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Theorem ablfac1eu 18680

Description: The factorization of ablfac1b 18677 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 482	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
3		ablfac1eu.2	. . . 4 ⊢ (𝜑 → dom 𝑇 = 𝐴)
4	2, 3	dprdf2 18614	. . 3 ⊢ (𝜑 → 𝑇:𝐴⟶(SubGrp‘𝐺))
5	4	ffnd 6185	. 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 18677	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
13	6	fvexi 6345	. . . . . . 7 ⊢ 𝐵 ∈ V
14	13	rabex 4947	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
15	14, 8	dmmpti 6162	. . . . 5 ⊢ dom 𝑆 = 𝐴
16	15	a1i 11	. . . 4 ⊢ (𝜑 → dom 𝑆 = 𝐴)
17	12, 16	dprdf2 18614	. . 3 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
18	17	ffnd 6185	. 2 ⊢ (𝜑 → 𝑆 Fn 𝐴)
19	10	adantr 466	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐵 ∈ Fin)
20	17	ffvelrnda 6504	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
21	6	subgss 17803	. . . . 5 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑆‘𝑞) ⊆ 𝐵)
22	20, 21	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ 𝐵)
23	19, 22	ssfid 8343	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ Fin)
24	4	ffvelrnda 6504	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ∈ (SubGrp‘𝐺))
25	6	subgss 17803	. . . . . 6 ⊢ ((𝑇‘𝑞) ∈ (SubGrp‘𝐺) → (𝑇‘𝑞) ⊆ 𝐵)
26	24, 25	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ 𝐵)
27	24	adantr 466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑇‘𝑞) ∈ (SubGrp‘𝐺))
28	19, 26	ssfid 8343	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ∈ Fin)
29	28	adantr 466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑇‘𝑞) ∈ Fin)
30		simpr 471	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → 𝑥 ∈ (𝑇‘𝑞))
31	7	odsubdvds 18193	. . . . . . 7 ⊢ (((𝑇‘𝑞) ∈ (SubGrp‘𝐺) ∧ (𝑇‘𝑞) ∈ Fin ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∥ (♯‘(𝑇‘𝑞)))
32	27, 29, 30, 31	syl3anc 1476	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∥ (♯‘(𝑇‘𝑞)))
33		ablfac1eu.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))
34	11	sselda 3752	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
35		prmz 15596	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
36	34, 35	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℤ)
37		ablfac1eu.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℕ₀)
38	37	nn0zd 11687	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℤ)
39		ablgrp 18405	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
40	9, 39	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ Grp)
41	6	grpbn0 17659	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
42	40, 41	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ≠ ∅)
43		hashnncl 13359	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
44	10, 43	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
45	42, 44	mpbird 247	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
46	45	adantr 466	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ)
47	34, 46	pccld 15762	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
48	47	nn0zd 11687	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℤ)
49	6	lagsubg 17864	. . . . . . . . . . . . 13 ⊢ (((𝑇‘𝑞) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (♯‘(𝑇‘𝑞)) ∥ (♯‘𝐵))
50	24, 19, 49	syl2anc 573	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∥ (♯‘𝐵))
51	33, 50	eqbrtrrd 4811	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑𝐶) ∥ (♯‘𝐵))
52	46	nnzd 11688	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) ∈ ℤ)
53		pcdvdsb 15780	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℙ ∧ (♯‘𝐵) ∈ ℤ ∧ 𝐶 ∈ ℕ₀) → (𝐶 ≤ (𝑞 pCnt (♯‘𝐵)) ↔ (𝑞↑𝐶) ∥ (♯‘𝐵)))
54	34, 52, 37, 53	syl3anc 1476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐶 ≤ (𝑞 pCnt (♯‘𝐵)) ↔ (𝑞↑𝐶) ∥ (♯‘𝐵)))
55	51, 54	mpbird 247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ≤ (𝑞 pCnt (♯‘𝐵)))
56		eluz2 11899	. . . . . . . . . 10 ⊢ ((𝑞 pCnt (♯‘𝐵)) ∈ (ℤ_≥‘𝐶) ↔ (𝐶 ∈ ℤ ∧ (𝑞 pCnt (♯‘𝐵)) ∈ ℤ ∧ 𝐶 ≤ (𝑞 pCnt (♯‘𝐵))))
57	38, 48, 55, 56	syl3anbrc 1428	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ (ℤ_≥‘𝐶))
