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Theorem ablfac1eu 20039

Description: The factorization of ablfac1b 20036 is unique, in that any other factorization into prime power factors (even if the exponents are different) must be equal to 𝑆. (Contributed by Mario Carneiro, 21-Apr-2016.)

**Hypotheses**
Ref	Expression
ablfac1.b	⊢ 𝐵 = (Base‘𝐺)
ablfac1.o	⊢ 𝑂 = (od‘𝐺)
ablfac1.s	⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
ablfac1.g	⊢ (𝜑 → 𝐺 ∈ Abel)
ablfac1.f	⊢ (𝜑 → 𝐵 ∈ Fin)
ablfac1.1	⊢ (𝜑 → 𝐴 ⊆ ℙ)
ablfac1c.d	⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
ablfac1.2	⊢ (𝜑 → 𝐷 ⊆ 𝐴)
ablfac1eu.1	⊢ (𝜑 → (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝐵))
ablfac1eu.2	⊢ (𝜑 → dom 𝑇 = 𝐴)
ablfac1eu.3	⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℕ₀)
ablfac1eu.4	⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))

**Assertion**
Ref	Expression
ablfac1eu	⊢ (𝜑 → 𝑇 = 𝑆)

Distinct variable groups: 𝑞,𝑝,𝑤,𝑥,𝐵 𝐷,𝑝,𝑞,𝑥 𝜑,𝑝,𝑞,𝑤,𝑥 𝑆,𝑞 𝐴,𝑝,𝑞,𝑥 𝑂,𝑝,𝑞,𝑥 𝑇,𝑞,𝑥 𝐺,𝑝,𝑞,𝑥 Allowed substitution hints: 𝐴(𝑤) 𝐶(𝑥,𝑤,𝑞,𝑝) 𝐷(𝑤) 𝑆(𝑥,𝑤,𝑝) 𝑇(𝑤,𝑝) 𝐺(𝑤) 𝑂(𝑤)

Proof of Theorem ablfac1eu Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablfac1eu.1	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝐵))
2	1	simpld 494	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
3		ablfac1eu.2	. . . 4 ⊢ (𝜑 → dom 𝑇 = 𝐴)
4	2, 3	dprdf2 19973	. . 3 ⊢ (𝜑 → 𝑇:𝐴⟶(SubGrp‘𝐺))
5	4	ffnd 6658	. 2 ⊢ (𝜑 → 𝑇 Fn 𝐴)
6		ablfac1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
7		ablfac1.o	. . . . 5 ⊢ 𝑂 = (od‘𝐺)
8		ablfac1.s	. . . . 5 ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
9		ablfac1.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Abel)
10		ablfac1.f	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ Fin)
11		ablfac1.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
12	6, 7, 8, 9, 10, 11	ablfac1b 20036	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
13	6	fvexi 6843	. . . . . . 7 ⊢ 𝐵 ∈ V
14	13	rabex 5269	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
15	14, 8	dmmpti 6631	. . . . 5 ⊢ dom 𝑆 = 𝐴
16	15	a1i 11	. . . 4 ⊢ (𝜑 → dom 𝑆 = 𝐴)
17	12, 16	dprdf2 19973	. . 3 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
18	17	ffnd 6658	. 2 ⊢ (𝜑 → 𝑆 Fn 𝐴)
19	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐵 ∈ Fin)
20	17	ffvelcdmda 7025	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
21	6	subgss 19092	. . . . 5 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑆‘𝑞) ⊆ 𝐵)
22	20, 21	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ 𝐵)
23	19, 22	ssfid 9168	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ Fin)
24	4	ffvelcdmda 7025	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ∈ (SubGrp‘𝐺))
25	6	subgss 19092	. . . . . 6 ⊢ ((𝑇‘𝑞) ∈ (SubGrp‘𝐺) → (𝑇‘𝑞) ⊆ 𝐵)
26	24, 25	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ 𝐵)
27	26	sselda 3917	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → 𝑥 ∈ 𝐵)
28	6, 7	odcl 19500	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝑂‘𝑥) ∈ ℕ₀)
29	27, 28	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∈ ℕ₀)
30	29	nn0zd 12538	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∈ ℤ)
31	19, 26	ssfid 9168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ∈ Fin)
32		hashcl 14307	. . . . . . . . 9 ⊢ ((𝑇‘𝑞) ∈ Fin → (♯‘(𝑇‘𝑞)) ∈ ℕ₀)
33	31, 32	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∈ ℕ₀)
34	33	nn0zd 12538	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∈ ℤ)
35	34	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (♯‘(𝑇‘𝑞)) ∈ ℤ)
36	11	sselda 3917	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
37		prmnn 16632	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
38	36, 37	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℕ)
39		ablgrp 19749	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
40	9, 39	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ Grp)
41	6	grpbn0 18931	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
42	40, 41	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ ∅)
43		hashnncl 14317	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
44	10, 43	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
45	42, 44	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
46	45	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ)
47	36, 46	pccld 16810	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
