Theorem gsummoncoe1 22368

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 3254	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝐾)
4	3	fmpttd 7096	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐴):ℕ0⟶𝐾)
5		gsummonply1.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑅)
6	5	fvexi 6881	. . . . . . 7 ⊢ 𝐾 ∈ V
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ V)
8		nn0ex 12487	. . . . . 6 ⊢ ℕ0 ∈ V
9		elmapg 8820	. . . . . 6 ⊢ ((𝐾 ∈ V ∧ ℕ0 ∈ V) → ((𝑘 ∈ ℕ0 ↦ 𝐴) ∈ (𝐾 ↑m ℕ0) ↔ (𝑘 ∈ ℕ0 ↦ 𝐴):ℕ0⟶𝐾))
10	7, 8, 9	sylancl 595	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ 𝐴) ∈ (𝐾 ↑m ℕ0) ↔ (𝑘 ∈ ℕ0 ↦ 𝐴):ℕ0⟶𝐾))
11	4, 10	mpbird 259	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐴) ∈ (𝐾 ↑m ℕ0))
12		gsummonply1.0	. . . . 5 ⊢ 0 = (0g‘𝑅)
13	12	fvexi 6881	. . . 4 ⊢ 0 ∈ V
14		fsuppmapnn0ub 14008	. . . 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 595	. . 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 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
18	2	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐾)
19		rspcsbela 4392	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐾) → ⦋𝑥 / 𝑘⦌𝐴 ∈ 𝐾)
20	17, 18, 19	syl2anc 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ⦋𝑥 / 𝑘⦌𝐴 ∈ 𝐾)
21		eqid 2762	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐴) = (𝑘 ∈ ℕ0 ↦ 𝐴)
22	21	fvmpts 6979	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐴 ∈ 𝐾) → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐴)
23	17, 20, 22	syl2anc 593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐴)
24	23	eqeq1d 2764	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ↔ ⦋𝑥 / 𝑘⦌𝐴 = 0 ))
25	24	imbi2d 342	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑠 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ) ↔ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )))
26	25	biimpd 231	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑠 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐴)‘𝑥) = 0 ) → (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )))
27	26	ralimdva 3174	. . . 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 2762	. . . . . . . . 9 ⊢ (0g‘𝑃) = (0g‘𝑃)
30		gsummonply1.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Ring)
31		gsummonply1.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
32	31	ply1ring 22306	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
33		ringcmn 20328	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
34	30, 32, 33	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ CMnd)
35	34	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑃 ∈ CMnd)
36	30	3ad2ant1 1146	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → 𝑅 ∈ Ring)
37		simp3 1151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → 𝐴 ∈ 𝐾)
38		simp2 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → 𝑘 ∈ ℕ0)
39		gsummonply1.x	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = (var1‘𝑅)
40		gsummonply1.m	. . . . . . . . . . . . . . 15 ⊢ ∗ = ( ·𝑠 ‘𝑃)
41		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
42		gsummonply1.e	. . . . . . . . . . . . . . 15 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
43	5, 31, 39, 40, 41, 42, 28	ply1tmcl 22332	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
44	36, 37, 38, 43	syl3anc 1390	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
45	44	3expia 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴 ∈ 𝐾 → (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵))
46	45	ralimdva 3174	. . . . . . . . . . 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 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ ℕ0 (𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
49		simplr 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑠 ∈ ℕ0)
50		nfv 1934	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑠 < 𝑥
51		nfcsb1v 3876	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴
52	51	nfeq1 2939	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐴 = 0
53	50, 52	nfim 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )
54		nfv 1934	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 )
55		breq2 5104	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (𝑠 < 𝑥 ↔ 𝑠 < 𝑘))
56		csbeq1 3855	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝑘 / 𝑘⦌𝐴)
57	56	eqeq1d 2764	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (⦋𝑥 / 𝑘⦌𝐴 = 0 ↔ ⦋𝑘 / 𝑘⦌𝐴 = 0 ))
58	55, 57	imbi12d 346	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → ((𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 )))
59	53, 54, 58	cbvralw 3304	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) ↔ ∀𝑘 ∈ ℕ0 (𝑠 < 𝑘 → ⦋𝑘 / 𝑘⦌𝐴 = 0 ))
60		csbid 3865	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑘 / 𝑘⦌𝐴 = 𝐴
61	60	eqeq1i 2767	. . . . . . . . . . . . . 14 ⊢ (⦋𝑘 / 𝑘⦌𝐴 = 0 ↔ 𝐴 = 0 )
62		oveq1 7403	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = ( 0 ∗ (𝑘 ↑ 𝑋)))
63	31	ply1sca 22311	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
64	30, 63	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