58		dvdsexp 15258	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℤ ∧ 𝐶 ∈ ℕ₀ ∧ (𝑞 pCnt (♯‘𝐵)) ∈ (ℤ_≥‘𝐶)) → (𝑞↑𝐶) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
59	36, 37, 57, 58	syl3anc 1476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑𝐶) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
60	33, 59	eqbrtrd 4809	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
61	60	adantr 466	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (♯‘(𝑇‘𝑞)) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
62	26	sselda 3752	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → 𝑥 ∈ 𝐵)
63	6, 7	odcl 18162	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (𝑂‘𝑥) ∈ ℕ₀)
64	62, 63	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∈ ℕ₀)
65	64	nn0zd 11687	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∈ ℤ)
66		hashcl 13349	. . . . . . . . . 10 ⊢ ((𝑇‘𝑞) ∈ Fin → (♯‘(𝑇‘𝑞)) ∈ ℕ₀)
67	28, 66	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∈ ℕ₀)
68	67	nn0zd 11687	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∈ ℤ)
69	68	adantr 466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (♯‘(𝑇‘𝑞)) ∈ ℤ)
70		prmnn 15595	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
71	34, 70	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℕ)
72	71, 47	nnexpcld 13237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℕ)
73	72	nnzd 11688	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℤ)
74	73	adantr 466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℤ)
75		dvdstr 15227	. . . . . . 7 ⊢ (((𝑂‘𝑥) ∈ ℤ ∧ (♯‘(𝑇‘𝑞)) ∈ ℤ ∧ (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℤ) → (((𝑂‘𝑥) ∥ (♯‘(𝑇‘𝑞)) ∧ (♯‘(𝑇‘𝑞)) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))) → (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))))
76	65, 69, 74, 75	syl3anc 1476	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (((𝑂‘𝑥) ∥ (♯‘(𝑇‘𝑞)) ∧ (♯‘(𝑇‘𝑞)) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))) → (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))))
77	32, 61, 76	mp2and 679	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
78	26, 77	ssrabdv 3830	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
79		id 22	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → 𝑝 = 𝑞)
80		oveq1 6803	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → (𝑝 pCnt (♯‘𝐵)) = (𝑞 pCnt (♯‘𝐵)))
81	79, 80	oveq12d 6814	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
82	81	breq2d 4799	. . . . . . 7 ⊢ (𝑝 = 𝑞 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))))
83	82	rabbidv 3339	. . . . . 6 ⊢ (𝑝 = 𝑞 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
84	83, 8, 14	fvmpt3i 6431	. . . . 5 ⊢ (𝑞 ∈ 𝐴 → (𝑆‘𝑞) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
85	84	adantl 467	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
86	78, 85	sseqtr4d 3791	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ (𝑆‘𝑞))
87	72	nnnn0d 11558	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℕ₀)
88		pcdvds 15775	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
89	34, 46, 88	syl2anc 573	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
90	2	adantr 466	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd 𝑇)
91	3	adantr 466	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom 𝑇 = 𝐴)
92		ablfac1.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
93	92	adantr 466	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
94	90, 91, 93	dprdres 18635	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑇)))
95	94	simpld 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd (𝑇 ↾ 𝐷))
96	4	adantr 466	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇:𝐴⟶(SubGrp‘𝐺))
97	96, 93	fssresd 6212	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
98	97	fdmd 6193	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom (𝑇 ↾ 𝐷) = 𝐷)
99		difssd 3889	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐷 ∖ {𝑞}) ⊆ 𝐷)
100	95, 98, 99	dprdres 18635	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})) ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷))))
101	100	simpld 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))