48	38, 47	nnexpcld 14196	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℕ)
49	48	nnzd 12539	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℤ)
50	49	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℤ)
51	24	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑇‘𝑞) ∈ (SubGrp‘𝐺))
52	31	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑇‘𝑞) ∈ Fin)
53		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → 𝑥 ∈ (𝑇‘𝑞))
54	7	odsubdvds 19535	. . . . . . 7 ⊢ (((𝑇‘𝑞) ∈ (SubGrp‘𝐺) ∧ (𝑇‘𝑞) ∈ Fin ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∥ (♯‘(𝑇‘𝑞)))
55	51, 52, 53, 54	syl3anc 1374	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∥ (♯‘(𝑇‘𝑞)))
56		ablfac1eu.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))
57		prmz 16633	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
58	36, 57	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℤ)
59		ablfac1eu.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℕ₀)
60	59	nn0zd 12538	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℤ)
61	47	nn0zd 12538	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℤ)
62	6	lagsubg 19159	. . . . . . . . . . . . 13 ⊢ (((𝑇‘𝑞) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (♯‘(𝑇‘𝑞)) ∥ (♯‘𝐵))
63	24, 19, 62	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∥ (♯‘𝐵))
64	56, 63	eqbrtrrd 5098	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑𝐶) ∥ (♯‘𝐵))
65	46	nnzd 12539	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) ∈ ℤ)
66		pcdvdsb 16829	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℙ ∧ (♯‘𝐵) ∈ ℤ ∧ 𝐶 ∈ ℕ₀) → (𝐶 ≤ (𝑞 pCnt (♯‘𝐵)) ↔ (𝑞↑𝐶) ∥ (♯‘𝐵)))
67	36, 65, 59, 66	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐶 ≤ (𝑞 pCnt (♯‘𝐵)) ↔ (𝑞↑𝐶) ∥ (♯‘𝐵)))
68	64, 67	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ≤ (𝑞 pCnt (♯‘𝐵)))
69		eluz2 12783	. . . . . . . . . 10 ⊢ ((𝑞 pCnt (♯‘𝐵)) ∈ (ℤ_≥‘𝐶) ↔ (𝐶 ∈ ℤ ∧ (𝑞 pCnt (♯‘𝐵)) ∈ ℤ ∧ 𝐶 ≤ (𝑞 pCnt (♯‘𝐵))))
70	60, 61, 68, 69	syl3anbrc 1345	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ (ℤ_≥‘𝐶))
71		dvdsexp 16286	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℤ ∧ 𝐶 ∈ ℕ₀ ∧ (𝑞 pCnt (♯‘𝐵)) ∈ (ℤ_≥‘𝐶)) → (𝑞↑𝐶) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
72	58, 59, 70, 71	syl3anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑𝐶) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
73	56, 72	eqbrtrd 5096	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
74	73	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (♯‘(𝑇‘𝑞)) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
75	30, 35, 50, 55, 74	dvdstrd 16253	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
76	26, 75	ssrabdv 4006	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
77		id 22	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → 𝑝 = 𝑞)
78		oveq1 7363	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → (𝑝 pCnt (♯‘𝐵)) = (𝑞 pCnt (♯‘𝐵)))
79	77, 78	oveq12d 7374	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
80	79	breq2d 5086	. . . . . . 7 ⊢ (𝑝 = 𝑞 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))))
81	80	rabbidv 3394	. . . . . 6 ⊢ (𝑝 = 𝑞 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
82	81, 8, 14	fvmpt3i 6942	. . . . 5 ⊢ (𝑞 ∈ 𝐴 → (𝑆‘𝑞) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
83	82	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
84	76, 83	sseqtrrd 3954	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ (𝑆‘𝑞))
85	48	nnnn0d 12487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℕ₀)
86		pcdvds 16824	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
87	36, 46, 86	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
88	2	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd 𝑇)
89	3	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom 𝑇 = 𝐴)
90		ablfac1.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
91	90	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
92	88, 89, 91	dprdres 19994	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑇)))
93	92	simpld 494	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd (𝑇 ↾ 𝐷))
94	4	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇:𝐴⟶(SubGrp‘𝐺))
95	94, 91	fssresd 6696	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
96	95	fdmd 6667	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom (𝑇 ↾ 𝐷) = 𝐷)
