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Theorem gsummoncoe1 22048

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 3246	. . . . . 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 6904	. . . . . . 7 ⊢ 𝐾 ∈ V
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ V)
8		nn0ex 12482	. . . . . 6 ⊢ ℕ₀ ∈ V
9		elmapg 8835	. . . . . 6 ⊢ ((𝐾 ∈ V ∧ ℕ₀ ∈ V) → ((𝑘 ∈ ℕ₀ ↦ 𝐴) ∈ (𝐾 ↑_m ℕ₀) ↔ (𝑘 ∈ ℕ₀ ↦ 𝐴):ℕ₀⟶𝐾))
10	7, 8, 9	sylancl 584	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ ℕ₀ ↦ 𝐴) ∈ (𝐾 ↑_m ℕ₀) ↔ (𝑘 ∈ ℕ₀ ↦ 𝐴):ℕ₀⟶𝐾))
11	4, 10	mpbird 256	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ₀ ↦ 𝐴) ∈ (𝐾 ↑_m ℕ₀))
12		gsummonply1.0	. . . . 5 ⊢ 0 = (0_g‘𝑅)
13	12	fvexi 6904	. . . 4 ⊢ 0 ∈ V
14		fsuppmapnn0ub 13964	. . . 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 584	. . 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 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → 𝑥 ∈ ℕ₀)
18	2	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → ∀𝑘 ∈ ℕ₀ 𝐴 ∈ 𝐾)
19		rspcsbela 4434	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ₀ ∧ ∀𝑘 ∈ ℕ₀ 𝐴 ∈ 𝐾) → ⦋𝑥 / 𝑘⦌𝐴 ∈ 𝐾)
20	17, 18, 19	syl2anc 582	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → ⦋𝑥 / 𝑘⦌𝐴 ∈ 𝐾)
21		eqid 2730	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ ↦ 𝐴) = (𝑘 ∈ ℕ₀ ↦ 𝐴)
22	21	fvmpts 7000	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ₀ ∧ ⦋𝑥 / 𝑘⦌𝐴 ∈ 𝐾) → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐴)
23	17, 20, 22	syl2anc 582	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐴)
24	23	eqeq1d 2732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑥 ∈ ℕ₀) → (((𝑘 ∈ ℕ₀ ↦ 𝐴)‘𝑥) = 0 ↔ ⦋𝑥 / 𝑘⦌𝐴 = 0 ))
25	24	imbi2d 339	. . . . . 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 3165	. . . 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 2730	. . . . . . . . 9 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
30		gsummonply1.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Ring)
31		gsummonply1.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly₁‘𝑅)
32	31	ply1ring 21990	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
33		ringcmn 20170	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
34	30, 32, 33	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ CMnd)
35	34	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑃 ∈ CMnd)
36	30	3ad2ant1 1131	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝐾) → 𝑅 ∈ Ring)
37		simp3 1136	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝐾) → 𝐴 ∈ 𝐾)
38		simp2 1135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝐾) → 𝑘 ∈ ℕ₀)
39		gsummonply1.x	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = (var₁‘𝑅)
40		gsummonply1.m	. . . . . . . . . . . . . . 15 ⊢ ∗ = ( ·_𝑠 ‘𝑃)
41		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
42		gsummonply1.e	. . . . . . . . . . . . . . 15 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
43	5, 31, 39, 40, 41, 42, 28	ply1tmcl 22014	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ₀) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
44	36, 37, 38, 43	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝐾) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
45	44	3expia 1119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴 ∈ 𝐾 → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵))
46	45	ralimdva 3165	. . . . . . . . . . 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 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ ℕ₀ (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
49		simplr 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑠 ∈ ℕ₀)
50		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑠 < 𝑥
51		nfcsb1v 3917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴
52	51	nfeq1 2916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴 = 0
53	50, 52	nfim 1897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )
54		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 )
55		breq2 5151	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑘))
56		csbeq1 3895	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝑘 / 𝑘⦌𝐴)
57	56	eqeq1d 2732	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (⦋𝑥 / 𝑘⦌𝐴 = 0 ↔ ⦋𝑘 / 𝑘⦌𝐴 = 0 ))
58	55, 57	imbi12d 343	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → ((𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 )))
59	53, 54, 58	cbvralw 3301	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ ∀𝑘 ∈ ℕ₀ (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ))
60		csbid 3905	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑘 / 𝑘⦌𝐴 = 𝐴
61	60	eqeq1i 2735	. . . . . . . . . . . . . 14 ⊢ (⦋𝑘 / 𝑘⦌𝐴 = 0 ↔ 𝐴 = 0 )