102		dprdsubg 18631	. . . . . . . . . . 11 ⊢ (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺))
103	101, 102	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺))
104	6	lagsubg 17864	. . . . . . . . . 10 ⊢ (((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∥ (♯‘𝐵))
105	103, 19, 104	syl2anc 573	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∥ (♯‘𝐵))
106		eqid 2771	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
107	106	subg0cl 17810	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))
108	103, 107	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (0_g‘𝐺) ∈ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))
109	108	ne0d 4070	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ≠ ∅)
110	6	dprdssv 18623	. . . . . . . . . . . . . 14 ⊢ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ⊆ 𝐵
111		ssfi 8340	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ⊆ 𝐵) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ Fin)
112	19, 110, 111	sylancl 574	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ Fin)
113		hashnncl 13359	. . . . . . . . . . . . 13 ⊢ ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ Fin → ((♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℕ ↔ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ≠ ∅))
114	112, 113	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℕ ↔ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ≠ ∅))
115	109, 114	mpbird 247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℕ)
116	115	nnzd 11688	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℤ)
117		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑞 → 𝑥 = 𝑞)
118		sneq 4327	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑞 → {𝑥} = {𝑞})
119	118	difeq2d 3879	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑞 → (𝐷 ∖ {𝑥}) = (𝐷 ∖ {𝑞}))
120	119	reseq2d 5533	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑞 → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})) = ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))
121	120	oveq2d 6812	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑞 → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥}))) = (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))
122	121	fveq2d 6337	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑞 → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))) = (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
123	117, 122	breq12d 4800	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))) ↔ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
124	123	notbid 307	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑞 → (¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))) ↔ ¬ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
125		eqid 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ (𝑂‘𝑦) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) = (𝑝 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ (𝑂‘𝑦) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
126	9	adantr 466	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐺 ∈ Abel)
127	10	adantr 466	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐵 ∈ Fin)
128		ablfac1c.d	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
129		ssrab2 3836	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ⊆ ℙ
130	128, 129	eqsstri 3784	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 ⊆ ℙ
131	130	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐷 ⊆ ℙ)
132		ssid 3773	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 ⊆ 𝐷
133	132	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐷 ⊆ 𝐷)
134	2, 3, 92	dprdres 18635	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑇)))
135	134	simpld 482	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ 𝐷))
136		dprdsubg 18631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺dom DProd (𝑇 ↾ 𝐷) → (𝐺 DProd (𝑇 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
137	135, 136	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
138		difssd 3889	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ 𝐴)
