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Theorem psdpw 22344

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 7435	. . . 4 ⊢ (𝑛 = 1 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘(1 ↑ 𝐹)))
3		id 23	. . . . . 6 ⊢ (𝑛 = 1 → 𝑛 = 1)
4		oveq1 7419	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
5		1m1e0 12319	. . . . . . . 8 ⊢ (1 − 1) = 0
6	4, 5	eqtrdi 2813	. . . . . . 7 ⊢ (𝑛 = 1 → (𝑛 − 1) = 0)
7	6	oveq1d 7427	. . . . . 6 ⊢ (𝑛 = 1 → ((𝑛 − 1) ↑ 𝐹) = (0 ↑ 𝐹))
8	3, 7	oveq12d 7430	. . . . 5 ⊢ (𝑛 = 1 → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = (1 · (0 ↑ 𝐹)))
9	8	oveq1d 7427	. . . 4 ⊢ (𝑛 = 1 → ((𝑛 · ((𝑛 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = ((1 · (0 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
10	2, 9	eqeq12d 2778	. . 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 7435	. . . 4 ⊢ (𝑛 = 𝑚 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)))
12		id 23	. . . . . 6 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚)
13		oveq1 7419	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑛 − 1) = (𝑚 − 1))
14	13	oveq1d 7427	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑛 − 1) ↑ 𝐹) = ((𝑚 − 1) ↑ 𝐹))
15	12, 14	oveq12d 7430	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = (𝑚 · ((𝑚 − 1) ↑ 𝐹)))
16	15	oveq1d 7427	. . . 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 2778	. . 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 7435	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘((𝑚 + 1) ↑ 𝐹)))
19		id 23	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → 𝑛 = (𝑚 + 1))
20		oveq1 7419	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 − 1) = ((𝑚 + 1) − 1))
21	20	oveq1d 7427	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛 − 1) ↑ 𝐹) = (((𝑚 + 1) − 1) ↑ 𝐹))
22	19, 21	oveq12d 7430	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = ((𝑚 + 1) · (((𝑚 + 1) − 1) ↑ 𝐹)))
23	22	oveq1d 7427	. . . 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 2778	. . 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 7435	. . . 4 ⊢ (𝑛 = 𝑁 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑛 ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑁 ↑ 𝐹)))
26		id 23	. . . . . 6 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
27		oveq1 7419	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
28	27	oveq1d 7427	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑛 − 1) ↑ 𝐹) = ((𝑁 − 1) ↑ 𝐹))
29	26, 28	oveq12d 7430	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 · ((𝑛 − 1) ↑ 𝐹)) = (𝑁 · ((𝑁 − 1) ↑ 𝐹)))
30	29	oveq1d 7427	. . . 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 2778	. . 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 2762	. . . . 5 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
35		psdpw.s	. . . . . . 7 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
36		psdpw.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
37		reldmpsr 22075	. . . . . . . . . 10 ⊢ Rel dom mPwSer
38	37, 35, 32	elbasov 17282	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
39	36, 38	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
40	39	simpld 499	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V)
41		psdpw.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ CRing)
42	35, 40, 41	psrcrng 22132	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
43	42	crngringd 20334	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
44	41	crnggrpd 20335	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp)
45	44	grpmgmd 19034	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Mgm)
46		psdpw.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
47	35, 32, 45, 46, 36	psdcl 22335	. . . . 5 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
48	32, 33, 34, 43, 47	ringlidmd 20362	. . . 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 20227	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
51	49, 34	ringidval 20271	. . . . . . . . 9 ⊢ (1_r‘𝑆) = (0_g‘𝑀)
52		psdpw.e	. . . . . . . . 9 ⊢ ↑ = (._g‘𝑀)