97		difssd 4069	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐷 ∖ {𝑞}) ⊆ 𝐷)
98	93, 96, 97	dprdres 19994	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})) ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷))))
99	98	simpld 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))
100		dprdsubg 19990	. . . . . . . . . . 11 ⊢ (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺))
101	99, 100	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺))
102	6	lagsubg 19159	. . . . . . . . . 10 ⊢ (((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∥ (♯‘𝐵))
103	101, 19, 102	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∥ (♯‘𝐵))
104		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
105	104	subg0cl 19099	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))
106	101, 105	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (0_g‘𝐺) ∈ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))
107	106	ne0d 4272	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ≠ ∅)
108	6	dprdssv 19982	. . . . . . . . . . . . . 14 ⊢ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ⊆ 𝐵
109		ssfi 9096	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ⊆ 𝐵) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ Fin)
110	19, 108, 109	sylancl 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ Fin)
111		hashnncl 14317	. . . . . . . . . . . . 13 ⊢ ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ Fin → ((♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℕ ↔ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ≠ ∅))
112	110, 111	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℕ ↔ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ≠ ∅))
113	107, 112	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℕ)
114	113	nnzd 12539	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℤ)
115		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑞 → 𝑥 = 𝑞)
116		sneq 4567	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑞 → {𝑥} = {𝑞})
117	116	difeq2d 4059	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑞 → (𝐷 ∖ {𝑥}) = (𝐷 ∖ {𝑞}))
118	117	reseq2d 5933	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑞 → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})) = ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))
119	118	oveq2d 7372	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑞 → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥}))) = (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))
120	119	fveq2d 6833	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑞 → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))) = (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
121	115, 120	breq12d 5087	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))) ↔ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
122	121	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑞 → (¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))) ↔ ¬ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
123		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ (𝑂‘𝑦) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) = (𝑝 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ (𝑂‘𝑦) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
124	9	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐺 ∈ Abel)
125	10	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐵 ∈ Fin)
126		ablfac1c.d	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
127	126	ssrab3 4015	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 ⊆ ℙ
128	127	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐷 ⊆ ℙ)
129		ssidd 3940	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐷 ⊆ 𝐷)
130	2, 3, 90	dprdres 19994	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑇)))
131	130	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ 𝐷))
132		dprdsubg 19990	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺dom DProd (𝑇 ↾ 𝐷) → (𝐺 DProd (𝑇 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
133	131, 132	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
134		difssd 4069	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ 𝐴)
135	2, 3, 134	dprdres 19994	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐺dom DProd (𝑇 ↾ (𝐴 ∖ 𝐷)) ∧ (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ⊆ (𝐺 DProd 𝑇)))
136	135	simpld 494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))