65	64	fveq2d 6871	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
66	12, 65	eqtrid 2809	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 = (0g‘(Scalar‘𝑃)))
67	66	ad2antrr 736	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 0 = (0g‘(Scalar‘𝑃)))
68	67	oveq1d 7411	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ( 0 ∗ (𝑘 ↑ 𝑋)) = ((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)))
69	31	ply1lmod 22310	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
70	30, 69	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ LMod)
71	70	ad2antrr 736	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ LMod)
72		eqid 2762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘𝑃) = (Base‘𝑃)
73	41, 72	mgpbas 20191	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
74	41	ringmgp 20285	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
75	30, 32, 74	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
76	75	ad2antrr 736	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd)
77		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
78	39, 31, 72	vr1cl 22276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
79	30, 78	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑃))
80	79	ad2antrr 736	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑋 ∈ (Base‘𝑃))
81	73, 42, 76, 77, 80	mulgnn0cld 19137	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑃))
82		eqid 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
83		eqid 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
84	72, 82, 40, 83, 29	lmod0vs 20959	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ LMod ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑃)) → ((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
85	71, 81, 84	syl2anc 593	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((0g‘(Scalar‘𝑃)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
86	68, 85	eqtrd 2797	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ( 0 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
87	62, 86	sylan9eqr 2819	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝐴 = 0 ) → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))
88	87	ex 416	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝐴 = 0 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃)))
89	61, 88	biimtrid 244	. . . . . . . . . . . . 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 3174	. . . . . . . . . . 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 244	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ) → ∀𝑘 ∈ ℕ0 (𝑠 < 𝑘 → (𝐴 ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑃))))
93	92	imp 410	. . . . . . . . 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 20026	. . . . . . . 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 6871	. . . . . . 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 6869	. . . . . 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 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝑅 ∈ Ring)
98		gsummonply1.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℕ0)
99	98	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → 𝐿 ∈ ℕ0)
100		elfznn0 13625	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑠) → 𝑘 ∈ ℕ0)
101		simpll 776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝜑)
102	3	adantlr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ 𝐾)
103	101, 77, 102	3jca 1141	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾))
104	100, 103	sylan2 602	. . . . . . . . . 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 3154	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → ∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
107	106	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → ∀𝑘 ∈ (0...𝑠)(𝐴 ∗ (𝑘 ↑ 𝑋)) ∈ 𝐵)
108		fzfid 13986	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (0...𝑠) ∈ Fin)
109	31, 28, 97, 99, 107, 108	coe1fzgsumd 22364	. . . . . 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 1934	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑠 ∈ ℕ0)
111		nfcv 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ0
112	111, 53	nfralw 3309	. . . . . . . . . 10 ⊢ Ⅎ𝑘∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )
113	110, 112	nfan 1919	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 ))
114	30	ad3antrrr 740	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝑅 ∈ Ring)
115	3	expcom 417	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (𝜑 → 𝐴 ∈ 𝐾))
116	115, 100	syl11 33	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾))
117	116	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (𝑘 ∈ (0...𝑠) → 𝐴 ∈ 𝐾))
118	117	imp 410	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ 𝐾)
119	100	adantl 485	. . . . . . . . . . 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 22333	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0) → (coe1‘(𝐴 ∗ (𝑘 ↑ 𝑋))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑘, 𝐴, 0 )))