62		oveq1 7418	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = ( 0 ∗ (𝑘 ↑ 𝑋)))
63	31	ply1sca 21995	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
64	30, 63	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
65	64	fveq2d 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0_g‘𝑅) = (0_g‘(Scalar‘𝑃)))
66	12, 65	eqtrid 2782	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 = (0_g‘(Scalar‘𝑃)))
67	66	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 0 = (0_g‘(Scalar‘𝑃)))
68	67	oveq1d 7426	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ( 0 ∗ (𝑘 ↑ 𝑋)) = ((0_g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)))
69	31	ply1lmod 21994	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
70	30, 69	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ LMod)
71	70	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑃 ∈ LMod)
72		eqid 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘𝑃) = (Base‘𝑃)
73	41, 72	mgpbas 20034	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
74	41	ringmgp 20133	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
75	30, 32, 74	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
76	75	ad2antrr 722	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (mulGrp‘𝑃) ∈ Mnd)
77		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
78	39, 31, 72	vr1cl 21960	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
79	30, 78	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑃))
80	79	ad2antrr 722	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑋 ∈ (Base‘𝑃))
81	73, 42, 76, 77, 80	mulgnn0cld 19011	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑃))
82		eqid 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
83		eqid 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘(Scalar‘𝑃)) = (0_g‘(Scalar‘𝑃))
84	72, 82, 40, 83, 29	lmod0vs 20649	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ LMod ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑃)) → ((0_g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃))
85	71, 81, 84	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((0_g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃))
86	68, 85	eqtrd 2770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ( 0 ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃))
87	62, 86	sylan9eqr 2792	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝐴 = 0 ) → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0_g‘𝑃))
88	87	ex 411	. . . . . . . . . . . . . 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 3165	. . . . . . . . . . 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 405	. . . . . . . . 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 19895	. . . . . . . 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 6894	. . . . . . 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 6892	. . . . . 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 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑅 ∈ Ring)
98		gsummonply1.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℕ₀)
99	98	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝐿 ∈ ℕ₀)
100		elfznn0 13598	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑠) → 𝑘 ∈ ℕ₀)
101		simpll 763	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝜑)
102	3	adantlr 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ 𝐾)
103	101, 77, 102	3jca 1126	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝜑 ∧ 𝑘 ∈ ℕ₀ ∧ 𝐴 ∈ 𝐾))
104	100, 103	sylan2 591	. . . . . . . . . 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 3144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → ∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
107	106	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
108		fzfid 13942	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (0...𝑠) ∈ Fin)
109	31, 28, 97, 99, 107, 108	coe1fzgsumd 22046	. . . . . 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 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑠 ∈ ℕ₀)
111		nfcv 2901	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ₀
112	111, 53	nfralw 3306	. . . . . . . . . 10 ⊢ Ⅎ𝑘∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )
113	110, 112	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ))
114	30	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝑅 ∈ Ring)
115	3	expcom 412	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → (𝜑 → 𝐴 ∈ 𝐾))
116	115, 100	syl11 33	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾))
117	116	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾))
118	117	imp 405	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ 𝐾)
