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Theorem psdpw 22091

Description: Power rule for partial derivative of power series. (Contributed by SN, 25-Apr-2025.)

**Hypotheses**
Ref	Expression
psdpw.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psdpw.b	⊢ 𝐵 = (Base‘𝑆)
psdpw.g	⊢ · = (._g‘𝑆)
psdpw.t	⊢ ∙ = (._r‘𝑆)
psdpw.m	⊢ 𝑀 = (mulGrp‘𝑆)
psdpw.e	⊢ ↑ = (._g‘𝑀)
psdpw.r	⊢ (𝜑 → 𝑅 ∈ CRing)
psdpw.x	⊢ (𝜑 → 𝑋 ∈ 𝐼)
psdpw.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
psdpw.n	⊢ (𝜑 → 𝑁 ∈ ℕ)

**Assertion**
Ref	Expression
psdpw	⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑁 ↑ 𝐹)) = ((𝑁 · ((𝑁 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))

Proof of Theorem psdpw Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psdpw.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		fvoveq1 7375	. . . 4 ⊢ (𝑛 = 1 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘(1 ↑ 𝐹)))
3		id 22	. . . . . 6 ⊢ (𝑛 = 1 → 𝑛 = 1)
4		oveq1 7359	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
5		1m1e0 12203	. . . . . . . 8 ⊢ (1 − 1) = 0
6	4, 5	eqtrdi 2782	. . . . . . 7 ⊢ (𝑛 = 1 → (𝑛 − 1) = 0)
7	6	oveq1d 7367	. . . . . 6 ⊢ (𝑛 = 1 → ((𝑛 − 1) ↑ 𝐹) = (0 ↑ 𝐹))
8	3, 7	oveq12d 7370	. . . . 5 ⊢ (𝑛 = 1 → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = (1 · (0 ↑ 𝐹)))
9	8	oveq1d 7367	. . . 4 ⊢ (𝑛 = 1 → ((𝑛 · ((𝑛 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = ((1 · (0 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
10	2, 9	eqeq12d 2747	. . 3 ⊢ (𝑛 = 1 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = ((𝑛 · ((𝑛 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) ↔ (((𝐼 mPSDer 𝑅)‘𝑋)‘(1 ↑ 𝐹)) = ((1 · (0 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))))
11		fvoveq1 7375	. . . 4 ⊢ (𝑛 = 𝑚 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)))
12		id 22	. . . . . 6 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚)
13		oveq1 7359	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑛 − 1) = (𝑚 − 1))
14	13	oveq1d 7367	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑛 − 1) ↑ 𝐹) = ((𝑚 − 1) ↑ 𝐹))
15	12, 14	oveq12d 7370	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = (𝑚 · ((𝑚 − 1) ↑ 𝐹)))
16	15	oveq1d 7367	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝑛 · ((𝑛 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
17	11, 16	eqeq12d 2747	. . 3 ⊢ (𝑛 = 𝑚 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = ((𝑛 · ((𝑛 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) ↔ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))))
18		fvoveq1 7375	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘((𝑚 + 1) ↑ 𝐹)))
19		id 22	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → 𝑛 = (𝑚 + 1))
20		oveq1 7359	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 − 1) = ((𝑚 + 1) − 1))
21	20	oveq1d 7367	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛 − 1) ↑ 𝐹) = (((𝑚 + 1) − 1) ↑ 𝐹))
22	19, 21	oveq12d 7370	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = ((𝑚 + 1) · (((𝑚 + 1) − 1) ↑ 𝐹)))
23	22	oveq1d 7367	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛 · ((𝑛 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = (((𝑚 + 1) · (((𝑚 + 1) − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
24	18, 23	eqeq12d 2747	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = ((𝑛 · ((𝑛 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) ↔ (((𝐼 mPSDer 𝑅)‘𝑋)‘((𝑚 + 1) ↑ 𝐹)) = (((𝑚 + 1) · (((𝑚 + 1) − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))))
25		fvoveq1 7375	. . . 4 ⊢ (𝑛 = 𝑁 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑁 ↑ 𝐹)))
26		id 22	. . . . . 6 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
27		oveq1 7359	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
28	27	oveq1d 7367	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑛 − 1) ↑ 𝐹) = ((𝑁 − 1) ↑ 𝐹))
29	26, 28	oveq12d 7370	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = (𝑁 · ((𝑁 − 1) ↑ 𝐹)))
30	29	oveq1d 7367	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 · ((𝑛 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = ((𝑁 · ((𝑁 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