137		dprdsubg 19990	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺dom DProd (𝑇 ↾ (𝐴 ∖ 𝐷)) → (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ∈ (SubGrp‘𝐺))
138	136, 137	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ∈ (SubGrp‘𝐺))
139		difss 4068	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∖ 𝐷) ⊆ 𝐴
140		fssres 6695	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑇:𝐴⟶(SubGrp‘𝐺) ∧ (𝐴 ∖ 𝐷) ⊆ 𝐴) → (𝑇 ↾ (𝐴 ∖ 𝐷)):(𝐴 ∖ 𝐷)⟶(SubGrp‘𝐺))
141	4, 139, 140	sylancl 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑇 ↾ (𝐴 ∖ 𝐷)):(𝐴 ∖ 𝐷)⟶(SubGrp‘𝐺))
142	141	fdmd 6667	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom (𝑇 ↾ (𝐴 ∖ 𝐷)) = (𝐴 ∖ 𝐷))
143		fvres 6848	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑞 ∈ (𝐴 ∖ 𝐷) → ((𝑇 ↾ (𝐴 ∖ 𝐷))‘𝑞) = (𝑇‘𝑞))
144	143	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑞 ∈ (𝐴 ∖ 𝐷)) → ((𝑇 ↾ (𝐴 ∖ 𝐷))‘𝑞) = (𝑇‘𝑞))
145		eldif 3895	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑞 ∈ (𝐴 ∖ 𝐷) ↔ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷))
146	31	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑇‘𝑞) ∈ Fin)
147	104	subg0cl 19099	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑇‘𝑞) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝑇‘𝑞))
148	24, 147	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (0_g‘𝐺) ∈ (𝑇‘𝑞))
149	148	snssd 4720	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {(0_g‘𝐺)} ⊆ (𝑇‘𝑞))
150	149	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} ⊆ (𝑇‘𝑞))
151		fvex 6842	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0_g‘𝐺) ∈ V
152		hashsng 14320	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0_g‘𝐺) ∈ V → (♯‘{(0_g‘𝐺)}) = 1)
153	151, 152	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (♯‘{(0_g‘𝐺)}) = 1
154	56	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))
155	36	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∈ ℙ)
156		iddvdsexp 16237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑞 ∈ ℤ ∧ 𝐶 ∈ ℕ) → 𝑞 ∥ (𝑞↑𝐶))
157	58, 156	sylan 581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∥ (𝑞↑𝐶))
158	64	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → (𝑞↑𝐶) ∥ (♯‘𝐵))
159	56, 34	eqeltrrd 2836	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑𝐶) ∈ ℤ)
160		dvdstr 16252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑞 ∈ ℤ ∧ (𝑞↑𝐶) ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → ((𝑞 ∥ (𝑞↑𝐶) ∧ (𝑞↑𝐶) ∥ (♯‘𝐵)) → 𝑞 ∥ (♯‘𝐵)))
161	58, 159, 65, 160	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞 ∥ (𝑞↑𝐶) ∧ (𝑞↑𝐶) ∥ (♯‘𝐵)) → 𝑞 ∥ (♯‘𝐵)))
162	161	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → ((𝑞 ∥ (𝑞↑𝐶) ∧ (𝑞↑𝐶) ∥ (♯‘𝐵)) → 𝑞 ∥ (♯‘𝐵)))
163	157, 158, 162	mp2and 700	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∥ (♯‘𝐵))
164		breq1 5077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑤 = 𝑞 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑞 ∥ (♯‘𝐵)))
165	164, 126	elrab2 3634	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑞 ∈ 𝐷 ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ (♯‘𝐵)))
166	155, 163, 165	sylanbrc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∈ 𝐷)
167	166	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐶 ∈ ℕ → 𝑞 ∈ 𝐷))
168	167	con3d 152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞 ∈ 𝐷 → ¬ 𝐶 ∈ ℕ))
169	168	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ¬ 𝐶 ∈ ℕ)
170	59	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝐶 ∈ ℕ₀)
171		elnn0 12428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐶 ∈ ℕ₀ ↔ (𝐶 ∈ ℕ ∨ 𝐶 = 0))
172	170, 171	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐶 ∈ ℕ ∨ 𝐶 = 0))
173	172	ord 865	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (¬ 𝐶 ∈ ℕ → 𝐶 = 0))
174	169, 173	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝐶 = 0)
175	174	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑞↑𝐶) = (𝑞↑0))
176	38	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝑞 ∈ ℕ)
177	176	nncnd 12179	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝑞 ∈ ℂ)
178	177	exp0d 14091	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑞↑0) = 1)
179	154, 175, 178	3eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (♯‘(𝑇‘𝑞)) = 1)
180	153, 179	eqtr4id 2789	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (♯‘{(0_g‘𝐺)}) = (♯‘(𝑇‘𝑞)))