53	50, 51, 52	mulg0 19146	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → (0 ↑ 𝐹) = (1_r‘𝑆))
54	36, 53	syl 18	. . . . . . 7 ⊢ (𝜑 → (0 ↑ 𝐹) = (1_r‘𝑆))
55	54	oveq2d 7428	. . . . . 6 ⊢ (𝜑 → (1 · (0 ↑ 𝐹)) = (1 · (1_r‘𝑆)))
56	32, 34, 43	ringidcld 20356	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑆) ∈ 𝐵)
57		psdpw.g	. . . . . . . 8 ⊢ · = (._g‘𝑆)
58	32, 57	mulg1 19153	. . . . . . 7 ⊢ ((1_r‘𝑆) ∈ 𝐵 → (1 · (1_r‘𝑆)) = (1_r‘𝑆))
59	56, 58	syl 18	. . . . . 6 ⊢ (𝜑 → (1 · (1_r‘𝑆)) = (1_r‘𝑆))
60	55, 59	eqtrd 2797	. . . . 5 ⊢ (𝜑 → (1 · (0 ↑ 𝐹)) = (1_r‘𝑆))
61	60	oveq1d 7427	. . . 4 ⊢ (𝜑 → ((1 · (0 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)) = ((1_r‘𝑆) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
62	50, 52	mulg1 19153	. . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (1 ↑ 𝐹) = 𝐹)
63	36, 62	syl 18	. . . . 5 ⊢ (𝜑 → (1 ↑ 𝐹) = 𝐹)
64	63	fveq2d 6885	. . . 4 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(1 ↑ 𝐹)) = (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))
65	48, 61, 64	3eqtr4rd 2808	. . 3 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(1 ↑ 𝐹)) = ((1 · (0 ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
66		simpr 489	. . . . . . . 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 7427	. . . . . . 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 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ CRing)
69	42	crnggrpd 20335	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ Grp)
70	69	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ Grp)
71		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
72	71	nnzd 12623	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
73	43	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ Ring)
74	49	ringmgp 20327	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Ring → 𝑀 ∈ Mnd)
75	73, 74	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑀 ∈ Mnd)
76		nnm1nn0 12551	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ₀)
77	76	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 − 1) ∈ ℕ₀)
78	36	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹 ∈ 𝐵)
79	50, 52, 75, 77, 78	mulgnn0cld 19167	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 − 1) ↑ 𝐹) ∈ 𝐵)
80	32, 57, 70, 72, 79	mulgcld 19168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∈ 𝐵)
81	47	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
82	32, 33, 68, 80, 81, 78	crng32d 20348	. . . . . . . 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 485	. . . . . . 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 20399	. . . . . . . . . . 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 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ 𝐹) = (𝑚 · (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹)))
86	49, 33	mgpplusg 20226	. . . . . . . . . . . . . 14 ⊢ ∙ = (+_g‘𝑀)
87	50, 52, 86	mulgnn0p1 19157	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ Mnd ∧ (𝑚 − 1) ∈ ℕ₀ ∧ 𝐹 ∈ 𝐵) → (((𝑚 − 1) + 1) ↑ 𝐹) = (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹))
88	75, 77, 78, 87	syl3anc 1397	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 − 1) + 1) ↑ 𝐹) = (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹))
89	71	nncnd 12255	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
90		npcan1 11645	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℂ → ((𝑚 − 1) + 1) = 𝑚)
91	89, 90	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 − 1) + 1) = 𝑚)
92	91	oveq1d 7427	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 − 1) + 1) ↑ 𝐹) = (𝑚 ↑ 𝐹))
93	88, 92	eqtr3d 2799	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹) = (𝑚 ↑ 𝐹))
94	93	oveq2d 7428	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 · (((𝑚 − 1) ↑ 𝐹) ∙ 𝐹)) = (𝑚 · (𝑚 ↑ 𝐹)))
95	85, 94	eqtrd 2797	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ 𝐹) = (𝑚 · (𝑚 ↑ 𝐹)))
96	95	oveq1d 7427	. . . . . . . 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 485	. . . . . . 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 2801	. . . . . 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 7427	. . . . 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 2762	. . . . . 6 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
