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Theorem psdpw 22242

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 7419	. . . 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 7403	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
5		1m1e0 12300	. . . . . . . 8 ⊢ (1 − 1) = 0
6	4, 5	eqtrdi 2814	. . . . . . 7 ⊢ (𝑛 = 1 → (𝑛 − 1) = 0)
7	6	oveq1d 7411	. . . . . 6 ⊢ (𝑛 = 1 → ((𝑛 − 1) ↑ 𝐹) = (0 ↑ 𝐹))
8	3, 7	oveq12d 7414	. . . . 5 ⊢ (𝑛 = 1 → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = (1 · (0 ↑ 𝐹)))
9	8	oveq1d 7411	. . . 4 ⊢ (𝑛 = 1 → ((𝑛 · ((𝑛 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = ((1 · (0 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
10	2, 9	eqeq12d 2779	. . 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 7419	. . . 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 7403	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑛 − 1) = (𝑚 − 1))
14	13	oveq1d 7411	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑛 − 1) ↑ 𝐹) = ((𝑚 − 1) ↑ 𝐹))
15	12, 14	oveq12d 7414	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = (𝑚 · ((𝑚 − 1) ↑ 𝐹)))
16	15	oveq1d 7411	. . . 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 2779	. . 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 7419	. . . 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 7403	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 − 1) = ((𝑚 + 1) − 1))
21	20	oveq1d 7411	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛 − 1) ↑ 𝐹) = (((𝑚 + 1) − 1) ↑ 𝐹))
22	19, 21	oveq12d 7414	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = ((𝑚 + 1) · (((𝑚 + 1) − 1) ↑ 𝐹)))
23	22	oveq1d 7411	. . . 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 2779	. . 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 7419	. . . 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 7403	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
28	27	oveq1d 7411	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑛 − 1) ↑ 𝐹) = ((𝑁 − 1) ↑ 𝐹))
29	26, 28	oveq12d 7414	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = (𝑁 · ((𝑁 − 1) ↑ 𝐹)))
30	29	oveq1d 7411	. . . 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 2779	. . 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 2763	. . . . 5 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
35		psdpw.s	. . . . . . 7 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
36		psdpw.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
37		reldmpsr 21973	. . . . . . . . . 10 ⊢ Rel dom mPwSer
38	37, 35, 32	elbasov 17262	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
39	36, 38	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
40	39	simpld 498	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V)
41		psdpw.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ CRing)
42	35, 40, 41	psrcrng 22030	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
43	42	crngringd 20306	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
44	41	crnggrpd 20307	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp)
45	44	grpmgmd 19013	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Mgm)
46		psdpw.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
47	35, 32, 45, 46, 36	psdcl 22233	. . . . 5 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
48	32, 33, 34, 43, 47	ringlidmd 20332	. . . 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 20201	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
51	49, 34	ringidval 20243	. . . . . . . . 9 ⊢ (1_r‘𝑆) = (0_g‘𝑀)
52		psdpw.e	. . . . . . . . 9 ⊢ ↑ = (._g‘𝑀)
53	50, 51, 52	mulg0 19126	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → (0 ↑ 𝐹) = (1_r‘𝑆))
54	36, 53	syl 17	. . . . . . 7 ⊢ (𝜑 → (0 ↑ 𝐹) = (1_r‘𝑆))
55	54	oveq2d 7412	. . . . . 6 ⊢ (𝜑 → (1 · (0 ↑ 𝐹)) = (1 · (1_r‘𝑆)))
56	32, 34, 43	ringidcld 20326	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑆) ∈ 𝐵)
57		psdpw.g	. . . . . . . 8 ⊢ · = (._g‘𝑆)
58	32, 57	mulg1 19133	. . . . . . 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 2798	. . . . 5 ⊢ (𝜑 → (1 · (0 ↑ 𝐹)) = (1_r‘𝑆))
61	60	oveq1d 7411	. . . 4 ⊢ (𝜑 → ((1 · (0 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = ((1_r‘𝑆) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
62	50, 52	mulg1 19133	. . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (1 ↑ 𝐹) = 𝐹)
63	36, 62	syl 17	. . . . 5 ⊢ (𝜑 → (1 ↑ 𝐹) = 𝐹)
64	63	fveq2d 6871	. . . 4 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(1 ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))
65	48, 61, 64	3eqtr4rd 2809	. . 3 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(1 ↑ 𝐹)) = ((1 · (0 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
66		simpr 488	. . . . . . . 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 7411	. . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ CRing)
69	42	crnggrpd 20307	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ Grp)
70	69	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ Grp)
71		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
72	71	nnzd 12604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
73	43	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ Ring)
74	49	ringmgp 20299	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Ring → 𝑀 ∈ Mnd)
75	73, 74	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑀 ∈ Mnd)
76		nnm1nn0 12532	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ₀)
77	76	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 − 1) ∈ ℕ₀)
78	36	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 ∈ 𝐵)
79	50, 52, 75, 77, 78	mulgnn0cld 19147	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 − 1) ↑ 𝐹) ∈ 𝐵)
80	32, 57, 70, 72, 79	mulgcld 19148	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∈ 𝐵)
81	47	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
82	32, 33, 68, 80, 81, 78	crng32d 20319	. . . . . . . 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 484	. . . . . . 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 20369	. . . . . . . . . . 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 1393	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ 𝐹) = (𝑚 · (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹)))
86	49, 33	mgpplusg 20200	. . . . . . . . . . . . . 14 ⊢ ∙ = (+_g‘𝑀)
87	50, 52, 86	mulgnn0p1 19137	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ Mnd ∧ (𝑚 − 1) ∈ ℕ₀ ∧ 𝐹 ∈ 𝐵) → (((𝑚 − 1) + 1) ↑ 𝐹) = (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹))
88	75, 77, 78, 87	syl3anc 1392	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 − 1) + 1) ↑ 𝐹) = (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹))
89	71	nncnd 12236	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
90		npcan1 11623	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℂ → ((𝑚 − 1) + 1) = 𝑚)
91	89, 90	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 − 1) + 1) = 𝑚)
92	91	oveq1d 7411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 − 1) + 1) ↑ 𝐹) = (𝑚 ↑ 𝐹))
93	88, 92	eqtr3d 2800	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹) = (𝑚 ↑ 𝐹))
94	93	oveq2d 7412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 · (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹)) = (𝑚 · (𝑚 ↑ 𝐹)))
95	85, 94	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ 𝐹) = (𝑚 · (𝑚 ↑ 𝐹)))
96	95	oveq1d 7411	. . . . . . . 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 484	. . . . . . 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 2802	. . . . . 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 7411	. . . . 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 2763	. . . . . 6 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
101	41	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → 𝑅 ∈ CRing)
102	46	ad2antrr 736	. . . . . 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 18785	. . . . . . . . . 10 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mgm)
105	103, 104	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ Mgm)
106	105	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑀 ∈ Mgm)
107	50, 52	mulgnncl 19141	. . . . . . . 8 ⊢ ((𝑀 ∈ Mgm ∧ 𝑚 ∈ ℕ ∧ 𝐹 ∈ 𝐵) → (𝑚 ↑ 𝐹) ∈ 𝐵)
108	106, 71, 78, 107	syl3anc 1392	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 ↑ 𝐹) ∈ 𝐵)
109	108	adantr 484	. . . . . 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 736	. . . . . 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 22238	. . . . 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 19134	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 ↑ 𝐹) ∈ 𝐵) → ((𝑚 + 1) · (𝑚 ↑ 𝐹)) = ((𝑚 · (𝑚 ↑ 𝐹))(+_g‘𝑆)(𝑚 ↑ 𝐹)))
113	71, 108, 112	syl2anc 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) · (𝑚 ↑ 𝐹)) = ((𝑚 · (𝑚 ↑ 𝐹))(+_g‘𝑆)(𝑚 ↑ 𝐹)))
114	113	oveq1d 7411	. . . . . . 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 19148	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 · (𝑚 ↑ 𝐹)) ∈ 𝐵)
116	32, 100, 33, 73, 115, 108, 81	ringdird 20324	. . . . . . 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 2798	. . . . . 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 484	. . . . 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 2808	. . . 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 778	. . . . . 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 19134	. . . . . 6 ⊢ ((𝑚 ∈ ℕ ∧ 𝐹 ∈ 𝐵) → ((𝑚 + 1) ↑ 𝐹) = ((𝑚 ↑ 𝐹) ∙ 𝐹))
122	120, 110, 121	syl2anc 593	. . . . 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 6871	. . . 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 12236	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → 𝑚 ∈ ℂ)
125		pncan1 11622	. . . . . . . 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 7411	. . . . . 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 7412	. . . . 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 7411	. . . 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 2808	. . 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 12240	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑁 ↑ 𝐹)) = ((𝑁 · ((𝑁 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
132	1, 131	mpdan 697	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 399 = wceq 1561 ∈ wcel 2143 Vcvv 3455 ‘cfv 6521 (class class class)co 7396 ℂcc 11082 0cc0 11084 1c1 11085 + caddc 11087 − cmin 11425 ℕcn 12220 ℕ₀cn0 12491 ℤcz 12578 Basecbs 17255 +_gcplusg 17296 ._rcmulr 17297 Mgmcmgm 18682 Mndcmnd 18778 Grpcgrp 18985 ._gcmg 19119 mulGrpcmgp 20196 1_rcur 20241 Ringcrg 20293 CRingccrg 20294 mPwSer cmps 21963 mPSDer cpsd 22206