181		snfi 8979	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ {(0_g‘𝐺)} ∈ Fin
182		hashen 14298	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (({(0_g‘𝐺)} ∈ Fin ∧ (𝑇‘𝑞) ∈ Fin) → ((♯‘{(0_g‘𝐺)}) = (♯‘(𝑇‘𝑞)) ↔ {(0_g‘𝐺)} ≈ (𝑇‘𝑞)))
183	181, 146, 182	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ((♯‘{(0_g‘𝐺)}) = (♯‘(𝑇‘𝑞)) ↔ {(0_g‘𝐺)} ≈ (𝑇‘𝑞)))
184	180, 183	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} ≈ (𝑇‘𝑞))
185		fisseneq 9162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑇‘𝑞) ∈ Fin ∧ {(0_g‘𝐺)} ⊆ (𝑇‘𝑞) ∧ {(0_g‘𝐺)} ≈ (𝑇‘𝑞)) → {(0_g‘𝐺)} = (𝑇‘𝑞))
186	146, 150, 184, 185	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} = (𝑇‘𝑞))
187	104	subg0cl 19099	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐺 DProd (𝑇 ↾ 𝐷)) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd (𝑇 ↾ 𝐷)))
188	133, 187	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (0_g‘𝐺) ∈ (𝐺 DProd (𝑇 ↾ 𝐷)))
189	188	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (0_g‘𝐺) ∈ (𝐺 DProd (𝑇 ↾ 𝐷)))
190	189	snssd 4720	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
191	186, 190	eqsstrrd 3952	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑇‘𝑞) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
192	145, 191	sylan2b 595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑞 ∈ (𝐴 ∖ 𝐷)) → (𝑇‘𝑞) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
193	144, 192	eqsstrd 3951	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑞 ∈ (𝐴 ∖ 𝐷)) → ((𝑇 ↾ (𝐴 ∖ 𝐷))‘𝑞) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
194	136, 142, 133, 193	dprdlub 19992	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
195		eqid 2735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
196	195	lsmss2 19631	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 DProd (𝑇 ↾ 𝐷)) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷))) → ((𝐺 DProd (𝑇 ↾ 𝐷))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))) = (𝐺 DProd (𝑇 ↾ 𝐷)))
197	133, 138, 194, 196	syl3anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ 𝐷))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))) = (𝐺 DProd (𝑇 ↾ 𝐷)))
198		disjdif 4402	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∩ (𝐴 ∖ 𝐷)) = ∅
199	198	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐷 ∩ (𝐴 ∖ 𝐷)) = ∅)
200		undif2 4407	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∪ (𝐴 ∖ 𝐷)) = (𝐷 ∪ 𝐴)
201		ssequn1 4117	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐷 ⊆ 𝐴 ↔ (𝐷 ∪ 𝐴) = 𝐴)
202	90, 201	sylib 218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐷 ∪ 𝐴) = 𝐴)
203	200, 202	eqtr2id 2783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 = (𝐷 ∪ (𝐴 ∖ 𝐷)))
204	4, 199, 203, 195, 2	dprdsplit 20014	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐺 DProd (𝑇 ↾ 𝐷))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))))
205	1	simprd 495	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd 𝑇) = 𝐵)
206	204, 205	eqtr3d 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ 𝐷))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))) = 𝐵)
207	197, 206	eqtr3d 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵)
208	131, 207	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵))
209	208	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵))
210	4, 90	fssresd 6696	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑇 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
211	210	fdmd 6667	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom (𝑇 ↾ 𝐷) = 𝐷)
212	211	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → dom (𝑇 ↾ 𝐷) = 𝐷)
213	90	sselda 3917	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → 𝑞 ∈ 𝐴)
214	213, 59	syldan 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → 𝐶 ∈ ℕ₀)
215	214	adantlr 716	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℙ) ∧ 𝑞 ∈ 𝐷) → 𝐶 ∈ ℕ₀)
216		fvres 6848	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 ∈ 𝐷 → ((𝑇 ↾ 𝐷)‘𝑞) = (𝑇‘𝑞))
217	216	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → ((𝑇 ↾ 𝐷)‘𝑞) = (𝑇‘𝑞))
218	217	fveq2d 6833	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → (♯‘((𝑇 ↾ 𝐷)‘𝑞)) = (♯‘(𝑇‘𝑞)))
219	213, 56	syldan 592	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))
220	218, 219	eqtrd 2770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → (♯‘((𝑇 ↾ 𝐷)‘𝑞)) = (𝑞↑𝐶))