119	100	adantl 480	. . . . . . . . . . 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 22015	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ₀) → (coe₁‘(𝐴 ∗ (𝑘 ↑ 𝑋))) = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 𝑘, 𝐴, 0 )))
121	114, 118, 119, 120	syl3anc 1369	. . . . . . . . . 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 2734	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐿 → (𝑛 = 𝑘 ↔ 𝐿 = 𝑘))
123	122	ifbid 4550	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐿 → if(𝑛 = 𝑘, 𝐴, 0 ) = if(𝐿 = 𝑘, 𝐴, 0 ))
124	123	adantl 480	. . . . . . . . . 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 726	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐿 ∈ ℕ₀)
126	5, 12	ring0cl 20155	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
127	30, 126	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ 𝐾)
128	127	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 0 ∈ 𝐾)
129	118, 128	ifcld 4573	. . . . . . . . . 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 7004	. . . . . . . . 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 5245	. . . . . . . 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 7427	. . . . . . 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 5151	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (𝑠 < 𝑥 ↔ 𝑠 < 𝐿))
134		csbeq1 3895	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐿 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝐿 / 𝑘⦌𝐴)
135	134	eqeq1d 2732	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (⦋𝑥 / 𝑘⦌𝐴 = 0 ↔ ⦋𝐿 / 𝑘⦌𝐴 = 0 ))
136	133, 135	imbi12d 343	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐿 → ((𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ (𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 )))
137	136	rspcva 3609	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ))
138		nfv 1915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿))
139		nfcsb1v 3917	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴
140	139	nfeq1 2916	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴 = 0
141	138, 140	nfan 1900	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 )
142		elfz2nn0 13596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ (0...𝑠) ↔ (𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑠))
143		nn0re 12485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
144	143	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → 𝑘 ∈ ℝ)
145		nn0re 12485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑠 ∈ ℕ₀ → 𝑠 ∈ ℝ)
146	145	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) → 𝑠 ∈ ℝ)
147	146	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → 𝑠 ∈ ℝ)
148		nn0re 12485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℝ)
149	148	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → 𝐿 ∈ ℝ)
150		lelttr 11308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑘 ∈ ℝ ∧ 𝑠 ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿))
151	144, 147, 149, 150	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿))
152		animorr 975	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) ∧ 𝑘 < 𝐿) → (𝐿 < 𝑘 ∨ 𝑘 < 𝐿))
153		df-ne 2939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐿 ≠ 𝑘 ↔ ¬ 𝐿 = 𝑘)
154	143	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) → 𝑘 ∈ ℝ)
155		lttri2 11300	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝐿 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
156	148, 154, 155	syl2anr 595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
157	156	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) ∧ 𝑘 < 𝐿) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
158	153, 157	bitr3id 284	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) ∧ 𝑘 < 𝐿) → (¬ 𝐿 = 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
159	152, 158	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ∈ ℕ₀) ∧ 𝑘 < 𝐿) → ¬ 𝐿 = 𝑘)
160	159	ex 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀) → (𝐿 ∈ ℕ₀ → (𝑘 ≤ 𝑠 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))))
163	162	expimpd 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑠) → ((𝑠 ∈ ℕ₀ ∧ 𝐿 ∈ ℕ₀) → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))
166	165	3adant2 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿) → (𝜑 → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘)))
173	172	impcom 406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))
174	173	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))
175	174	imp 405	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → ¬ 𝐿 = 𝑘)
176	175	iffalsed 4538	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → if(𝐿 = 𝑘, 𝐴, 0 ) = 0 )
177	141, 176	mpteq2da 5245	. . . . . . . . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . . . . . . . . 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 20137	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
