (𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 )))
137	136	rspcva 3611	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ))
138		nfv 1918	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿))
139		nfcsb1v 3919	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴
140	139	nfeq1 2919	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴 = 0
141	138, 140	nfan 1903	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 )
142		elfz2nn0 13592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ (0...𝑠) ↔ (𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑠))
143		nn0re 12481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
144	143	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → 𝑘 ∈ ℝ)
145		nn0re 12481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑠 ∈ ℕ₀ → 𝑠 ∈ ℝ)
146	145	adantl 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) → 𝑠 ∈ ℝ)
147	146	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → 𝑠 ∈ ℝ)
148		nn0re 12481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℝ)
149	148	adantl 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → 𝐿 ∈ ℝ)
150		lelttr 11304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑘 ∈ ℝ ∧ 𝑠 ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿))
151	144, 147, 149, 150	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿))
152		animorr 978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) ∧ 𝑘 < 𝐿) → (𝐿 < 𝑘 ∨ 𝑘 < 𝐿))
153		df-ne 2942	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐿 ≠ 𝑘 ↔ ¬ 𝐿 = 𝑘)
154	143	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) → 𝑘 ∈ ℝ)
155		lttri2 11296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝐿 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
156	148, 154, 155	syl2anr 598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
157	156	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) ∧ 𝑘 < 𝐿) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
158	153, 157	bitr3id 285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) ∧ 𝑘 < 𝐿) → (¬ 𝐿 = 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
159	152, 158	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) ∧ 𝑘 < 𝐿) → ¬ 𝐿 = 𝑘)
160	159	ex 414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → (𝑘 < 𝐿 → ¬ 𝐿 = 𝑘))
161	151, 160	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → ¬ 𝐿 = 𝑘))
162	161	exp4b 432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) → (𝐿 ∈ ℕ₀ → (𝑘 ≤ 𝑠 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))))
163	162	expimpd 455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 ∈ ℕ₀ → ((𝑠 ∈ ℕ₀ ∧ 𝐿 ∈ ℕ₀) → (𝑘 ≤ 𝑠 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))))
164	163	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 ≤ 𝑠 → ((𝑠 ∈ ℕ₀ ∧ 𝐿 ∈ ℕ₀) → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))))
165	164	imp 408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑠) → ((𝑠 ∈ ℕ₀ ∧ 𝐿 ∈ ℕ₀) → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))
166	165	3adant2 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑠) → ((𝑠 ∈ ℕ₀ ∧ 𝐿 ∈ ℕ₀) → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))
167	142, 166	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ (0...𝑠) → ((𝑠 ∈ ℕ₀ ∧ 𝐿 ∈ ℕ₀) → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))
168	167	expd 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ (0...𝑠) → (𝑠 ∈ ℕ₀ → (𝐿 ∈ ℕ₀ → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))))
169	98, 168	syl7 74	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ (0...𝑠) → (𝑠 ∈ ℕ₀ → (𝜑 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))))
170	169	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑠 ∈ ℕ₀ → (𝑘 ∈ (0...𝑠) → (𝜑 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))))
171	170	com24 95	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 ∈ ℕ₀ → (𝑠 < 𝐿 → (𝜑 → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))))
172	171	imp 408	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿) → (𝜑 → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘)))
173	172	impcom 409	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))
174	173	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))
175	174	imp 408	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → ¬ 𝐿 = 𝑘)
176	175	iffalsed 4540	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → if(𝐿 = 𝑘, 𝐴, 0 ) = 0 )
177	141, 176	mpteq2da 5247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 )) = (𝑘 ∈ (0...𝑠) ↦ 0 ))
178	177	oveq2d 7425	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ 0 )))
179		ringmnd 20066	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
180	30, 179	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑅 ∈ Mnd)
181	180	adantr 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) → 𝑅 ∈ Mnd)
182		ovex 7442	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...𝑠) ∈ V
183	12	gsumz 18717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑠) ∈ V) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
184	181, 182, 183	sylancl 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
185	184	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
186		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⦋𝐿 / 𝑘⦌𝐴 = 0 → ⦋𝐿 / 𝑘⦌𝐴 = 0 )
187	186	eqcomd 2739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⦋𝐿 / 𝑘⦌𝐴 = 0 → 0 = ⦋𝐿 / 𝑘⦌𝐴)
188	187	adantl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → 0 = ⦋𝐿 / 𝑘⦌𝐴)
189	178, 185, 188	3eqtrd 2777	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)
190	189	ex 414	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) → (⦋𝐿 / 𝑘⦌𝐴 = 0 → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))
191	190	expr 458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (𝑠 < 𝐿 → (⦋𝐿 / 𝑘⦌𝐴 = 0 → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)))
192	191	a2d 29	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → ((𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝐿 → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)))