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-ifp 1075 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 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-tpos 8206 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 9306 df-sup 9386 df-oi 9456 df-card 9909 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-nn 12221 df-2 12290 df-3 12291 df-4 12292 df-5 12293 df-6 12294 df-7 12295 df-8 12296 df-9 12297 df-n0 12492 df-z 12579 df-dec 12699 df-uz 12850 df-fz 13523 df-fzo 13670 df-seq 14025 df-hash 14354 df-struct 17193 df-sets 17210 df-slot 17228 df-ndx 17240 df-base 17256 df-ress 17277 df-plusg 17309 df-mulr 17310 df-sca 17312 df-vsca 17313 df-ip 17314 df-tset 17315 df-ple 17316 df-ds 17318 df-hom 17320 df-cco 17321 df-0g 17480 df-gsum 17481 df-prds 17486 df-pws 17488 df-mre 17624 df-mrc 17625 df-acs 17627 df-mgm 18684 df-sgrp 18763 df-mnd 18779 df-mhm 18827 df-submnd 18828 df-grp 18988 df-minusg 18989 df-mulg 19120 df-ghm 19264 df-cntz 19367 df-cmn 19832 df-abl 19833 df-mgp 20197 df-rng 20209 df-ur 20242 df-ring 20295 df-cring 20296 df-oppr 20396 df-psr 21968 df-psd 22228

This theorem is referenced by: (None)

W3C validator