139	2, 3, 138	dprdres 18635	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐺dom DProd (𝑇 ↾ (𝐴 ∖ 𝐷)) ∧ (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ⊆ (𝐺 DProd 𝑇)))
140	139	simpld 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))
141		dprdsubg 18631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺dom DProd (𝑇 ↾ (𝐴 ∖ 𝐷)) → (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ∈ (SubGrp‘𝐺))
142	140, 141	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ∈ (SubGrp‘𝐺))
143		difss 3888	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∖ 𝐷) ⊆ 𝐴
144		fssres 6211	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑇:𝐴⟶(SubGrp‘𝐺) ∧ (𝐴 ∖ 𝐷) ⊆ 𝐴) → (𝑇 ↾ (𝐴 ∖ 𝐷)):(𝐴 ∖ 𝐷)⟶(SubGrp‘𝐺))
145	4, 143, 144	sylancl 574	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑇 ↾ (𝐴 ∖ 𝐷)):(𝐴 ∖ 𝐷)⟶(SubGrp‘𝐺))
146	145	fdmd 6193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom (𝑇 ↾ (𝐴 ∖ 𝐷)) = (𝐴 ∖ 𝐷))
147		fvres 6350	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑞 ∈ (𝐴 ∖ 𝐷) → ((𝑇 ↾ (𝐴 ∖ 𝐷))‘𝑞) = (𝑇‘𝑞))
148	147	adantl 467	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑞 ∈ (𝐴 ∖ 𝐷)) → ((𝑇 ↾ (𝐴 ∖ 𝐷))‘𝑞) = (𝑇‘𝑞))
149		eldif 3733	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑞 ∈ (𝐴 ∖ 𝐷) ↔ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷))
150	28	adantrr 696	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑇‘𝑞) ∈ Fin)
151	106	subg0cl 17810	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑇‘𝑞) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝑇‘𝑞))
152	24, 151	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (0_g‘𝐺) ∈ (𝑇‘𝑞))
153	152	snssd 4476	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {(0_g‘𝐺)} ⊆ (𝑇‘𝑞))
154	153	adantrr 696	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} ⊆ (𝑇‘𝑞))
155	33	adantrr 696	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))
156	34	adantr 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∈ ℙ)
157		iddvdsexp 15214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑞 ∈ ℤ ∧ 𝐶 ∈ ℕ) → 𝑞 ∥ (𝑞↑𝐶))
158	36, 157	sylan 569	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∥ (𝑞↑𝐶))
159	51	adantr 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → (𝑞↑𝐶) ∥ (♯‘𝐵))
160	33, 68	eqeltrrd 2851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑𝐶) ∈ ℤ)
161		dvdstr 15227	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑞 ∈ ℤ ∧ (𝑞↑𝐶) ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → ((𝑞 ∥ (𝑞↑𝐶) ∧ (𝑞↑𝐶) ∥ (♯‘𝐵)) → 𝑞 ∥ (♯‘𝐵)))
162	36, 160, 52, 161	syl3anc 1476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞 ∥ (𝑞↑𝐶) ∧ (𝑞↑𝐶) ∥ (♯‘𝐵)) → 𝑞 ∥ (♯‘𝐵)))
163	162	adantr 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → ((𝑞 ∥ (𝑞↑𝐶) ∧ (𝑞↑𝐶) ∥ (♯‘𝐵)) → 𝑞 ∥ (♯‘𝐵)))
164	158, 159, 163	mp2and 679	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∥ (♯‘𝐵))
165		breq1 4790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑤 = 𝑞 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑞 ∥ (♯‘𝐵)))
166	165, 128	elrab2 3518	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑞 ∈ 𝐷 ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ (♯‘𝐵)))
167	156, 164, 166	sylanbrc 572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∈ 𝐷)
168	167	ex 397	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐶 ∈ ℕ → 𝑞 ∈ 𝐷))
169	168	con3d 149	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞 ∈ 𝐷 → ¬ 𝐶 ∈ ℕ))
170	169	impr 442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ¬ 𝐶 ∈ ℕ)
171	37	adantrr 696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝐶 ∈ ℕ₀)
172		elnn0 11501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐶 ∈ ℕ₀ ↔ (𝐶 ∈ ℕ ∨ 𝐶 = 0))
173	171, 172	sylib 208	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐶 ∈ ℕ ∨ 𝐶 = 0))
174	173	ord 853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (¬ 𝐶 ∈ ℕ → 𝐶 = 0))
175	170, 174	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝐶 = 0)
176	175	oveq2d 6812	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑞↑𝐶) = (𝑞↑0))
177	71	adantrr 696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝑞 ∈ ℕ)
178	177	nncnd 11242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝑞 ∈ ℂ)
