Theorem gsummoncoe1 22273

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 = (0g‘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 Σg (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 3229	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝐾)
4	3	fmpttd 7067	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐴):ℕ0⟶𝐾)
5		gsummonply1.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑅)
6	5	fvexi 6854	. . . . . . 7 ⊢ 𝐾 ∈ V
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ V)
8		nn0ex 12443	. . . . . 6 ⊢ ℕ0 ∈ V
9		elmapg 8786	. . . . . 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 = (0g‘𝑅)
13	12	fvexi 6854	. . . 4 ⊢ 0 ∈ V
14		fsuppmapnn0ub 13957	. . . 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 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
18	2	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐾)
19		rspcsbela 4378	. . . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐴) = (𝑘 ∈ ℕ0 ↦ 𝐴)
22	21	fvmpts 6951	. . . . . . . . 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 2738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ↔ ⦋𝑥 / 𝑘⦌𝐴 = 0 ))
25	24	imbi2d 340	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑠 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ) ↔ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )))
26	25	biimpd 229	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑠 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ) → (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )))
27	26	ralimdva 3149	. . . 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 2736	. . . . . . . . 9 ⊢ (0g‘𝑃) = (0g‘𝑃)
30		gsummonply1.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Ring)
31		gsummonply1.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
32	31	ply1ring 22211	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
33		ringcmn 20263	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
34	30, 32, 33	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ CMnd)
35	34	ad2antrr 727	. . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
42		gsummonply1.e	. . . . . . . . . . . . . . 15 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
43	5, 31, 39, 40, 41, 42, 28	ply1tmcl 22237	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
44	36, 37, 38, 43	syl3anc 1374	. . . . . . . . . . . . 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 3149	. . . . . . . . . . 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 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ ℕ0 (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
49		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑠 ∈ ℕ0)
50		nfv 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑠 < 𝑥
51		nfcsb1v 3861	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴
52	51	nfeq1 2914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴 = 0
53	50, 52	nfim 1898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )
54		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 )
55		breq2 5089	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑘))
56		csbeq1 3840	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝑘 / 𝑘⦌𝐴)
57	56	eqeq1d 2738	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (⦋𝑥 / 𝑘⦌𝐴 = 0 ↔ ⦋𝑘 / 𝑘⦌𝐴 = 0 ))
58	55, 57	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → ((𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 )))
59	53, 54, 58	cbvralw 3279	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ ∀𝑘 ∈ ℕ0 (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ))
60		csbid 3850	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑘 / 𝑘⦌𝐴 = 𝐴
61	60	eqeq1i 2741	. . . . . . . . . . . . . 14 ⊢ (⦋𝑘 / 𝑘⦌𝐴 = 0 ↔ 𝐴 = 0 )
62		oveq1 7374	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = ( 0 ∗ (𝑘 ↑ 𝑋)))
63	31	ply1sca 22216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
64	30, 63	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
65	64	fveq2d 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
66	12, 65	eqtrid 2783	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 = (0g‘(Scalar‘𝑃)))
67	66	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 0 = (0g‘(Scalar‘𝑃)))
68	67	oveq1d 7382	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ( 0 ∗ (𝑘 ↑ 𝑋)) = ((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)))
69	31	ply1lmod 22215	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
70	30, 69	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ LMod)
71	70	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ LMod)
72		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘𝑃) = (Base‘𝑃)
73	41, 72	mgpbas 20126	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
74	41	ringmgp 20220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
75	30, 32, 74	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
76	75	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd)
77		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
78	39, 31, 72	vr1cl 22181	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
79	30, 78	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑃))
80	79	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑋 ∈ (Base‘𝑃))
81	73, 42, 76, 77, 80	mulgnn0cld 19071	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑃))
82		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
83		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
84	72, 82, 40, 83, 29	lmod0vs 20890	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ LMod ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑃)) → ((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
85	71, 81, 84	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
86	68, 85	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ( 0 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
87	62, 86	sylan9eqr 2793	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝐴 = 0 ) → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
88	87	ex 412	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)))
89	61, 88	biimtrid 242	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (⦋𝑘 / 𝑘⦌𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)))
90	89	imim2d 57	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))))
