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Theorem ablfac1eu 20008

Description: The factorization of ablfac1b 20005 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 19942	. . 3 ⊢ (𝜑 → 𝑇:𝐴⟶(SubGrp‘𝐺))
5	4	ffnd 6661	. 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 20005	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
13	6	fvexi 6846	. . . . . . 7 ⊢ 𝐵 ∈ V
14	13	rabex 5274	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
15	14, 8	dmmpti 6634	. . . . 5 ⊢ dom 𝑆 = 𝐴
16	15	a1i 11	. . . 4 ⊢ (𝜑 → dom 𝑆 = 𝐴)
17	12, 16	dprdf2 19942	. . 3 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
18	17	ffnd 6661	. 2 ⊢ (𝜑 → 𝑆 Fn 𝐴)
19	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐵 ∈ Fin)
20	17	ffvelcdmda 7028	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
21	6	subgss 19061	. . . . 5 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑆‘𝑞) ⊆ 𝐵)
22	20, 21	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ 𝐵)
23	19, 22	ssfid 9170	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ Fin)
24	4	ffvelcdmda 7028	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ∈ (SubGrp‘𝐺))
25	6	subgss 19061	. . . . . 6 ⊢ ((𝑇‘𝑞) ∈ (SubGrp‘𝐺) → (𝑇‘𝑞) ⊆ 𝐵)
26	24, 25	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ 𝐵)
27	26	sselda 3922	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → 𝑥 ∈ 𝐵)
28	6, 7	odcl 19469	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝑂‘𝑥) ∈ ℕ₀)
29	27, 28	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∈ ℕ₀)
30	29	nn0zd 12514	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∈ ℤ)
31	19, 26	ssfid 9170	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ∈ Fin)
32		hashcl 14280	. . . . . . . . 9 ⊢ ((𝑇‘𝑞) ∈ Fin → (♯‘(𝑇‘𝑞)) ∈ ℕ₀)
33	31, 32	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∈ ℕ₀)
34	33	nn0zd 12514	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∈ ℤ)
35	34	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (♯‘(𝑇‘𝑞)) ∈ ℤ)
36	11	sselda 3922	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
37		prmnn 16602	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
38	36, 37	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℕ)
39		ablgrp 19718	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
40	9, 39	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ Grp)
41	6	grpbn0 18900	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
42	40, 41	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ ∅)
43		hashnncl 14290	. . . . . . . . . . . . 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 16779	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
48	38, 47	nnexpcld 14169	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℕ)
49	48	nnzd 12515	. . . . . . 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 19504	. . . . . . 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 16603	. . . . . . . . . 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 12514	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℤ)
61	47	nn0zd 12514	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℤ)
62	6	lagsubg 19128	. . . . . . . . . . . . 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 5110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑𝐶) ∥ (♯‘𝐵))
65	46	nnzd 12515	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) ∈ ℤ)
66		pcdvdsb 16798	. . . . . . . . . . . 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 12758	. . . . . . . . . 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 16256	. . . . . . . . 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 5108	. . . . . . 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 16223	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
76	26, 75	ssrabdv 4014	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
77		id 22	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → 𝑝 = 𝑞)
78		oveq1 7365	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → (𝑝 pCnt (♯‘𝐵)) = (𝑞 pCnt (♯‘𝐵)))
79	77, 78	oveq12d 7376	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
80	79	breq2d 5098	. . . . . . 7 ⊢ (𝑝 = 𝑞 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))))
81	80	rabbidv 3397	. . . . . 6 ⊢ (𝑝 = 𝑞 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
82	81, 8, 14	fvmpt3i 6945	. . . . 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 3960	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ (𝑆‘𝑞))
85	48	nnnn0d 12463	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℕ₀)
86		pcdvds 16793	. . . . . . . . . 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 19963	. . . . . . . . . . . . . 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 6699	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
96	95	fdmd 6670	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom (𝑇 ↾ 𝐷) = 𝐷)
97		difssd 4078	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐷 ∖ {𝑞}) ⊆ 𝐷)
98	93, 96, 97	dprdres 19963	. . . . . . . . . . . 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 19959	. . . . . . . . . . 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 19128	. . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
105	104	subg0cl 19068	. . . . . . . . . . . . . 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 4283	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ≠ ∅)
108	6	dprdssv 19951	. . . . . . . . . . . . . 14 ⊢ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ⊆ 𝐵
109		ssfi 9098	. . . . . . . . . . . . . 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 14290	. . . . . . . . . . . . 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 12515	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℤ)
115		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑞 → 𝑥 = 𝑞)
116		sneq 4578	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑞 → {𝑥} = {𝑞})
117	116	difeq2d 4067	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑞 → (𝐷 ∖ {𝑥}) = (𝐷 ∖ {𝑞}))
118	117	reseq2d 5936	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑞 → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})) = ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))