179	178	exp0d 13209	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑞↑0) = 1)
180	155, 176, 179	3eqtrd 2809	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (♯‘(𝑇‘𝑞)) = 1)
181		fvex 6344	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0_g‘𝐺) ∈ V
182		hashsng 13361	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0_g‘𝐺) ∈ V → (♯‘{(0_g‘𝐺)}) = 1)
183	181, 182	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (♯‘{(0_g‘𝐺)}) = 1
184	180, 183	syl6reqr 2824	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (♯‘{(0_g‘𝐺)}) = (♯‘(𝑇‘𝑞)))
185		snfi 8198	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ {(0_g‘𝐺)} ∈ Fin
186		hashen 13339	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (({(0_g‘𝐺)} ∈ Fin ∧ (𝑇‘𝑞) ∈ Fin) → ((♯‘{(0_g‘𝐺)}) = (♯‘(𝑇‘𝑞)) ↔ {(0_g‘𝐺)} ≈ (𝑇‘𝑞)))
187	185, 150, 186	sylancr 575	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ((♯‘{(0_g‘𝐺)}) = (♯‘(𝑇‘𝑞)) ↔ {(0_g‘𝐺)} ≈ (𝑇‘𝑞)))
188	184, 187	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} ≈ (𝑇‘𝑞))
189		fisseneq 8331	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑇‘𝑞) ∈ Fin ∧ {(0_g‘𝐺)} ⊆ (𝑇‘𝑞) ∧ {(0_g‘𝐺)} ≈ (𝑇‘𝑞)) → {(0_g‘𝐺)} = (𝑇‘𝑞))
190	150, 154, 188, 189	syl3anc 1476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} = (𝑇‘𝑞))
191	106	subg0cl 17810	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐺 DProd (𝑇 ↾ 𝐷)) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd (𝑇 ↾ 𝐷)))
192	137, 191	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (0_g‘𝐺) ∈ (𝐺 DProd (𝑇 ↾ 𝐷)))
193	192	adantr 466	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (0_g‘𝐺) ∈ (𝐺 DProd (𝑇 ↾ 𝐷)))
194	193	snssd 4476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
195	190, 194	eqsstr3d 3789	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑇‘𝑞) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
196	149, 195	sylan2b 581	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑞 ∈ (𝐴 ∖ 𝐷)) → (𝑇‘𝑞) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
197	148, 196	eqsstrd 3788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑞 ∈ (𝐴 ∖ 𝐷)) → ((𝑇 ↾ (𝐴 ∖ 𝐷))‘𝑞) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
198	140, 146, 137, 197	dprdlub 18633	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
199		eqid 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
200	199	lsmss2 18288	. . . . . . . . . . . . . . . . . . . 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)))
201	137, 142, 198, 200	syl3anc 1476	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ 𝐷))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))) = (𝐺 DProd (𝑇 ↾ 𝐷)))
202		disjdif 4183	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∩ (𝐴 ∖ 𝐷)) = ∅
203	202	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐷 ∩ (𝐴 ∖ 𝐷)) = ∅)
204		undif2 4187	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∪ (𝐴 ∖ 𝐷)) = (𝐷 ∪ 𝐴)
205		ssequn1 3934	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐷 ⊆ 𝐴 ↔ (𝐷 ∪ 𝐴) = 𝐴)
206	92, 205	sylib 208	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐷 ∪ 𝐴) = 𝐴)
207	204, 206	syl5req 2818	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 = (𝐷 ∪ (𝐴 ∖ 𝐷)))
208	4, 203, 207, 199, 2	dprdsplit 18655	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐺 DProd (𝑇 ↾ 𝐷))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))))
209	1	simprd 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd 𝑇) = 𝐵)
210	208, 209	eqtr3d 2807	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ 𝐷))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))) = 𝐵)
211	201, 210	eqtr3d 2807	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵)
212	135, 211	jca 501	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵))
213	212	adantr 466	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵))
214	4, 92	fssresd 6212	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑇 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
215	214	fdmd 6193	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom (𝑇 ↾ 𝐷) = 𝐷)
216	215	adantr 466	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → dom (𝑇 ↾ 𝐷) = 𝐷)
217	92	sselda 3752	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → 𝑞 ∈ 𝐴)