31	25, 30	eqeq12d 2747	. . 3 ⊢ (𝑛 = 𝑁 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = ((𝑛 · ((𝑛 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) ↔ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑁 ↑ 𝐹)) = ((𝑁 · ((𝑁 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))))
32		psdpw.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
33		psdpw.t	. . . . 5 ⊢ ∙ = (._r‘𝑆)
34		eqid 2731	. . . . 5 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
35		psdpw.s	. . . . . . 7 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
36		psdpw.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
37		reldmpsr 21857	. . . . . . . . . 10 ⊢ Rel dom mPwSer
38	37, 35, 32	elbasov 17133	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
39	36, 38	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
40	39	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V)
41		psdpw.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ CRing)
42	35, 40, 41	psrcrng 21915	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
43	42	crngringd 20170	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
44	41	crnggrpd 20171	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp)
45	44	grpmgmd 18880	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Mgm)
46		psdpw.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
47	35, 32, 45, 46, 36	psdcl 22082	. . . . 5 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
48	32, 33, 34, 43, 47	ringlidmd 20196	. . . 4 ⊢ (𝜑 → ((1_r‘𝑆) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))
49		psdpw.m	. . . . . . . . . 10 ⊢ 𝑀 = (mulGrp‘𝑆)
50	49, 32	mgpbas 20069	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
51	49, 34	ringidval 20107	. . . . . . . . 9 ⊢ (1_r‘𝑆) = (0_g‘𝑀)
52		psdpw.e	. . . . . . . . 9 ⊢ ↑ = (._g‘𝑀)
53	50, 51, 52	mulg0 18993	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → (0 ↑ 𝐹) = (1_r‘𝑆))
54	36, 53	syl 17	. . . . . . 7 ⊢ (𝜑 → (0 ↑ 𝐹) = (1_r‘𝑆))
55	54	oveq2d 7368	. . . . . 6 ⊢ (𝜑 → (1 · (0 ↑ 𝐹)) = (1 · (1_r‘𝑆)))
56	32, 34, 43	ringidcld 20190	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑆) ∈ 𝐵)
57		psdpw.g	. . . . . . . 8 ⊢ · = (._g‘𝑆)
58	32, 57	mulg1 19000	. . . . . . 7 ⊢ ((1_r‘𝑆) ∈ 𝐵 → (1 · (1_r‘𝑆)) = (1_r‘𝑆))
59	56, 58	syl 17	. . . . . 6 ⊢ (𝜑 → (1 · (1_r‘𝑆)) = (1_r‘𝑆))
60	55, 59	eqtrd 2766	. . . . 5 ⊢ (𝜑 → (1 · (0 ↑ 𝐹)) = (1_r‘𝑆))
61	60	oveq1d 7367	. . . 4 ⊢ (𝜑 → ((1 · (0 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = ((1_r‘𝑆) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
62	50, 52	mulg1 19000	. . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (1 ↑ 𝐹) = 𝐹)
63	36, 62	syl 17	. . . . 5 ⊢ (𝜑 → (1 ↑ 𝐹) = 𝐹)
64	63	fveq2d 6832	. . . 4 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(1 ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))
65	48, 61, 64	3eqtr4rd 2777	. . 3 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(1 ↑ 𝐹)) = ((1 · (0 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
66		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
67	66	oveq1d 7367	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → ((((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) ∙ 𝐹) = (((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) ∙ 𝐹))
68	42	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ CRing)
69	42	crnggrpd 20171	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ Grp)
70	69	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ Grp)
71		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
72	71	nnzd 12501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
73	43	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ Ring)
74	49	ringmgp 20163	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Ring → 𝑀 ∈ Mnd)
75	73, 74	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑀 ∈ Mnd)
76		nnm1nn0 12428	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ₀)
77	76	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 − 1) ∈ ℕ₀)
78	36	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 ∈ 𝐵)
79	50, 52, 75, 77, 78	mulgnn0cld 19014	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 − 1) ↑ 𝐹) ∈ 𝐵)
80	32, 57, 70, 72, 79	mulgcld 19015	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∈ 𝐵)
81	47	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