221	220	adantlr 716	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℙ) ∧ 𝑞 ∈ 𝐷) → (♯‘((𝑇 ↾ 𝐷)‘𝑞)) = (𝑞↑𝐶))
222		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝑥 ∈ ℙ)
223		fzfid 13924	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...(♯‘𝐵)) ∈ Fin)
224		prmnn 16632	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℕ)
225	224	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ ℕ)
226		prmz 16633	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℤ)
227		dvdsle 16268	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℤ ∧ (♯‘𝐵) ∈ ℕ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
228	226, 45, 227	syl2anr 598	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
229	228	3impia 1118	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ≤ (♯‘𝐵))
230	45	nnzd 12539	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
231	230	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (♯‘𝐵) ∈ ℤ)
232		fznn 13535	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐵) ∈ ℤ → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
233	231, 232	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
234	225, 229, 233	mpbir2and 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ (1...(♯‘𝐵)))
235	234	rabssdv 4007	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ⊆ (1...(♯‘𝐵)))
236	126, 235	eqsstrid 3955	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 ⊆ (1...(♯‘𝐵)))
237	223, 236	ssfid 9168	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐷 ∈ Fin)
238	237	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐷 ∈ Fin)
239	6, 7, 123, 124, 125, 128, 126, 129, 209, 212, 215, 221, 222, 238	ablfac1eulem 20038	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → ¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))))
240	239	ralrimiva 3127	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ ℙ ¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))))
241	240	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ∀𝑥 ∈ ℙ ¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))))
242	122, 241, 36	rspcdva 3563	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ¬ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
243		coprm 16670	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℙ ∧ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℤ) → (¬ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ↔ (𝑞 gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1))
244	36, 114, 243	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ↔ (𝑞 gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1))
245	242, 244	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1)
246		rpexp1i 16682	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℤ ∧ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℤ ∧ (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀) → ((𝑞 gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1 → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1))
247	58, 114, 47, 246	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞 gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1 → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1))
248	245, 247	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1)
249		coprmdvds2 16612	. . . . . . . . . 10 ⊢ ((((𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℤ ∧ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) ∧ ((𝑞↑(𝑞 pCnt (♯‘𝐵))) gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1) → (((𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘𝐵) ∧ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∥ (♯‘𝐵)) → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) ∥ (♯‘𝐵)))
250	49, 114, 65, 248, 249	syl31anc 1376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (((𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘𝐵) ∧ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∥ (♯‘𝐵)) → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) ∥ (♯‘𝐵)))
251	87, 103, 250	mp2and 700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) ∥ (♯‘𝐵))
252		eqid 2735	. . . . . . . . . 10 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
253		inss1 4167	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∩ {𝑞}) ⊆ 𝐷
254	253	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐷 ∩ {𝑞}) ⊆ 𝐷)
255	93, 96, 254	dprdres 19994	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷))))