218	217, 37	syldan 579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → 𝐶 ∈ ℕ₀)
219	218	adantlr 694	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℙ) ∧ 𝑞 ∈ 𝐷) → 𝐶 ∈ ℕ₀)
220		fvres 6350	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 ∈ 𝐷 → ((𝑇 ↾ 𝐷)‘𝑞) = (𝑇‘𝑞))
221	220	adantl 467	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → ((𝑇 ↾ 𝐷)‘𝑞) = (𝑇‘𝑞))
222	221	fveq2d 6337	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → (♯‘((𝑇 ↾ 𝐷)‘𝑞)) = (♯‘(𝑇‘𝑞)))
223	217, 33	syldan 579	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))
224	222, 223	eqtrd 2805	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → (♯‘((𝑇 ↾ 𝐷)‘𝑞)) = (𝑞↑𝐶))
225	224	adantlr 694	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℙ) ∧ 𝑞 ∈ 𝐷) → (♯‘((𝑇 ↾ 𝐷)‘𝑞)) = (𝑞↑𝐶))
226		simpr 471	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝑥 ∈ ℙ)
227		fzfid 12980	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...(♯‘𝐵)) ∈ Fin)
228		prmnn 15595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℕ)
229	228	3ad2ant2 1128	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ ℕ)
230		prmz 15596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℤ)
231		dvdsle 15241	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℤ ∧ (♯‘𝐵) ∈ ℕ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
232	230, 45, 231	syl2anr 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
233	232	3impia 1109	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ≤ (♯‘𝐵))
234	45	nnzd 11688	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
235	234	3ad2ant1 1127	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (♯‘𝐵) ∈ ℤ)
236		fznn 12615	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐵) ∈ ℤ → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
237	235, 236	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
238	229, 233, 237	mpbir2and 692	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ (1...(♯‘𝐵)))
239	238	rabssdv 3831	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ⊆ (1...(♯‘𝐵)))
240	128, 239	syl5eqss 3798	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 ⊆ (1...(♯‘𝐵)))
241	227, 240	ssfid 8343	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐷 ∈ Fin)
242	241	adantr 466	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐷 ∈ Fin)
243	6, 7, 125, 126, 127, 131, 128, 133, 213, 216, 219, 225, 226, 242	ablfac1eulem 18679	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → ¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))))
244	243	ralrimiva 3115	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ ℙ ¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))))
245	244	adantr 466	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ∀𝑥 ∈ ℙ ¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))))
246	124, 245, 34	rspcdva 3466	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ¬ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
247		coprm 15630	. . . . . . . . . . . . 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))
248	34, 116, 247	syl2anc 573	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ↔ (𝑞 gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1))
249	246, 248	mpbid 222	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1)
250		rpexp1i 15640	. . . . . . . . . . . 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))
251	36, 116, 47, 250	syl3anc 1476	. . . . . . . . . . 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))
252	249, 251	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1)
253		coprmdvds2 15575	. . . . . . . . . 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)))
254	73, 116, 52, 252, 253	syl31anc 1479	. . . . . . . . 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)))
255	89, 105, 254	mp2and 679	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) ∥ (♯‘𝐵))
256		eqid 2771	. . . . . . . . . 10 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
257		inss1 3981	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∩ {𝑞}) ⊆ 𝐷
258	257	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐷 ∩ {𝑞}) ⊆ 𝐷)