193	192	ex 414	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑠 ∈ ℕ₀ → ((𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝐿 → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))))
194	193	com13 88	. . . . . . . . . . . . . 14 ⊢ ((𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑠 ∈ ℕ₀ → (𝜑 → (𝑠 < 𝐿 → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))))
195	137, 194	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑠 ∈ ℕ₀ → (𝜑 → (𝑠 < 𝐿 → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))))
196	195	ex 414	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℕ₀ → (∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) → (𝑠 ∈ ℕ₀ → (𝜑 → (𝑠 < 𝐿 → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)))))
197	196	com24 95	. . . . . . . . . . 11 ⊢ (𝐿 ∈ ℕ₀ → (𝜑 → (𝑠 ∈ ℕ₀ → (∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝐿 → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)))))
198	98, 197	mpcom 38	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ ℕ₀ → (∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝐿 → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))))
199	198	imp31 419	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑠 < 𝐿 → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))
200	199	com12 32	. . . . . . . 8 ⊢ (𝑠 < 𝐿 → (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))
201		pm3.2 471	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (¬ 𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿)))
202	201	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (¬ 𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿)))
203	180	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿) → 𝑅 ∈ Mnd)
204	182	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿) → (0...𝑠) ∈ V)
205	98	nn0red 12533	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℝ)
206		lenlt 11292	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (𝐿 ≤ 𝑠 ↔ ¬ 𝑠 < 𝐿))
207	205, 145, 206	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (𝐿 ≤ 𝑠 ↔ ¬ 𝑠 < 𝐿))
208	98	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈ ℕ₀)
209		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ≤ 𝑠) → 𝑠 ∈ ℕ₀)
210		simpr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ≤ 𝑠) → 𝐿 ≤ 𝑠)
211		elfz2nn0 13592	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (0...𝑠) ↔ (𝐿 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀ ∧ 𝐿 ≤ 𝑠))
212	208, 209, 210, 211	syl3anbrc 1344	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈ (0...𝑠))
213	212	ex 414	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (𝐿 ≤ 𝑠 → 𝐿 ∈ (0...𝑠)))
214	207, 213	sylbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (¬ 𝑠 < 𝐿 → 𝐿 ∈ (0...𝑠)))
215	214	imp 408	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿) → 𝐿 ∈ (0...𝑠))
216		eqcom 2740	. . . . . . . . . . . 12 ⊢ (𝐿 = 𝑘 ↔ 𝑘 = 𝐿)
217		ifbi 4551	. . . . . . . . . . . 12 ⊢ ((𝐿 = 𝑘 ↔ 𝑘 = 𝐿) → if(𝐿 = 𝑘, 𝐴, 0 ) = if(𝑘 = 𝐿, 𝐴, 0 ))
218	216, 217	ax-mp 5	. . . . . . . . . . 11 ⊢ if(𝐿 = 𝑘, 𝐴, 0 ) = if(𝑘 = 𝐿, 𝐴, 0 )
219	218	mpteq2i 5254	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 )) = (𝑘 ∈ (0...𝑠) ↦ if(𝑘 = 𝐿, 𝐴, 0 ))
220	3, 5	eleqtrdi 2844	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ (Base‘𝑅))
221	220	ex 414	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ ℕ₀ → 𝐴 ∈ (Base‘𝑅)))
222	221	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (𝑘 ∈ ℕ₀ → 𝐴 ∈ (Base‘𝑅)))
223	222, 100	impel 507	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ (Base‘𝑅))
224	223	ralrimiva 3147	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → ∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅))
225	224	adantr 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿) → ∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅))
226	12, 203, 204, 215, 219, 225	gsummpt1n0 19833	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)
227	202, 226	syl6com 37	. . . . . . . 8 ⊢ (¬ 𝑠 < 𝐿 → (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))
228	200, 227	pm2.61i 182	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)
229	132, 228	eqtrd 2773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ ((coe₁‘(𝐴 ∗ (𝑘 ↑ 𝑋)))‘𝐿))) = ⦋𝐿 / 𝑘⦌𝐴)
230	96, 109, 229	3eqtrd 2777	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ((coe₁‘(𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴)
231	230	ex 414	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) → ((coe₁‘(𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴))
232	27, 231	syld 47	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = 0 ) → ((coe₁‘(𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴))
233	232	rexlimdva 3156	. 2 ⊢ (𝜑 → (∃𝑠 ∈ ℕ₀ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = 0 ) → ((coe₁‘(𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴))
234	16, 233	mpd 15	1 ⊢ (𝜑 → ((coe₁‘(𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ⦋csb 3894 ifcif 4529 class class class wbr 5149 ↦ cmpt 5232 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ↑_m cmap 8820 finSupp cfsupp 9361 ℝcr 11109 0cc0 11110 < clt 11248 ≤ cle 11249 ℕ₀cn0 12472 ...cfz 13484 Basecbs 17144 Scalarcsca 17200 ·_𝑠 cvsca 17201 0_gc0g 17385 Σ_g cgsu 17386 Mndcmnd 18625 ._gcmg 18950 CMndccmn 19648 mulGrpcmgp 19987 Ringcrg 20056 LModclmod 20471 var₁cv1 21700 Poly₁cpl1 21701 coe₁cco1 21702

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-ofr 7671 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-hom 17221 df-cco 17222 df-0g 17387 df-gsum 17388 df-prds 17393 df-pws 17395 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-ghm 19090 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-subrg 20317 df-lmod 20473 df-lss 20543 df-psr 21462 df-mvr 21463 df-mpl 21464 df-opsr 21466 df-psr1 21704 df-vr1 21705 df-ply1 21706 df-coe1 21707

This theorem is referenced by: gsumply1eq 21829 pm2mpf1lem 22296 pm2mpcoe1 22302 pm2mpmhmlem2 22321 cayleyhamilton1 22394 gsummoncoe1fzo 32668 ply1mulgsum 47071

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