121	114, 118, 119, 120	syl3anc 1390	. . . . . . . . . 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 2766	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐿 → (𝑛 = 𝑘 ↔ 𝐿 = 𝑘))
123	122	ifbid 4504	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐿 → if(𝑛 = 𝑘, 𝐴, 0 ) = if(𝐿 = 𝑘, 𝐴, 0 ))
124	123	adantl 485	. . . . . . . . . 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 740	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 𝐿 ∈ ℕ0)
126	5, 12	ring0cl 20313	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
127	30, 126	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ 𝐾)
128	127	ad3antrrr 740	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) ∧ 𝑘 ∈ (0...𝑠)) → 0 ∈ 𝐾)
129	118, 128	ifcld 4527	. . . . . . . . . 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 6983	. . . . . . . . 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 5192	. . . . . . . 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 7412	. . . . . . 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 5104	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (𝑠 < 𝑥 ↔ 𝑠 < 𝐿))
134		csbeq1 3855	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐿 → ⦋𝑥 / 𝑘⦌𝐴 = ⦋𝐿 / 𝑘⦌𝐴)
135	134	eqeq1d 2764	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (⦋𝑥 / 𝑘⦌𝐴 = 0 ↔ ⦋𝐿 / 𝑘⦌𝐴 = 0 ))
136	133, 135	imbi12d 346	. . . . . . . . . . . . . . 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 1934	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿))
139		nfcsb1v 3876	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴
140	139	nfeq1 2939	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘⦋𝐿 / 𝑘⦌𝐴 = 0
141	138, 140	nfan 1919	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 )
142		elfz2nn0 13623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ (0...𝑠) ↔ (𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑘 ≤ 𝑠))
143		nn0re 12490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
144	143	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝑘 ∈ ℝ)
145		nn0re 12490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ)
146	145	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → 𝑠 ∈ ℝ)
147	146	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝑠 ∈ ℝ)
148		nn0re 12490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
149	148	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℝ)
150		lelttr 11273	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑘 ∈ ℝ ∧ 𝑠 ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿))
151	144, 147, 149, 150	syl3anc 1390	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → ((𝑘 ≤ 𝑠 ∧ 𝑠 < 𝐿) → 𝑘 < 𝐿))
152		animorr 992	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 < 𝐿) → (𝐿 < 𝑘 ∨ 𝑘 < 𝐿))
153		df-ne 2958	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐿 ≠ 𝑘 ↔ ¬ 𝐿 = 𝑘)
154	143	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → 𝑘 ∈ ℝ)
155		lttri2 11265	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝐿 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
156	148, 154, 155	syl2anr 606	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
157	156	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 < 𝐿) → (𝐿 ≠ 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
158	153, 157	bitr3id 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 < 𝐿) → (¬ 𝐿 = 𝑘 ↔ (𝐿 < 𝑘 ∨ 𝑘 < 𝐿)))
159	152, 158	mpbird 259	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 < 𝐿) → ¬ 𝐿 = 𝑘)
160	159	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (𝑘 ≤ 𝑠 → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘))))
163	162	expimpd 457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑘 ≤ 𝑠) → ((𝑠 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))
166	165	3adant2 1144	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑘 ≤ 𝑠) → ((𝑠 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))
167	142, 166	sylbi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ (0...𝑠) → ((𝑠 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑠 < 𝐿 → ¬ 𝐿 = 𝑘)))
168	167	expd 419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿) → (𝜑 → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘)))
173	172	impcom 411	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))
174	173	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → (𝑘 ∈ (0...𝑠) → ¬ 𝐿 = 𝑘))
175	174	imp 410	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → ¬ 𝐿 = 𝑘)
176	175	iffalsed 4491	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) ∧ 𝑘 ∈ (0...𝑠)) → if(𝐿 = 𝑘, 𝐴, 0 ) = 0 )
177	141, 176	mpteq2da 5192	. . . . . . . . . . . . . . . . . . . . 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 7412	. . . . . . . . . . . . . . . . . . . 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 20289	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
180	30, 179	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑅 ∈ Mnd)
181	180	adantr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → 𝑅 ∈ Mnd)
182		ovex 7429	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...𝑠) ∈ V