91	90	ralimdva 3149	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑘 ∈ ℕ0 (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ) → ∀𝑘 ∈ ℕ0 (𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))))
92	59, 91	biimtrid 242	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) → ∀𝑘 ∈ ℕ0 (𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))))
93	92	imp 406	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ ℕ0 (𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)))
94	28, 29, 35, 48, 49, 93	gsummptnn0fz 19961	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))) = (𝑃 Σg (𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))
95	94	fveq2d 6844	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋))))) = (coe1‘(𝑃 Σg (𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋))))))
96	95	fveq1d 6842	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ((coe1‘(𝑃 Σg (𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿))
97	30	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑅 ∈ Ring)
98		gsummonply1.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℕ0)
99	98	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝐿 ∈ ℕ0)
100		elfznn0 13574	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑠) → 𝑘 ∈ ℕ0)
101		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝜑)
102	3	adantlr 716	. . . . . . . . . . . 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 3129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → ∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
107	106	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
108		fzfid 13935	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (0...𝑠) ∈ Fin)
109	31, 28, 97, 99, 107, 108	coe1fzgsumd 22269	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ((coe1‘(𝑃 Σg (𝑘 ∈ (0...𝑠) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ ((coe1‘(𝐴 ∗ (𝑘 ↑ 𝑋)))‘𝐿))))
110		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑠 ∈ ℕ0)
111		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ0
112	111, 53	nfralw 3284	. . . . . . . . . 10 ⊢ Ⅎ𝑘∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )
113	110, 112	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ))
114	30	ad3antrrr 731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝑅 ∈ Ring)
115	3	expcom 413	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (𝜑 → 𝐴 ∈ 𝐾))
116	115, 100	syl11 33	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾))
117	116	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾))
118	117	imp 406	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ 𝐾)
119	100	adantl 481	. . . . . . . . . . 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 22238	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0) → (coe1‘(𝐴 ∗ (𝑘 ↑ 𝑋))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑘, 𝐴, 0 )))
121	114, 118, 119, 120	syl3anc 1374	. . . . . . . . . 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 2740	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐿 → (𝑛 = 𝑘 ↔ 𝐿 = 𝑘))
123	122	ifbid 4490	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐿 → if(𝑛 = 𝑘, 𝐴, 0 ) = if(𝐿 = 𝑘, 𝐴, 0 ))
124	123	adantl 481	. . . . . . . . . 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 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐿 ∈ ℕ0)
126	5, 12	ring0cl 20248	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
127	30, 126	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ 𝐾)
128	127	ad3antrrr 731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 0 ∈ 𝐾)
129	118, 128	ifcld 4513	. . . . . . . . . 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 6955	. . . . . . . . 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 5177	. . . . . . . 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 7383	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ ((coe1‘(𝐴 ∗ (𝑘 ↑ 𝑋)))‘𝐿))) = (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))))
133		breq2 5089	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (𝑠 < 𝑥 ↔ 𝑠 < 𝐿))
134		csbeq1 3840	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐿 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝐿 / 𝑘⦌𝐴)
135	134	eqeq1d 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (⦋𝑥 / 𝑘⦌𝐴 = 0 ↔ ⦋𝐿 / 𝑘⦌𝐴 = 0 ))
136	133, 135	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐿 → ((𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ (𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 )))
137	136	rspcva 3562	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ))
138		nfv 1916	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿))
139		nfcsb1v 3861	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴
140	139	nfeq1 2914	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴 = 0
141	138, 140	nfan 1901	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 )
142		elfz2nn0 13572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ (0...𝑠) ↔ (𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑘 ≤ 𝑠))
143		nn0re 12446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
144	143	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝑘 ∈ ℝ)
145		nn0re 12446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ)
146	145	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → 𝑠 ∈ ℝ)
147	146	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝑠 ∈ ℝ)
148		nn0re 12446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
149	148	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℝ)
150		lelttr 11236	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑘 ∈ ℝ ∧ 𝑠 ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿))
151	144, 147, 149, 150	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿))
152		animorr 981	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 < 𝐿) → (𝐿 < 𝑘 ∨ 𝑘 < 𝐿))
153		df-ne 2933	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐿 ≠ 𝑘 ↔ ¬ 𝐿 = 𝑘)
154	143	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → 𝑘 ∈ ℝ)
155		lttri2 11228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (𝑘 ≤ 𝑠 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))))
163	162	expimpd 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ (0...𝑠) → ((𝑠 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))
168	167	expd 415	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿) → (𝜑 → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘)))