180	30, 179	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑅 ∈ Mnd)
181	180	adantr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) → 𝑅 ∈ Mnd)
182		ovex 7444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...𝑠) ∈ V
183	12	gsumz 18753	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑠) ∈ V) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
184	181, 182, 183	sylancl 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) → (𝑅 Σ_g (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
185	184	adantr 479	. . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⦋𝐿 / 𝑘⦌𝐴 = 0 → 0 = ⦋𝐿 / 𝑘⦌𝐴)
188	187	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ₀ ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → 0 = ⦋𝐿 / 𝑘⦌𝐴)
189	178, 185, 188	3eqtrd 2774	. . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . 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 455	. . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 416	. . . . . . . . 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		pm3.2 468	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (¬ 𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿)))
202	201	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ∀𝑥 ∈ ℕ₀ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (¬ 𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿)))
203	180	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿) → 𝑅 ∈ Mnd)
204	182	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿) → (0...𝑠) ∈ V)
205	98	nn0red 12537	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℝ)
206		lenlt 11296	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (𝐿 ≤ 𝑠 ↔ ¬ 𝑠 < 𝐿))
207	205, 145, 206	syl2an 594	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (𝐿 ≤ 𝑠 ↔ ¬ 𝑠 < 𝐿))
208	98	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈ ℕ₀)
209		simplr 765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ≤ 𝑠) → 𝑠 ∈ ℕ₀)
210		simpr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ≤ 𝑠) → 𝐿 ≤ 𝑠)
211		elfz2nn0 13596	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (0...𝑠) ↔ (𝐿 ∈ ℕ₀ ∧ 𝑠 ∈ ℕ₀ ∧ 𝐿 ≤ 𝑠))
212	208, 209, 210, 211	syl3anbrc 1341	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈ (0...𝑠))
213	212	ex 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (𝐿 ≤ 𝑠 → 𝐿 ∈ (0...𝑠)))
214	207, 213	sylbird 259	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (¬ 𝑠 < 𝐿 → 𝐿 ∈ (0...𝑠)))
215	214	imp 405	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿) → 𝐿 ∈ (0...𝑠))
216		eqcom 2737	. . . . . . . . . . . 12 ⊢ (𝐿 = 𝑘 ↔ 𝑘 = 𝐿)
217		ifbi 4549	. . . . . . . . . . . 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 5252	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 )) = (𝑘 ∈ (0...𝑠) ↦ if(𝑘 = 𝐿, 𝐴, 0 ))
220	3, 5	eleqtrdi 2841	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ (Base‘𝑅))
221	220	ex 411	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ ℕ₀ → 𝐴 ∈ (Base‘𝑅)))
222	221	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → (𝑘 ∈ ℕ₀ → 𝐴 ∈ (Base‘𝑅)))
223	222, 100	impel 504	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ (Base‘𝑅))
224	223	ralrimiva 3144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ₀) → ∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅))
225	224	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ₀) ∧ ¬ 𝑠 < 𝐿) → ∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅))
226	12, 203, 204, 215, 219, 225	gsummpt1n0 19874	. . . . . . . . 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 2770	. . . . . 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 2774	. . . . 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 411	. . . 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 3153	. 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 394 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ⦋csb 3892 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ↑_m cmap 8822 finSupp cfsupp 9363 ℝcr 11111 0cc0 11112 < clt 11252 ≤ cle 11253 ℕ₀cn0 12476 ...cfz 13488 Basecbs 17148 Scalarcsca 17204 ·_𝑠 cvsca 17205 0_gc0g 17389 Σ_g cgsu 17390 Mndcmnd 18659 ._gcmg 18986 CMndccmn 19689 mulGrpcmgp 20028 Ringcrg 20127 LModclmod 20614 var₁cv1 21919 Poly₁cpl1 21920 coe₁cco1 21921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-subrng 20434 df-subrg 20459 df-lmod 20616 df-lss 20687 df-psr 21681 df-mvr 21682 df-mpl 21683 df-opsr 21685 df-psr1 21923 df-vr1 21924 df-ply1 21925 df-coe1 21926

This theorem is referenced by: gsumply1eq 22049 pm2mpf1lem 22516 pm2mpcoe1 22522 pm2mpmhmlem2 22541 cayleyhamilton1 22614 gsummoncoe1fzo 32943 ply1mulgsum 47158

W3C validator