256	255	simpld 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))
257		dprdsubg 19990	. . . . . . . . . . 11 ⊢ (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ (SubGrp‘𝐺))
258	256, 257	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ (SubGrp‘𝐺))
259		inass 4158	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∩ {𝑞}) ∩ (𝐷 ∖ {𝑞})) = (𝐷 ∩ ({𝑞} ∩ (𝐷 ∖ {𝑞})))
260		disjdif 4402	. . . . . . . . . . . . . 14 ⊢ ({𝑞} ∩ (𝐷 ∖ {𝑞})) = ∅
261	260	ineq2i 4148	. . . . . . . . . . . . 13 ⊢ (𝐷 ∩ ({𝑞} ∩ (𝐷 ∖ {𝑞}))) = (𝐷 ∩ ∅)
262		in0 4325	. . . . . . . . . . . . 13 ⊢ (𝐷 ∩ ∅) = ∅
263	259, 261, 262	3eqtri 2762	. . . . . . . . . . . 12 ⊢ ((𝐷 ∩ {𝑞}) ∩ (𝐷 ∖ {𝑞})) = ∅
264	263	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝐷 ∩ {𝑞}) ∩ (𝐷 ∖ {𝑞})) = ∅)
265	93, 96, 254, 97, 264, 104	dprddisj2 20005	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∩ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) = {(0_g‘𝐺)})
266	93, 96, 254, 97, 264, 252	dprdcntz2 20004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
267	6	dprdssv 19982	. . . . . . . . . . 11 ⊢ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ 𝐵
268		ssfi 9096	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ 𝐵) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ Fin)
269	19, 267, 268	sylancl 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ Fin)
270	195, 104, 252, 258, 101, 265, 266, 269, 110	lsmhash 19669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = ((♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
271		inundif 4409	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∩ {𝑞}) ∪ (𝐷 ∖ {𝑞})) = 𝐷
272	271	eqcomi 2744	. . . . . . . . . . . . 13 ⊢ 𝐷 = ((𝐷 ∩ {𝑞}) ∪ (𝐷 ∖ {𝑞}))
273	272	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐷 = ((𝐷 ∩ {𝑞}) ∪ (𝐷 ∖ {𝑞})))
274	95, 264, 273, 195, 93	dprdsplit 20014	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd (𝑇 ↾ 𝐷)) = ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
275	207	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵)
276	274, 275	eqtr3d 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) = 𝐵)
277	276	fveq2d 6833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = (♯‘𝐵))
278		snssi 4719	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ 𝐷 → {𝑞} ⊆ 𝐷)
279	278	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → {𝑞} ⊆ 𝐷)
280		sseqin2 4154	. . . . . . . . . . . . . . . 16 ⊢ ({𝑞} ⊆ 𝐷 ↔ (𝐷 ∩ {𝑞}) = {𝑞})
281	279, 280	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐷 ∩ {𝑞}) = {𝑞})
282	281	reseq2d 5933	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ((𝑇 ↾ 𝐷) ↾ {𝑞}))
283	282	oveq2d 7372	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ {𝑞})))
284	93	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → 𝐺dom DProd (𝑇 ↾ 𝐷))
285	211	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → dom (𝑇 ↾ 𝐷) = 𝐷)
286		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → 𝑞 ∈ 𝐷)
287	284, 285, 286	dpjlem 20017	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ {𝑞})) = ((𝑇 ↾ 𝐷)‘𝑞))
288	216	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → ((𝑇 ↾ 𝐷)‘𝑞) = (𝑇‘𝑞))
289	283, 287, 288	3eqtrd 2774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
290		simprr 773	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ¬ 𝑞 ∈ 𝐷)
291		disjsn 4645	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∩ {𝑞}) = ∅ ↔ ¬ 𝑞 ∈ 𝐷)
292	290, 291	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐷 ∩ {𝑞}) = ∅)
293	292	reseq2d 5933	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ((𝑇 ↾ 𝐷) ↾ ∅))
294		res0 5937	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ↾ 𝐷) ↾ ∅) = ∅
295	293, 294	eqtrdi 2786	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ∅)
296	295	oveq2d 7372	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝐺 DProd ∅))
297	104	dprd0 19997	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0_g‘𝐺)}))
298	40, 297	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0_g‘𝐺)}))
299	298	simprd 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺 DProd ∅) = {(0_g‘𝐺)})
300	299	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐺 DProd ∅) = {(0_g‘𝐺)})
301	296, 300, 186	3eqtrd 2774	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
302	301	anassrs 467	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ ¬ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