82	32, 33, 68, 80, 81, 78	crng32d 20183	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) ∙ 𝐹) = (((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ 𝐹) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
83	82	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) ∙ 𝐹) = (((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ 𝐹) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
84	32, 57, 33	mulgass2 20233	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Ring ∧ (𝑚 ∈ ℤ ∧ ((𝑚 − 1) ↑ 𝐹) ∈ 𝐵 ∧ 𝐹 ∈ 𝐵)) → ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ 𝐹) = (𝑚 · (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹)))
85	73, 72, 79, 78, 84	syl13anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ 𝐹) = (𝑚 · (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹)))
86	49, 33	mgpplusg 20068	. . . . . . . . . . . . . 14 ⊢ ∙ = (+_g‘𝑀)
87	50, 52, 86	mulgnn0p1 19004	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ Mnd ∧ (𝑚 − 1) ∈ ℕ₀ ∧ 𝐹 ∈ 𝐵) → (((𝑚 − 1) + 1) ↑ 𝐹) = (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹))
88	75, 77, 78, 87	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 − 1) + 1) ↑ 𝐹) = (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹))
89	71	nncnd 12147	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
90		npcan1 11548	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℂ → ((𝑚 − 1) + 1) = 𝑚)
91	89, 90	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 − 1) + 1) = 𝑚)
92	91	oveq1d 7367	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 − 1) + 1) ↑ 𝐹) = (𝑚 ↑ 𝐹))
93	88, 92	eqtr3d 2768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹) = (𝑚 ↑ 𝐹))
94	93	oveq2d 7368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 · (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹)) = (𝑚 · (𝑚 ↑ 𝐹)))
95	85, 94	eqtrd 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ 𝐹) = (𝑚 · (𝑚 ↑ 𝐹)))
96	95	oveq1d 7367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ 𝐹) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = ((𝑚 · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
97	96	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ 𝐹) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = ((𝑚 · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
98	67, 83, 97	3eqtrd 2770	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → ((((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) ∙ 𝐹) = ((𝑚 · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
99	98	oveq1d 7367	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (((((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) ∙ 𝐹)(+_g‘𝑆)((𝑚 ↑ 𝐹) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) = (((𝑚 · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))(+_g‘𝑆)((𝑚 ↑ 𝐹) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))))
100		eqid 2731	. . . . . 6 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
101	41	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → 𝑅 ∈ CRing)
102	46	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → 𝑋 ∈ 𝐼)
103	43, 74	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ Mnd)
104		mndmgm 18655	. . . . . . . . . 10 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mgm)
105	103, 104	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ Mgm)
106	105	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑀 ∈ Mgm)
107	50, 52	mulgnncl 19008	. . . . . . . 8 ⊢ ((𝑀 ∈ Mgm ∧ 𝑚 ∈ ℕ ∧ 𝐹 ∈ 𝐵) → (𝑚 ↑ 𝐹) ∈ 𝐵)
108	106, 71, 78, 107	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 ↑ 𝐹) ∈ 𝐵)
109	108	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (𝑚 ↑ 𝐹) ∈ 𝐵)
110	36	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → 𝐹 ∈ 𝐵)
111	35, 32, 100, 33, 101, 102, 109, 110	psdmul 22087	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (((𝐼 mPSDer 𝑅)‘𝑋)‘((𝑚 ↑ 𝐹) ∙ 𝐹)) = (((((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) ∙ 𝐹)(+_g‘𝑆)((𝑚 ↑ 𝐹) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))))
112	32, 57, 100	mulgnnp1 19001	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 ↑ 𝐹) ∈ 𝐵) → ((𝑚 + 1) · (𝑚 ↑ 𝐹)) = ((𝑚 · (𝑚 ↑ 𝐹))(+_g‘𝑆)(𝑚 ↑ 𝐹)))
113	71, 108, 112	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) · (𝑚 ↑ 𝐹)) = ((𝑚 · (𝑚 ↑ 𝐹))(+_g‘𝑆)(𝑚 ↑ 𝐹)))
