Theorem gsummoncoe1 21675

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 3234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝐾)
4	3	fmpttd 7063	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐴):ℕ0⟶𝐾)
5		gsummonply1.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑅)
6	5	fvexi 6856	. . . . . . 7 ⊢ 𝐾 ∈ V
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ V)
8		nn0ex 12419	. . . . . 6 ⊢ ℕ0 ∈ V
9		elmapg 8778	. . . . . 6 ⊢ ((𝐾 ∈ V ∧ ℕ0 ∈ V) → ((𝑘 ∈ ℕ0 ↦ 𝐴) ∈ (𝐾 ↑m ℕ0) ↔ (𝑘 ∈ ℕ0 ↦ 𝐴):ℕ0⟶𝐾))
10	7, 8, 9	sylancl 586	. . . . 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 = (0g‘𝑅)
13	12	fvexi 6856	. . . 4 ⊢ 0 ∈ V
14		fsuppmapnn0ub 13900	. . . 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 586	. . 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 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
18	2	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐾)
19		rspcsbela 4395	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐾) → ⦋𝑥 / 𝑘⦌𝐴 ∈ 𝐾)
20	17, 18, 19	syl2anc 584	. . . . . . . . 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 584	. . . . . . . 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 228	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑠 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ) → (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )))
27	26	ralimdva 3164	. . . 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 21619	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
33		ringcmn 20003	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
34	30, 32, 33	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ CMnd)
35	34	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑃 ∈ CMnd)
36	30	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → 𝑅 ∈ Ring)
37		simp3 1138	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → 𝐴 ∈ 𝐾)
38		simp2 1137	. . . . . . . . . . . . . 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 21643	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
44	36, 37, 38, 43	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
45	44	3expia 1121	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴 ∈ 𝐾 → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵))
46	45	ralimdva 3164	. . . . . . . . . . 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 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ ℕ0 (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
49		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑠 ∈ ℕ0)
50		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑠 < 𝑥
51		nfcsb1v 3880	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴
52	51	nfeq1 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴 = 0
53	50, 52	nfim 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )
54		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 )
55		breq2 5109	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑘))
56		csbeq1 3858	. . . . . . . . . . . . . 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 3289	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ ∀𝑘 ∈ ℕ0 (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ))
60		csbid 3868	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑘 / 𝑘⦌𝐴 = 𝐴
61	60	eqeq1i 2741	. . . . . . . . . . . . . 14 ⊢ (⦋𝑘 / 𝑘⦌𝐴 = 0 ↔ 𝐴 = 0 )
62		oveq1 7364	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = ( 0 ∗ (𝑘 ↑ 𝑋)))
63	31	ply1sca 21624	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
64	30, 63	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
65	64	fveq2d 6846	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
66	12, 65	eqtrid 2788	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 = (0g‘(Scalar‘𝑃)))
67	66	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 0 = (0g‘(Scalar‘𝑃)))
68	67	oveq1d 7372	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ( 0 ∗ (𝑘 ↑ 𝑋)) = ((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)))
69	31	ply1lmod 21623	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
70	30, 69	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ LMod)
71	70	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ LMod)
72		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘𝑃) = (Base‘𝑃)
73	41, 72	mgpbas 19902	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
74	41	ringmgp 19970	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
75	30, 32, 74	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
76	75	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd)
77		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
78	39, 31, 72	vr1cl 21588	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
79	30, 78	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑃))
80	79	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑋 ∈ (Base‘𝑃))
81	73, 42, 76, 77, 80	mulgnn0cld 18897	. . . . . . . . . . . . . . . . . 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 20355	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ LMod ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑃)) → ((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
85	71, 81, 84	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
86	68, 85	eqtrd 2776	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ( 0 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
87	62, 86	sylan9eqr 2798	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝐴 = 0 ) → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
88	87	ex 413	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)))
89	61, 88	biimtrid 241	. . . . . . . . . . . . 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 3164	. . . . . . . . . . 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 241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) → ∀𝑘 ∈ ℕ0 (𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))))
93	92	imp 407	. . . . . . . . 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 19763	. . . . . . . 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 6846	. . . . . . 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 6844	. . . . . 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 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑅 ∈ Ring)
98		gsummonply1.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℕ0)
99	98	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝐿 ∈ ℕ0)
100		elfznn0 13534	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑠) → 𝑘 ∈ ℕ0)
101		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝜑)
102	3	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝐾)
103	101, 77, 102	3jca 1128	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾))
104	100, 103	sylan2 593	. . . . . . . . . 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 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → ∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
107	106	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
108		fzfid 13878	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (0...𝑠) ∈ Fin)
109	31, 28, 97, 99, 107, 108	coe1fzgsumd 21673	. . . . . 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 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑠 ∈ ℕ0)
111		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ0
112	111, 53	nfralw 3294	. . . . . . . . . 10 ⊢ Ⅎ𝑘∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )
113	110, 112	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ))
114	30	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝑅 ∈ Ring)
115	3	expcom 414	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (𝜑 → 𝐴 ∈ 𝐾))
116	115, 100	syl11 33	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾))
117	116	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾))
118	117	imp 407	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ 𝐾)
119	100	adantl 482	. . . . . . . . . . 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 21644	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0) → (coe1‘(𝐴 ∗ (𝑘 ↑ 𝑋))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑘, 𝐴, 0 )))