303	289, 302	pm2.61dan 813	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
304	303	fveq2d 6833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))) = (♯‘(𝑇‘𝑞)))
305	304	oveq1d 7371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = ((♯‘(𝑇‘𝑞)) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
306	270, 277, 305	3eqtr3d 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) = ((♯‘(𝑇‘𝑞)) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
307	251, 306	breqtrd 5100	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) ∥ ((♯‘(𝑇‘𝑞)) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
308	113	nnne0d 12216	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ≠ 0)
309		dvdsmulcr 16243	. . . . . . . 8 ⊢ (((𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℤ ∧ (♯‘(𝑇‘𝑞)) ∈ ℤ ∧ ((♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℤ ∧ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ≠ 0)) → (((𝑞↑(𝑞 pCnt (♯‘𝐵))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) ∥ ((♯‘(𝑇‘𝑞)) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) ↔ (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝑇‘𝑞))))
310	49, 34, 114, 308, 309	syl112anc 1377	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (((𝑞↑(𝑞 pCnt (♯‘𝐵))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) ∥ ((♯‘(𝑇‘𝑞)) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) ↔ (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝑇‘𝑞))))
311	307, 310	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝑇‘𝑞)))
312		dvdseq 16272	. . . . . 6 ⊢ ((((♯‘(𝑇‘𝑞)) ∈ ℕ₀ ∧ (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℕ₀) ∧ ((♯‘(𝑇‘𝑞)) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∧ (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝑇‘𝑞)))) → (♯‘(𝑇‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
313	33, 85, 73, 311, 312	syl22anc 839	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
314	6, 7, 8, 9, 10, 11	ablfac1a 20035	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
315	313, 314	eqtr4d 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (♯‘(𝑆‘𝑞)))
316		hashen 14298	. . . . 5 ⊢ (((𝑇‘𝑞) ∈ Fin ∧ (𝑆‘𝑞) ∈ Fin) → ((♯‘(𝑇‘𝑞)) = (♯‘(𝑆‘𝑞)) ↔ (𝑇‘𝑞) ≈ (𝑆‘𝑞)))
317	31, 23, 316	syl2anc 585	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((♯‘(𝑇‘𝑞)) = (♯‘(𝑆‘𝑞)) ↔ (𝑇‘𝑞) ≈ (𝑆‘𝑞)))
318	315, 317	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ≈ (𝑆‘𝑞))
319		fisseneq 9162	. . 3 ⊢ (((𝑆‘𝑞) ∈ Fin ∧ (𝑇‘𝑞) ⊆ (𝑆‘𝑞) ∧ (𝑇‘𝑞) ≈ (𝑆‘𝑞)) → (𝑇‘𝑞) = (𝑆‘𝑞))
320	23, 84, 318, 319	syl3anc 1374	. 2 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) = (𝑆‘𝑞))
321	5, 18, 320	eqfnfvd 6975	1 ⊢ (𝜑 → 𝑇 = 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2930 ∀wral 3049 {crab 3387 Vcvv 3427 ∖ cdif 3882 ∪ cun 3883 ∩ cin 3884 ⊆ wss 3885 ∅c0 4263 {csn 4557 class class class wbr 5074 ↦ cmpt 5155 dom cdm 5620 ↾ cres 5622 ⟶wf 6483 ‘cfv 6487 (class class class)co 7356 ≈ cen 8879 Fincfn 8882 0cc0 11027 1c1 11028 · cmul 11032 ≤ cle 11169 ℕcn 12163 ℕ₀cn0 12426 ℤcz 12513 ℤ_≥cuz 12777 ...cfz 13450 ↑cexp 14012 ♯chash 14281 ∥ cdvds 16210 gcd cgcd 16452 ℙcprime 16629 pCnt cpc 16796 Basecbs 17168 0_gc0g 17391 Grpcgrp 18898 SubGrpcsubg 19085 Cntzccntz 19279 odcod 19488 LSSumclsm 19598 Abelcabl 19745 DProd cdprd 19959

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-disj 5042 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 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 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8100 df-tpos 8165 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-omul 8399 df-er 8632 df-ec 8634 df-qs 8638 df-map 8764 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9264 df-sup 9344 df-inf 9345 df-oi 9414 df-dju 9814 df-card 9852 df-acn 9855 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-n0 12427 df-xnn0 12500 df-z 12514 df-uz 12778 df-q 12888 df-rp 12932 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15439 df-sum 15638 df-dvds 16211 df-gcd 16453 df-prm 16630 df-pc 16797 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-0g 17393 df-gsum 17394 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-eqg 19090 df-ghm 19177 df-gim 19223 df-ga 19254 df-cntz 19281 df-oppg 19310 df-od 19492 df-lsm 19600 df-pj1 19601 df-cmn 19746 df-abl 19747 df-dprd 19961

This theorem is referenced by: (None)

W3C validator