173	172	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))
174	173	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))
175	174	imp 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → ¬ 𝐿 = 𝑘)
176	175	iffalsed 4477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → if(𝐿 = 𝑘, 𝐴, 0 ) = 0 )
177	141, 176	mpteq2da 5177	. . . . . . . . . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ 0 )))
179		ringmnd 20224	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
180	30, 179	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑅 ∈ Mnd)
181	180	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → 𝑅 ∈ Mnd)
182		ovex 7400	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...𝑠) ∈ V
183	12	gsumz 18804	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑠) ∈ V) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
184	181, 182, 183	sylancl 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
185	184	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
186		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⦋𝐿 / 𝑘⦌𝐴 = 0 → ⦋𝐿 / 𝑘⦌𝐴 = 0 )
187	186	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⦋𝐿 / 𝑘⦌𝐴 = 0 → 0 = ⦋𝐿 / 𝑘⦌𝐴)
188	187	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → 0 = ⦋𝐿 / 𝑘⦌𝐴)
189	178, 185, 188	3eqtrd 2775	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)
190	189	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → (⦋𝐿 / 𝑘⦌𝐴 = 0 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))
191	190	expr 456	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (𝑠 < 𝐿 → (⦋𝐿 / 𝑘⦌𝐴 = 0 → (𝑅 Σg (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 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)))
193	192	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑠 ∈ ℕ0 → ((𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑠 < 𝐿 → (𝑅 Σg (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 Σg (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 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))))
196	195	ex 412	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) → (𝑠 ∈ ℕ0 → (𝜑 → (𝑠 < 𝐿 → (𝑅 Σg (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 Σg (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 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))))
199	198	imp31 417	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑠 < 𝐿 → (𝑅 Σg (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 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴))
201		pm3.2 469	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿)))
202	201	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (¬ 𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿)))
203	180	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → 𝑅 ∈ Mnd)
204	182	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → (0...𝑠) ∈ V)
205	98	nn0red 12499	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℝ)
206		lenlt 11224	. . . . . . . . . . . . 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 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈ ℕ0)
209		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝑠 ∈ ℕ0)
210		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ≤ 𝑠)
211		elfz2nn0 13572	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (0...𝑠) ↔ (𝐿 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝐿 ≤ 𝑠))
212	208, 209, 210, 211	syl3anbrc 1345	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈ (0...𝑠))
213	212	ex 412	. . . . . . . . . . . 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 406	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → 𝐿 ∈ (0...𝑠))
216		eqcom 2743	. . . . . . . . . . . 12 ⊢ (𝐿 = 𝑘 ↔ 𝑘 = 𝐿)
217		ifbi 4489	. . . . . . . . . . . 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 5181	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 )) = (𝑘 ∈ (0...𝑠) ↦ if(𝑘 = 𝐿, 𝐴, 0 ))
220	3, 5	eleqtrdi 2846	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ (Base‘𝑅))
221	220	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ ℕ0 → 𝐴 ∈ (Base‘𝑅)))
222	221	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (𝑘 ∈ ℕ0 → 𝐴 ∈ (Base‘𝑅)))
223	222, 100	impel 505	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ (Base‘𝑅))
224	223	ralrimiva 3129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → ∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅))
225	224	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → ∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅))
226	12, 203, 204, 215, 219, 225	gsummpt1n0 19940	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → (𝑅 Σg (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 Σg (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 Σg (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 ))) = ⦋𝐿 / 𝑘⦌𝐴)
229	132, 228	eqtrd 2771	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ ((coe1‘(𝐴 ∗ (𝑘 ↑ 𝑋)))‘𝐿))) = ⦋𝐿 / 𝑘⦌𝐴)
230	96, 109, 229	3eqtrd 2775	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴)
231	230	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) → ((coe1‘(𝑃 Σg (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 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴))
233	232	rexlimdva 3138	. 2 ⊢ (𝜑 → (∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ) → ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴))
234	16, 233	mpd 15	1 ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  Vcvv 3429  ⦋csb 3837  ifcif 4466   class class class wbr 5085   ↦ cmpt 5166  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773   finSupp cfsupp 9274  ℝcr 11037  0cc0 11038   < clt 11179   ≤ cle 11180  ℕ₀cn0 12437  ...cfz 13461  Basecbs 17179  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402   Σ_g cgsu 17403  Mndcmnd 18702  ._gcmg 19043  CMndccmn 19755  mulGrpcmgp 20121  Ringcrg 20214  LModclmod 20855  var₁cv1 22139  Poly₁cpl1 22140  coe₁cco1 22141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-subrng 20523  df-subrg 20547  df-lmod 20857  df-lss 20927  df-psr 21889  df-mvr 21890  df-mpl 21891  df-opsr 21893  df-psr1 22143  df-vr1 22144  df-ply1 22145  df-coe1 22146

This theorem is referenced by:  gsumply1eq  22274  pm2mpf1lem  22759  pm2mpcoe1  22765  pm2mpmhmlem2  22784  cayleyhamilton1  22857  gsummoncoe1fzo  33657  ply1mulgsum  48866