121	114, 118, 119, 120	syl3anc 1371	. . . . . . . . . 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 4509	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐿 → if(𝑛 = 𝑘, 𝐴, 0 ) = if(𝐿 = 𝑘, 𝐴, 0 ))
124	123	adantl 482	. . . . . . . . . 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 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐿 ∈ ℕ0)
126	5, 12	ring0cl 19990	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
127	30, 126	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ 𝐾)
128	127	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 0 ∈ 𝐾)
129	118, 128	ifcld 4532	. . . . . . . . . 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 5203	. . . . . . . 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 7373	. . . . . . 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 5109	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (𝑠 < 𝑥 ↔ 𝑠 < 𝐿))
134		csbeq1 3858	. . . . . . . . . . . . . . . . 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 3579	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑠 < 𝐿 → ⦋𝐿 / 𝑘⦌𝐴 = 0 ))
138		nfv 1917	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿))
139		nfcsb1v 3880	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴
140	139	nfeq1 2922	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴 = 0
141	138, 140	nfan 1902	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 )
142		elfz2nn0 13532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ (0...𝑠) ↔ (𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑘 ≤ 𝑠))
143		nn0re 12422	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
144	143	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝑘 ∈ ℝ)
145		nn0re 12422	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ)
146	145	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → 𝑠 ∈ ℝ)
147	146	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝑠 ∈ ℝ)
148		nn0re 12422	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
149	148	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℝ)
150		lelttr 11245	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑘 ∈ ℝ ∧ 𝑠 ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿))
151	144, 147, 149, 150	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿))
152		animorr 977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 < 𝐿) → (𝐿 < 𝑘 ∨ 𝑘 < 𝐿))
153		df-ne 2944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐿 ≠ 𝑘 ↔ ¬ 𝐿 = 𝑘)
154	143	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → 𝑘 ∈ ℝ)
155		lttri2 11237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝐿 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
156	148, 154, 155	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
157	156	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (𝑘 ≤ 𝑠 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))))
163	162	expimpd 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑘 ≤ 𝑠) → ((𝑠 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))
166	165	3adant2 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿) → (𝜑 → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘)))
173	172	impcom 408	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))
174	173	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))
175	174	imp 407	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → ¬ 𝐿 = 𝑘)
176	175	iffalsed 4497	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → if(𝐿 = 𝑘, 𝐴, 0 ) = 0 )
177	141, 176	mpteq2da 5203	. . . . . . . . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . . . . . . . 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 19974	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
180	30, 179	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑅 ∈ Mnd)
181	180	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → 𝑅 ∈ Mnd)
182		ovex 7390	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...𝑠) ∈ V
183	12	gsumz 18646	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑠) ∈ V) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
184	181, 182, 183	sylancl 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
185	184	adantr 481	. . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → 0 = ⦋𝐿 / 𝑘⦌𝐴)
189	178, 185, 188	3eqtrd 2780	. . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 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 413	. . . . . . . . . . . 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 418	. . . . . . . . 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		pm3.2 470	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿)))
202	201	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (¬ 𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿)))
203	180	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → 𝑅 ∈ Mnd)
204	182	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → (0...𝑠) ∈ V)
205	98	nn0red 12474	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℝ)
206		lenlt 11233	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (𝐿 ≤ 𝑠 ↔ ¬ 𝑠 < 𝐿))
207	205, 145, 206	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (𝐿 ≤ 𝑠 ↔ ¬ 𝑠 < 𝐿))
208	98	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈ ℕ0)
209		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝑠 ∈ ℕ0)
210		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ≤ 𝑠)
211		elfz2nn0 13532	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (0...𝑠) ↔ (𝐿 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝐿 ≤ 𝑠))
212	208, 209, 210, 211	syl3anbrc 1343	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈ (0...𝑠))
213	212	ex 413	. . . . . . . . . . . 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 407	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → 𝐿 ∈ (0...𝑠))
216		eqcom 2743	. . . . . . . . . . . 12 ⊢ (𝐿 = 𝑘 ↔ 𝑘 = 𝐿)
217		ifbi 4508	. . . . . . . . . . . 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 5210	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 )) = (𝑘 ∈ (0...𝑠) ↦ if(𝑘 = 𝐿, 𝐴, 0 ))
220	3, 5	eleqtrdi 2848	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ (Base‘𝑅))
221	220	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ ℕ0 → 𝐴 ∈ (Base‘𝑅)))
222	221	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (𝑘 ∈ ℕ0 → 𝐴 ∈ (Base‘𝑅)))
223	222, 100	impel 506	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ (Base‘𝑅))
224	223	ralrimiva 3143	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → ∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅))
225	224	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → ∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅))
226	12, 203, 204, 215, 219, 225	gsummpt1n0 19742	. . . . . . . . 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 2776	. . . . . 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 2780	. . . . 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 413	. . . 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 3152	. 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 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  Vcvv 3445  ⦋csb 3855  ifcif 4486   class class class wbr 5105   ↦ cmpt 5188  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ↑_m cmap 8765   finSupp cfsupp 9305  ℝcr 11050  0cc0 11051   < clt 11189   ≤ cle 11190  ℕ₀cn0 12413  ...cfz 13424  Basecbs 17083  Scalarcsca 17136   ·_𝑠 cvsca 17137  0_gc0g 17321   Σ_g cgsu 17322  Mndcmnd 18556  ._gcmg 18872  CMndccmn 19562  mulGrpcmgp 19896  Ringcrg 19964  LModclmod 20322  var₁cv1 21547  Poly₁cpl1 21548  coe₁cco1 21549

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-subrg 20220  df-lmod 20324  df-lss 20393  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-vr1 21552  df-ply1 21553  df-coe1 21554

This theorem is referenced by:  gsumply1eq  21676  pm2mpf1lem  22143  pm2mpcoe1  22149  pm2mpmhmlem2  22168  cayleyhamilton1  22241  ply1mulgsum  46461