114	113	oveq1d 7367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 + 1) · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = (((𝑚 · (𝑚 ↑ 𝐹))(+_g‘𝑆)(𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
115	32, 57, 70, 72, 108	mulgcld 19015	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 · (𝑚 ↑ 𝐹)) ∈ 𝐵)
116	32, 100, 33, 73, 115, 108, 81	ringdird 20188	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 · (𝑚 ↑ 𝐹))(+_g‘𝑆)(𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = (((𝑚 · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))(+_g‘𝑆)((𝑚 ↑ 𝐹) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))))
117	114, 116	eqtrd 2766	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 + 1) · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = (((𝑚 · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))(+_g‘𝑆)((𝑚 ↑ 𝐹) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))))
118	117	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (((𝑚 + 1) · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = (((𝑚 · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))(+_g‘𝑆)((𝑚 ↑ 𝐹) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))))
119	99, 111, 118	3eqtr4d 2776	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (((𝐼 mPSDer 𝑅)‘𝑋)‘((𝑚 ↑ 𝐹) ∙ 𝐹)) = (((𝑚 + 1) · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
120		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → 𝑚 ∈ ℕ)
121	50, 52, 86	mulgnnp1 19001	. . . . . 6 ⊢ ((𝑚 ∈ ℕ ∧ 𝐹 ∈ 𝐵) → ((𝑚 + 1) ↑ 𝐹) = ((𝑚 ↑ 𝐹) ∙ 𝐹))
122	120, 110, 121	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → ((𝑚 + 1) ↑ 𝐹) = ((𝑚 ↑ 𝐹) ∙ 𝐹))
123	122	fveq2d 6832	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (((𝐼 mPSDer 𝑅)‘𝑋)‘((𝑚 + 1) ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘((𝑚 ↑ 𝐹) ∙ 𝐹)))
124	120	nncnd 12147	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → 𝑚 ∈ ℂ)
125		pncan1 11547	. . . . . . . 8 ⊢ (𝑚 ∈ ℂ → ((𝑚 + 1) − 1) = 𝑚)
126	124, 125	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → ((𝑚 + 1) − 1) = 𝑚)
127	126	oveq1d 7367	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (((𝑚 + 1) − 1) ↑ 𝐹) = (𝑚 ↑ 𝐹))
128	127	oveq2d 7368	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → ((𝑚 + 1) · (((𝑚 + 1) − 1) ↑ 𝐹)) = ((𝑚 + 1) · (𝑚 ↑ 𝐹)))
129	128	oveq1d 7367	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (((𝑚 + 1) · (((𝑚 + 1) − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = (((𝑚 + 1) · (𝑚 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
130	119, 123, 129	3eqtr4d 2776	. . 3 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → (((𝐼 mPSDer 𝑅)‘𝑋)‘((𝑚 + 1) ↑ 𝐹)) = (((𝑚 + 1) · (((𝑚 + 1) − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
131	10, 17, 24, 31, 65, 130	nnindd 12151	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑁 ↑ 𝐹)) = ((𝑁 · ((𝑁 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
132	1, 131	mpdan 687	1 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑁 ↑ 𝐹)) = ((𝑁 · ((𝑁 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 Vcvv 3436 ‘cfv 6487 (class class class)co 7352 ℂcc 11010 0cc0 11012 1c1 11013 + caddc 11015 − cmin 11350 ℕcn 12131 ℕ₀cn0 12387 ℤcz 12474 Basecbs 17126 +_gcplusg 17167 ._rcmulr 17168 Mgmcmgm 18552 Mndcmnd 18648 Grpcgrp 18852 ._gcmg 18986 mulGrpcmgp 20064 1_rcur 20105 Ringcrg 20157 CRingccrg 20158 mPwSer cmps 21847 mPSDer cpsd 22051

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-ofr 7617 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9252 df-sup 9332 df-oi 9402 df-card 9838 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-z 12475 df-dec 12595 df-uz 12739 df-fz 13414 df-fzo 13561 df-seq 13915 df-hash 14244 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-hom 17191 df-cco 17192 df-0g 17351 df-gsum 17352 df-prds 17357 df-pws 17359 df-mre 17494 df-mrc 17495 df-acs 17497 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-mhm 18697 df-submnd 18698 df-grp 18855 df-minusg 18856 df-mulg 18987 df-ghm 19131 df-cntz 19235 df-cmn 19700 df-abl 19701 df-mgp 20065 df-rng 20077 df-ur 20106 df-ring 20159 df-cring 20160 df-oppr 20261 df-psr 21852 df-psd 22077

This theorem is referenced by: (None)

W3C validator