Theorem gsummoncoe1 22282

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 3230	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝐾)
4	3	fmpttd 7059	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐴):ℕ0⟶𝐾)
5		gsummonply1.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑅)
6	5	fvexi 6846	. . . . . . 7 ⊢ 𝐾 ∈ V
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ V)
8		nn0ex 12432	. . . . . 6 ⊢ ℕ0 ∈ V
9		elmapg 8777	. . . . . 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 6846	. . . 4 ⊢ 0 ∈ V
14		fsuppmapnn0ub 13946	. . . 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 4379	. . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐴) = (𝑘 ∈ ℕ0 ↦ 𝐴)
22	21	fvmpts 6943	. . . . . . . . 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 2739	. . . . . . 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 3150	. . . 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 2737	. . . . . . . . 9 ⊢ (0g‘𝑃) = (0g‘𝑃)
30		gsummonply1.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Ring)
31		gsummonply1.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
32	31	ply1ring 22220	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
33		ringcmn 20252	. . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
42		gsummonply1.e	. . . . . . . . . . . . . . 15 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
43	5, 31, 39, 40, 41, 42, 28	ply1tmcl 22246	. . . . . . . . . . . . . 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 3150	. . . . . . . . . . 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 3862	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴
52	51	nfeq1 2915	. . . . . . . . . . . . 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 5090	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑘))
56		csbeq1 3841	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝑘 / 𝑘⦌𝐴)
57	56	eqeq1d 2739	. . . . . . . . . . . . 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 3280	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ ∀𝑘 ∈ ℕ0 (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ))
60		csbid 3851	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑘 / 𝑘⦌𝐴 = 𝐴
61	60	eqeq1i 2742	. . . . . . . . . . . . . 14 ⊢ (⦋𝑘 / 𝑘⦌𝐴 = 0 ↔ 𝐴 = 0 )
62		oveq1 7365	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = ( 0 ∗ (𝑘 ↑ 𝑋)))
63	31	ply1sca 22225	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
64	30, 63	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
65	64	fveq2d 6836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
66	12, 65	eqtrid 2784	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 = (0g‘(Scalar‘𝑃)))
67	66	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 0 = (0g‘(Scalar‘𝑃)))
68	67	oveq1d 7373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ( 0 ∗ (𝑘 ↑ 𝑋)) = ((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)))
69	31	ply1lmod 22224	. . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘𝑃) = (Base‘𝑃)
73	41, 72	mgpbas 20115	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
74	41	ringmgp 20209	. . . . . . . . . . . . . . . . . . . . 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 22190	. . . . . . . . . . . . . . . . . . . . 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 19060	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑃))
82		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
83		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
84	72, 82, 40, 83, 29	lmod0vs 20879	. . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ( 0 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
87	62, 86	sylan9eqr 2794	. . . . . . . . . . . . . . 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 3150	. . . . . . . . . . 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 19950	. . . . . . . 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 6836	. . . . . . 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 6834	. . . . . 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 13563	. . . . . . . . . . 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 3130	. . . . . . . 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 13924	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (0...𝑠) ∈ Fin)
109	31, 28, 97, 99, 107, 108	coe1fzgsumd 22278	. . . . . 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 2899	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ0
112	111, 53	nfralw 3285	. . . . . . . . . 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 22247	. . . . . . . . . . 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 2741	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐿 → (𝑛 = 𝑘 ↔ 𝐿 = 𝑘))
123	122	ifbid 4491	. . . . . . . . . . 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 20237	. . . . . . . . . . . . 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 4514	. . . . . . . . . 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 6947	. . . . . . . . 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 5178	. . . . . . . 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 7374	. . . . . . 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 5090	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (𝑠 < 𝑥 ↔ 𝑠 < 𝐿))
134		csbeq1 3841	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐿 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝐿 / 𝑘⦌𝐴)
135	134	eqeq1d 2739	. . . . . . . . . . . . . . . 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 3563	. . . . . . . . . . . . . 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 3862	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴
140	139	nfeq1 2915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴 = 0
141	138, 140	nfan 1901	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 )
142		elfz2nn0 13561	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ (0...𝑠) ↔ (𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑘 ≤ 𝑠))
143		nn0re 12435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
144	143	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝑘 ∈ ℝ)
145		nn0re 12435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
149	148	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℝ)
150		lelttr 11225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2934	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐿 ≠ 𝑘 ↔ ¬ 𝐿 = 𝑘)
154	143	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → 𝑘 ∈ ℝ)
155		lttri2 11217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4478	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → if(𝐿 = 𝑘, 𝐴, 0 ) = 0 )
177	141, 176	mpteq2da 5178	. . . . . . . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . . . . . . . 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 20213	. . . . . . . . . . . . . . . . . . . . . . . 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 7391	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...𝑠) ∈ V
183	12	gsumz 18793	. . . . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . 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 12488	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℝ)
206		lenlt 11213	. . . . . . . . . . . . 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 13561	. . . . . . . . . . . . . 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 2744	. . . . . . . . . . . 12 ⊢ (𝐿 = 𝑘 ↔ 𝑘 = 𝐿)
217		ifbi 4490	. . . . . . . . . . . 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 5182	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 )) = (𝑘 ∈ (0...𝑠) ↦ if(𝑘 = 𝐿, 𝐴, 0 ))
220	3, 5	eleqtrdi 2847	. . . . . . . . . . . . . . 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 3130	. . . . . . . . . . 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 19929	. . . . . . . . 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 2772	. . . . . 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 2776	. . . . 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 3139	. 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 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430  ⦋csb 3838  ifcif 4467   class class class wbr 5086   ↦ cmpt 5167  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358   ↑_m cmap 8764   finSupp cfsupp 9265  ℝcr 11026  0cc0 11027   < clt 11168   ≤ cle 11169  ℕ₀cn0 12426  ...cfz 13450  Basecbs 17168  Scalarcsca 17212   ·_𝑠 cvsca 17213  0_gc0g 17391   Σ_g cgsu 17392  Mndcmnd 18691  ._gcmg 19032  CMndccmn 19744  mulGrpcmgp 20110  Ringcrg 20203  LModclmod 20844  var₁cv1 22148  Poly₁cpl1 22149  coe₁cco1 22150