259	95, 98, 258	dprdres 18635	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷))))
260	259	simpld 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))
261		dprdsubg 18631	. . . . . . . . . . 11 ⊢ (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ (SubGrp‘𝐺))
262	260, 261	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ (SubGrp‘𝐺))
263		inass 3972	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∩ {𝑞}) ∩ (𝐷 ∖ {𝑞})) = (𝐷 ∩ ({𝑞} ∩ (𝐷 ∖ {𝑞})))
264		disjdif 4183	. . . . . . . . . . . . . 14 ⊢ ({𝑞} ∩ (𝐷 ∖ {𝑞})) = ∅
265	264	ineq2i 3962	. . . . . . . . . . . . 13 ⊢ (𝐷 ∩ ({𝑞} ∩ (𝐷 ∖ {𝑞}))) = (𝐷 ∩ ∅)
266		in0 4113	. . . . . . . . . . . . 13 ⊢ (𝐷 ∩ ∅) = ∅
267	263, 265, 266	3eqtri 2797	. . . . . . . . . . . 12 ⊢ ((𝐷 ∩ {𝑞}) ∩ (𝐷 ∖ {𝑞})) = ∅
268	267	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝐷 ∩ {𝑞}) ∩ (𝐷 ∖ {𝑞})) = ∅)
269	95, 98, 258, 99, 268, 106	dprddisj2 18646	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∩ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) = {(0_g‘𝐺)})
270	95, 98, 258, 99, 268, 256	dprdcntz2 18645	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
271	6	dprdssv 18623	. . . . . . . . . . 11 ⊢ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ 𝐵
272		ssfi 8340	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ 𝐵) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ Fin)
273	19, 271, 272	sylancl 574	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ Fin)
274	199, 106, 256, 262, 103, 269, 270, 273, 112	lsmhash 18325	. . . . . . . . 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}))))))
275		inundif 4189	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∩ {𝑞}) ∪ (𝐷 ∖ {𝑞})) = 𝐷
276	275	eqcomi 2780	. . . . . . . . . . . . 13 ⊢ 𝐷 = ((𝐷 ∩ {𝑞}) ∪ (𝐷 ∖ {𝑞}))
277	276	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐷 = ((𝐷 ∩ {𝑞}) ∪ (𝐷 ∖ {𝑞})))
278	97, 268, 277, 199, 95	dprdsplit 18655	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd (𝑇 ↾ 𝐷)) = ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
279	211	adantr 466	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵)
280	278, 279	eqtr3d 2807	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) = 𝐵)
281	280	fveq2d 6337	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = (♯‘𝐵))
282		snssi 4475	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ 𝐷 → {𝑞} ⊆ 𝐷)
283	282	adantl 467	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → {𝑞} ⊆ 𝐷)
284		sseqin2 3968	. . . . . . . . . . . . . . . 16 ⊢ ({𝑞} ⊆ 𝐷 ↔ (𝐷 ∩ {𝑞}) = {𝑞})
285	283, 284	sylib 208	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐷 ∩ {𝑞}) = {𝑞})
286	285	reseq2d 5533	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ((𝑇 ↾ 𝐷) ↾ {𝑞}))
287	286	oveq2d 6812	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ {𝑞})))
288	95	adantr 466	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → 𝐺dom DProd (𝑇 ↾ 𝐷))
289	215	ad2antrr 705	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → dom (𝑇 ↾ 𝐷) = 𝐷)
290		simpr 471	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → 𝑞 ∈ 𝐷)
291	288, 289, 290	dpjlem 18658	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ {𝑞})) = ((𝑇 ↾ 𝐷)‘𝑞))
292	220	adantl 467	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → ((𝑇 ↾ 𝐷)‘𝑞) = (𝑇‘𝑞))
293	287, 291, 292	3eqtrd 2809	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
294		simprr 756	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ¬ 𝑞 ∈ 𝐷)
295		disjsn 4384	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∩ {𝑞}) = ∅ ↔ ¬ 𝑞 ∈ 𝐷)
296	294, 295	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐷 ∩ {𝑞}) = ∅)
297	296	reseq2d 5533	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ((𝑇 ↾ 𝐷) ↾ ∅))
298		res0 5537	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ↾ 𝐷) ↾ ∅) = ∅
299	297, 298	syl6eq 2821	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ∅)
300	299	oveq2d 6812	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝐺 DProd ∅))
301	106	dprd0 18638	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0_g‘𝐺)}))
302	40, 301	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0_g‘𝐺)}))
303	302	simprd 483	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺 DProd ∅) = {(0_g‘𝐺)})