119	118	oveq2d 7374	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑞 → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥}))) = (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))
120	119	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑞 → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))) = (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
121	115, 120	breq12d 5099	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . 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 4023	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 ⊆ ℙ
128	127	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐷 ⊆ ℙ)
129		ssidd 3946	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐷 ⊆ 𝐷)
130	2, 3, 90	dprdres 19963	. . . . . . . . . . . . . . . . . . 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 19959	. . . . . . . . . . . . . . . . . . . . 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 4078	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ 𝐴)
135	2, 3, 134	dprdres 19963	. . . . . . . . . . . . . . . . . . . . . 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 19959	. . . . . . . . . . . . . . . . . . . . 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 4077	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∖ 𝐷) ⊆ 𝐴
140		fssres 6698	. . . . . . . . . . . . . . . . . . . . . . 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 6670	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom (𝑇 ↾ (𝐴 ∖ 𝐷)) = (𝐴 ∖ 𝐷))
143		fvres 6851	. . . . . . . . . . . . . . . . . . . . . . 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 3900	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑞 ∈ (𝐴 ∖ 𝐷) ↔ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷))
146	31	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑇‘𝑞) ∈ Fin)
147	104	subg0cl 19068	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4753	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6845	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0_g‘𝐺) ∈ V
152		hashsng 14293	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 16207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑𝐶) ∈ ℤ)
160		dvdstr 16222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5089	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑤 = 𝑞 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑞 ∥ (♯‘𝐵)))
165	164, 126	elrab2 3638	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12162	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝑞 ∈ ℂ)
178	177	exp0d 14064	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑞↑0) = 1)
179	154, 175, 178	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (♯‘(𝑇‘𝑞)) = 1)
180	153, 179	eqtr4id 2791	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (♯‘{(0_g‘𝐺)}) = (♯‘(𝑇‘𝑞)))
181		snfi 8981	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ {(0_g‘𝐺)} ∈ Fin
182		hashen 14271	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 9164	. . . . . . . . . . . . . . . . . . . . . . . . 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 19068	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4753	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
191	186, 190	eqsstrrd 3958	. . . . . . . . . . . . . . . . . . . . . . 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 3957	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑞 ∈ (𝐴 ∖ 𝐷)) → ((𝑇 ↾ (𝐴 ∖ 𝐷))‘𝑞) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
194	136, 142, 133, 193	dprdlub 19961	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
195		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
196	195	lsmss2 19600	. . . . . . . . . . . . . . . . . . . 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 4413	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∩ (𝐴 ∖ 𝐷)) = ∅
199	198	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐷 ∩ (𝐴 ∖ 𝐷)) = ∅)
200		undif2 4418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∪ (𝐴 ∖ 𝐷)) = (𝐷 ∪ 𝐴)
201		ssequn1 4127	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐷 ⊆ 𝐴 ↔ (𝐷 ∪ 𝐴) = 𝐴)
202	90, 201	sylib 218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐷 ∪ 𝐴) = 𝐴)
203	200, 202	eqtr2id 2785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 = (𝐷 ∪ (𝐴 ∖ 𝐷)))
204	4, 199, 203, 195, 2	dprdsplit 19983	. . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ 𝐷))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))) = 𝐵)
207	197, 206	eqtr3d 2774	. . . . . . . . . . . . . . . . . 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 6699	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑇 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
211	210	fdmd 6670	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom (𝑇 ↾ 𝐷) = 𝐷)
212	211	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → dom (𝑇 ↾ 𝐷) = 𝐷)
213	90	sselda 3922	. . . . . . . . . . . . . . . . . 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 6851	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 ∈ 𝐷 → ((𝑇 ↾ 𝐷)‘𝑞) = (𝑇‘𝑞))
217	216	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → ((𝑇 ↾ 𝐷)‘𝑞) = (𝑇‘𝑞))
218	217	fveq2d 6836	. . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . 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 13897	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...(♯‘𝐵)) ∈ Fin)
224		prmnn 16602	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℕ)
225	224	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ ℕ)
226		prmz 16603	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℤ)
227		dvdsle 16238	. . . . . . . . . . . . . . . . . . . . . . 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 12515	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
231	230	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (♯‘𝐵) ∈ ℤ)
232		fznn 13509	. . . . . . . . . . . . . . . . . . . . . 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 4015	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ⊆ (1...(♯‘𝐵)))
236	126, 235	eqsstrid 3961	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 ⊆ (1...(♯‘𝐵)))