183	12	gsumz 18870	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑠) ∈ V) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
184	181, 182, 183	sylancl 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ 0 )) = 0 )
185	184	adantr 484	. . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⦋𝐿 / 𝑘⦌𝐴 = 0 → 0 = ⦋𝐿 / 𝑘⦌𝐴)
188	187	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑠 < 𝐿)) ∧ ⦋𝐿 / 𝑘⦌𝐴 = 0 ) → 0 = ⦋𝐿 / 𝑘⦌𝐴)
189	178, 185, 188	3eqtrd 2801	. . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 421	. . . . . . . . 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		pm3.2 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿)))
202	201	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐴 = 0 )) → (¬ 𝑠 < 𝐿 → ((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿)))
203	180	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → 𝑅 ∈ Mnd)
204	182	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → (0...𝑠) ∈ V)
205	98	nn0red 12543	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℝ)
206		lenlt 11261	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (𝐿 ≤ 𝑠 ↔ ¬ 𝑠 < 𝐿))
207	205, 145, 206	syl2an 605	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (𝐿 ≤ 𝑠 ↔ ¬ 𝑠 < 𝐿))
208	98	ad2antrr 736	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈ ℕ0)
209		simplr 778	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝑠 ∈ ℕ0)
210		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ≤ 𝑠)
211		elfz2nn0 13623	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (0...𝑠) ↔ (𝐿 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝐿 ≤ 𝑠))
212	208, 209, 210, 211	syl3anbrc 1357	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝐿 ≤ 𝑠) → 𝐿 ∈ (0...𝑠))
213	212	ex 416	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (𝐿 ≤ 𝑠 → 𝐿 ∈ (0...𝑠)))
214	207, 213	sylbird 262	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (¬ 𝑠 < 𝐿 → 𝐿 ∈ (0...𝑠)))
215	214	imp 410	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → 𝐿 ∈ (0...𝑠))
216		eqcom 2769	. . . . . . . . . . . 12 ⊢ (𝐿 = 𝑘 ↔ 𝑘 = 𝐿)
217		ifbi 4503	. . . . . . . . . . . 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 5196	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑠) ↦ if(𝐿 = 𝑘, 𝐴, 0 )) = (𝑘 ∈ (0...𝑠) ↦ if(𝑘 = 𝐿, 𝐴, 0 ))
220	3, 5	eleqtrdi 2872	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ (Base‘𝑅))
221	220	ex 416	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ ℕ0 → 𝐴 ∈ (Base‘𝑅)))
222	221	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (𝑘 ∈ ℕ0 → 𝐴 ∈ (Base‘𝑅)))
223	222, 100	impel 513	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑠)) → 𝐴 ∈ (Base‘𝑅))
224	223	ralrimiva 3154	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → ∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅))
225	224	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ¬ 𝑠 < 𝐿) → ∀𝑘 ∈ (0...𝑠)𝐴 ∈ (Base‘𝑅))
226	12, 203, 204, 215, 219, 225	gsummpt1n0 20005	. . . . . . . . 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 183	. . . . . . 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 2797	. . . . . 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 2801	. . . . 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 416	. . . 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 3163	. 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 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  ∃wrex 3086  Vcvv 3454  ⦋csb 3852  ifcif 4480   class class class wbr 5100   ↦ cmpt 5181  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ↑_m cmap 8808   finSupp cfsupp 9307  ℝcr 11072  0cc0 11073   < clt 11216   ≤ cle 11217  ℕ₀cn0 12481  ...cfz 13512  Basecbs 17245  Scalarcsca 17289   ·_𝑠 cvsca 17290  0_gc0g 17468   Σ_g cgsu 17469  Mndcmnd 18768  ._gcmg 19109  CMndccmn 19820  mulGrpcmgp 20186  Ringcrg 20279  LModclmod 20924  var₁cv1 22235  Poly₁cpl1 22236  coe₁cco1 22237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-ofr 7661  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9308  df-sup 9388  df-oi 9458  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-z 12569  df-dec 12689  df-uz 12840  df-fz 13513  df-fzo 13660  df-seq 14015  df-hash 14344  df-struct 17183  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-mulr 17300  df-sca 17302  df-vsca 17303  df-ip 17304  df-tset 17305  df-ple 17306  df-ds 17308  df-hom 17310  df-cco 17311  df-0g 17470  df-gsum 17471  df-prds 17476  df-pws 17478  df-mre 17614  df-mrc 17615  df-acs 17617  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-mhm 18817  df-submnd 18818  df-grp 18978  df-minusg 18979  df-sbg 18980  df-mulg 19110  df-subg 19165  df-ghm 19254  df-cntz 19357  df-cmn 19822  df-abl 19823  df-mgp 20187  df-rng 20199  df-ur 20228  df-ring 20281  df-subrng 20592  df-subrg 20616  df-lmod 20926  df-lss 20996  df-psr 21958  df-mvr 21959  df-mpl 21960  df-opsr 21962  df-psr1 22239  df-vr1 22240  df-ply1 22241  df-coe1 22242

This theorem is referenced by:  gsumply1eq  22369  pm2mpf1lem  22851  pm2mpcoe1  22857  pm2mpmhmlem2  22876  cayleyhamilton1  22949  gsummoncoe1fzo  33790  ply1mulgsum  49009