304	303	adantr 466	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐺 DProd ∅) = {(0_g‘𝐺)})
305	300, 304, 190	3eqtrd 2809	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
306	305	anassrs 453	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ ¬ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
307	293, 306	pm2.61dan 814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
308	307	fveq2d 6337	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))) = (♯‘(𝑇‘𝑞)))
309	308	oveq1d 6811	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = ((♯‘(𝑇‘𝑞)) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
310	274, 281, 309	3eqtr3d 2813	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) = ((♯‘(𝑇‘𝑞)) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
311	255, 310	breqtrd 4813	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) ∥ ((♯‘(𝑇‘𝑞)) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
312	115	nnne0d 11271	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ≠ 0)
313		dvdsmulcr 15220	. . . . . . . 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))))
314	73, 68, 116, 312, 313	syl112anc 1480	. . . . . . 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))))
315	311, 314	mpbid 222	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝑇‘𝑞)))
316		dvdseq 15245	. . . . . 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))))
317	67, 87, 60, 315, 316	syl22anc 1477	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
318	6, 7, 8, 9, 10, 11	ablfac1a 18676	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
319	317, 318	eqtr4d 2808	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (♯‘(𝑆‘𝑞)))
320		hashen 13339	. . . . 5 ⊢ (((𝑇‘𝑞) ∈ Fin ∧ (𝑆‘𝑞) ∈ Fin) → ((♯‘(𝑇‘𝑞)) = (♯‘(𝑆‘𝑞)) ↔ (𝑇‘𝑞) ≈ (𝑆‘𝑞)))
321	28, 23, 320	syl2anc 573	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((♯‘(𝑇‘𝑞)) = (♯‘(𝑆‘𝑞)) ↔ (𝑇‘𝑞) ≈ (𝑆‘𝑞)))
322	319, 321	mpbid 222	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ≈ (𝑆‘𝑞))
323		fisseneq 8331	. . 3 ⊢ (((𝑆‘𝑞) ∈ Fin ∧ (𝑇‘𝑞) ⊆ (𝑆‘𝑞) ∧ (𝑇‘𝑞) ≈ (𝑆‘𝑞)) → (𝑇‘𝑞) = (𝑆‘𝑞))
324	23, 86, 322, 323	syl3anc 1476	. 2 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) = (𝑆‘𝑞))
325	5, 18, 324	eqfnfvd 6459	1 ⊢ (𝜑 → 𝑇 = 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 ∨ wo 836 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∀wral 3061 {crab 3065 Vcvv 3351 ∖ cdif 3720 ∪ cun 3721 ∩ cin 3722 ⊆ wss 3723 ∅c0 4063 {csn 4317 class class class wbr 4787 ↦ cmpt 4864 dom cdm 5250 ↾ cres 5252 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 ≈ cen 8110 Fincfn 8113 0cc0 10142 1c1 10143 · cmul 10147 ≤ cle 10281 ℕcn 11226 ℕ₀cn0 11499 ℤcz 11584 ℤ_≥cuz 11893 ...cfz 12533 ↑cexp 13067 ♯chash 13321 ∥ cdvds 15189 gcd cgcd 15424 ℙcprime 15592 pCnt cpc 15748 Basecbs 16064 0_gc0g 16308 Grpcgrp 17630 SubGrpcsubg 17796 Cntzccntz 17955 odcod 18151 LSSumclsm 18256 Abelcabl 18401 DProd cdprd 18600

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-disj 4756 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-of 7048 df-om 7217 df-1st 7319 df-2nd 7320 df-supp 7451 df-tpos 7508 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-omul 7722 df-er 7900 df-ec 7902 df-qs 7906 df-map 8015 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8436 df-sup 8508 df-inf 8509 df-oi 8575 df-card 8969 df-acn 8972 df-cda 9196 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-n0 11500 df-xnn0 11571 df-z 11585 df-uz 11894 df-q 11997 df-rp 12036 df-fz 12534 df-fzo 12674 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-sum 14625 df-dvds 15190 df-gcd 15425 df-prm 15593 df-pc 15749 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-0g 16310 df-gsum 16311 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-mhm 17543 df-submnd 17544 df-grp 17633 df-minusg 17634 df-sbg 17635 df-mulg 17749 df-subg 17799 df-eqg 17801 df-ghm 17866 df-gim 17909 df-ga 17930 df-cntz 17957 df-oppg 17983 df-od 18155 df-lsm 18258 df-pj1 18259 df-cmn 18402 df-abl 18403 df-dprd 18602

This theorem is referenced by: (None)

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