237	223, 236	ssfid 9170	. . . . . . . . . . . . . . . . 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 20007	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → ¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))))
240	239	ralrimiva 3130	. . . . . . . . . . . . . 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 3566	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ¬ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
243		coprm 16639	. . . . . . . . . . . . 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 16651	. . . . . . . . . . . 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 16582	. . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
253		inss1 4178	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∩ {𝑞}) ⊆ 𝐷
254	253	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐷 ∩ {𝑞}) ⊆ 𝐷)
255	93, 96, 254	dprdres 19963	. . . . . . . . . . . 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 19959	. . . . . . . . . . 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 4169	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∩ {𝑞}) ∩ (𝐷 ∖ {𝑞})) = (𝐷 ∩ ({𝑞} ∩ (𝐷 ∖ {𝑞})))
260		disjdif 4413	. . . . . . . . . . . . . 14 ⊢ ({𝑞} ∩ (𝐷 ∖ {𝑞})) = ∅
261	260	ineq2i 4158	. . . . . . . . . . . . 13 ⊢ (𝐷 ∩ ({𝑞} ∩ (𝐷 ∖ {𝑞}))) = (𝐷 ∩ ∅)
262		in0 4336	. . . . . . . . . . . . 13 ⊢ (𝐷 ∩ ∅) = ∅
263	259, 261, 262	3eqtri 2764	. . . . . . . . . . . 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 19974	. . . . . . . . . 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 19973	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
267	6	dprdssv 19951	. . . . . . . . . . 11 ⊢ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ 𝐵
268		ssfi 9098	. . . . . . . . . . 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 19638	. . . . . . . . 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 4420	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∩ {𝑞}) ∪ (𝐷 ∖ {𝑞})) = 𝐷
272	271	eqcomi 2746	. . . . . . . . . . . . 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 19983	. . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) = 𝐵)
277	276	fveq2d 6836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = (♯‘𝐵))
278		snssi 4752	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ 𝐷 → {𝑞} ⊆ 𝐷)
279	278	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → {𝑞} ⊆ 𝐷)
280		sseqin2 4164	. . . . . . . . . . . . . . . 16 ⊢ ({𝑞} ⊆ 𝐷 ↔ (𝐷 ∩ {𝑞}) = {𝑞})
281	279, 280	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐷 ∩ {𝑞}) = {𝑞})
282	281	reseq2d 5936	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ((𝑇 ↾ 𝐷) ↾ {𝑞}))
283	282	oveq2d 7374	. . . . . . . . . . . . 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 19986	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 4656	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∩ {𝑞}) = ∅ ↔ ¬ 𝑞 ∈ 𝐷)
292	290, 291	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐷 ∩ {𝑞}) = ∅)
293	292	reseq2d 5936	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ((𝑇 ↾ 𝐷) ↾ ∅))
294		res0 5940	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ↾ 𝐷) ↾ ∅) = ∅
295	293, 294	eqtrdi 2788	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ∅)
296	295	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝐺 DProd ∅))
297	104	dprd0 19966	. . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 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 6836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))) = (♯‘(𝑇‘𝑞)))
305	304	oveq1d 7373	. . . . . . . . 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 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) = ((♯‘(𝑇‘𝑞)) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
307	251, 306	breqtrd 5112	. . . . . . 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 12196	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ≠ 0)
309		dvdsmulcr 16213	. . . . . . . 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 16242	. . . . . 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 20004	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
315	313, 314	eqtr4d 2775	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (♯‘(𝑆‘𝑞)))
316		hashen 14271	. . . . 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 9164	. . 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 6978	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 2933 ∀wral 3052 {crab 3390 Vcvv 3430 ∖ cdif 3887 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5622 ↾ cres 5624 ⟶wf 6486 ‘cfv 6490 (class class class)co 7358 ≈ cen 8881 Fincfn 8884 0cc0 11027 1c1 11028 · cmul 11032 ≤ cle 11168 ℕcn 12146 ℕ₀cn0 12402 ℤcz 12489 ℤ_≥cuz 12752 ...cfz 13424 ↑cexp 13985 ♯chash 14254 ∥ cdvds 16180 gcd cgcd 16422 ℙcprime 16599 pCnt cpc 16765 Basecbs 17137 0_gc0g 17360 Grpcgrp 18867 SubGrpcsubg 19054 Cntzccntz 19248 odcod 19457 LSSumclsm 19567 Abelcabl 19714 DProd cdprd 19928

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 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-tpos 8167 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-omul 8401 df-er 8634 df-ec 8636 df-qs 8640 df-map 8766 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-inf 9347 df-oi 9416 df-dju 9814 df-card 9852 df-acn 9855 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-xnn0 12476 df-z 12490 df-uz 12753 df-q 12863 df-rp 12907 df-fz 13425 df-fzo 13572 df-fl 13713 df-mod 13791 df-seq 13926 df-exp 13986 df-fac 14198 df-bc 14227 df-hash 14255 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-clim 15412 df-sum 15611 df-dvds 16181 df-gcd 16423 df-prm 16600 df-pc 16766 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-0g 17362 df-gsum 17363 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18709 df-submnd 18710 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-eqg 19059 df-ghm 19146 df-gim 19192 df-ga 19223 df-cntz 19250 df-oppg 19279 df-od 19461 df-lsm 19569 df-pj1 19570 df-cmn 19715 df-abl 19716 df-dprd 19930

This theorem is referenced by: (None)

W3C validator