101	41	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → 𝑅 ∈ CRing)
102	46	ad2antrr 738	. . . . . 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 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ Mnd)
104		mndmgm 18805	. . . . . . . . . 10 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mgm)
105	103, 104	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ Mgm)
106	105	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑀 ∈ Mgm)
107	50, 52	mulgnncl 19161	. . . . . . . 8 ⊢ ((𝑀 ∈ Mgm ∧ 𝑚 ∈ ℕ ∧ 𝐹 ∈ 𝐵) → (𝑚 ↑ 𝐹) ∈ 𝐵)
108	106, 71, 78, 107	syl3anc 1397	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 ↑ 𝐹) ∈ 𝐵)
109	108	adantr 485	. . . . . 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 738	. . . . . 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 22340	. . . . 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 19154	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 ↑ 𝐹) ∈ 𝐵) → ((𝑚 + 1) · (𝑚 ↑ 𝐹)) = ((𝑚 · (𝑚 ↑ 𝐹))(+_g‘𝑆)(𝑚 ↑ 𝐹)))
113	71, 108, 112	syl2anc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) · (𝑚 ↑ 𝐹)) = ((𝑚 · (𝑚 ↑ 𝐹))(+_g‘𝑆)(𝑚 ↑ 𝐹)))
114	113	oveq1d 7427	. . . . . . 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 19168	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 · (𝑚 ↑ 𝐹)) ∈ 𝐵)
116	32, 100, 33, 73, 115, 108, 81	ringdird 20353	. . . . . . 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 2797	. . . . . 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 485	. . . . 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 2807	. . . 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 780	. . . . . 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 19154	. . . . . 6 ⊢ ((𝑚 ∈ ℕ ∧ 𝐹 ∈ 𝐵) → ((𝑚 + 1) ↑ 𝐹) = ((𝑚 ↑ 𝐹) ∙ 𝐹))
122	120, 110, 121	syl2anc 595	. . . . 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 6885	. . . 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 12255	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑚 ↑ 𝐹)) = ((𝑚 · ((𝑚 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹))) → 𝑚 ∈ ℂ)
125		pncan1 11644	. . . . . . . 8 ⊢ (𝑚 ∈ ℂ → ((𝑚 + 1) − 1) = 𝑚)
126	124, 125	syl 18	. . . . . . 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 7427	. . . . . 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 7428	. . . . 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 7427	. . . 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 2807	. . 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 12259	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑁 ↑ 𝐹)) = ((𝑁 · ((𝑁 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))
132	1, 131	mpdan 699	1 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑁 ↑ 𝐹)) = ((𝑁 · ((𝑁 − 1) ↑ 𝐹)) ∙ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 = wceq 1569 ∈ wcel 2142 Vcvv 3454 ‘cfv 6536 (class class class)co 7412 ℂcc 11104 0cc0 11106 1c1 11107 + caddc 11109 − cmin 11447 ℕcn 12239 ℕ₀cn0 12510 ℤcz 12597 Basecbs 17275 +_gcplusg 17316 ._rcmulr 17317 Mgmcmgm 18702 Mndcmnd 18798 Grpcgrp 19006 ._gcmg 19139 mulGrpcmgp 20222 1_rcur 20269 Ringcrg 20321 CRingccrg 20322 mPwSer cmps 22065 mPSDer cpsd 22308

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1078 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-ofr 7677 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-sup 9400 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-fz 13542 df-fzo 13690 df-seq 14045 df-hash 14374 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-hom 17340 df-cco 17341 df-0g 17500 df-gsum 17501 df-prds 17506 df-pws 17508 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-submnd 18848 df-grp 19009 df-minusg 19010 df-mulg 19140 df-ghm 19290 df-cntz 19393 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-cring 20324 df-oppr 20426 df-psr 22070 df-psd 22330

This theorem is used by: (None)

W3C validator