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8102  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-er 8634  df-map 8766  df-pm 8767  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fsupp 9266  df-sup 9346  df-oi 9416  df-card 9852  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-3 12234  df-4 12235  df-5 12236  df-6 12237  df-7 12238  df-8 12239  df-9 12240  df-n0 12427  df-z 12514  df-dec 12634  df-uz 12778  df-fz 13451  df-fzo 13598  df-seq 13953  df-hash 14282  df-struct 17106  df-sets 17123  df-slot 17141  df-ndx 17153  df-base 17169  df-ress 17190  df-plusg 17222  df-mulr 17223  df-sca 17225  df-vsca 17226  df-ip 17227  df-tset 17228  df-ple 17229  df-ds 17231  df-hom 17233  df-cco 17234  df-0g 17393  df-gsum 17394  df-prds 17399  df-pws 17401  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-mhm 18740  df-submnd 18741  df-grp 18901  df-minusg 18902  df-sbg 18903  df-mulg 19033  df-subg 19088  df-ghm 19177  df-cntz 19281  df-cmn 19746  df-abl 19747  df-mgp 20111  df-rng 20123  df-ur 20152  df-ring 20205  df-subrng 20512  df-subrg 20536  df-lmod 20846  df-lss 20916  df-psr 21897  df-mvr 21898  df-mpl 21899  df-opsr 21901  df-psr1 22152  df-vr1 22153  df-ply1 22154  df-coe1 22155

This theorem is referenced by:  gsumply1eq  22283  pm2mpf1lem  22768  pm2mpcoe1  22774  pm2mpmhmlem2  22793  cayleyhamilton1  22866  gsummoncoe1fzo  33677  ply1mulgsum  48863