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Theorem ablfac1eu 20289

Description: The factorization of ablfac1b 20286 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 500	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
3		ablfac1eu.2	. . . 4 ⊢ (𝜑 → dom 𝑇 = 𝐴)
4	2, 3	dprdf2 20223	. . 3 ⊢ (𝜑 → 𝑇:𝐴⟶(SubGrp‘𝐺))
5	4	ffnd 6710	. 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 20286	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
13	6	fvexi 6899	. . . . . . 7 ⊢ 𝐵 ∈ V
14	13	rabex 5299	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
15	14, 8	dmmpti 6683	. . . . 5 ⊢ dom 𝑆 = 𝐴
16	15	a1i 11	. . . 4 ⊢ (𝜑 → dom 𝑆 = 𝐴)
17	12, 16	dprdf2 20223	. . 3 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
18	17	ffnd 6710	. 2 ⊢ (𝜑 → 𝑆 Fn 𝐴)
19	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐵 ∈ Fin)
20	17	ffvelcdmda 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
21	6	subgss 19337	. . . . 5 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑆‘𝑞) ⊆ 𝐵)
22	20, 21	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ 𝐵)
23	19, 22	ssfid 9260	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ Fin)
24	4	ffvelcdmda 7084	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ∈ (SubGrp‘𝐺))
25	6	subgss 19337	. . . . . 6 ⊢ ((𝑇‘𝑞) ∈ (SubGrp‘𝐺) → (𝑇‘𝑞) ⊆ 𝐵)
26	24, 25	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ 𝐵)
27	26	sselda 3908	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → 𝑥 ∈ 𝐵)
28	6, 7	odcl 19750	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝑂‘𝑥) ∈ ℕ₀)
29	27, 28	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∈ ℕ₀)
30	29	nn0zd 12718	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∈ ℤ)
31	19, 26	ssfid 9260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ∈ Fin)
32		hashcl 14500	. . . . . . . . 9 ⊢ ((𝑇‘𝑞) ∈ Fin → (♯‘(𝑇‘𝑞)) ∈ ℕ₀)
33	31, 32	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∈ ℕ₀)
34	33	nn0zd 12718	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∈ ℤ)
35	34	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (♯‘(𝑇‘𝑞)) ∈ ℤ)
36	11	sselda 3908	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
37		prmnn 16849	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
38	36, 37	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℕ)
39		ablgrp 19999	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
40	9, 39	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ Grp)
41	6	grpbn0 19177	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
42	40, 41	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ ∅)
43		hashnncl 14510	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
44	10, 43	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
45	42, 44	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
46	45	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ)
47	36, 46	pccld 17028	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ₀)
48	38, 47	nnexpcld 14389	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℕ)
49	48	nnzd 12719	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℤ)
50	49	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℤ)
51	24	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑇‘𝑞) ∈ (SubGrp‘𝐺))
52	31	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑇‘𝑞) ∈ Fin)
53		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → 𝑥 ∈ (𝑇‘𝑞))
54	7	odsubdvds 19785	. . . . . . 7 ⊢ (((𝑇‘𝑞) ∈ (SubGrp‘𝐺) ∧ (𝑇‘𝑞) ∈ Fin ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∥ (♯‘(𝑇‘𝑞)))
55	51, 52, 53, 54	syl3anc 1398	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∥ (♯‘(𝑇‘𝑞)))
56		ablfac1eu.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))
57		prmz 16850	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
58	36, 57	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℤ)
59		ablfac1eu.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℕ₀)
60	59	nn0zd 12718	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℤ)
61	47	nn0zd 12718	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℤ)
62	6	lagsubg 19410	. . . . . . . . . . . . 13 ⊢ (((𝑇‘𝑞) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (♯‘(𝑇‘𝑞)) ∥ (♯‘𝐵))
63	24, 19, 62	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∥ (♯‘𝐵))
64	56, 63	eqbrtrrd 5128	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑𝐶) ∥ (♯‘𝐵))
65	46	nnzd 12719	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) ∈ ℤ)
66		pcdvdsb 17047	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℙ ∧ (♯‘𝐵) ∈ ℤ ∧ 𝐶 ∈ ℕ₀) → (𝐶 ≤ (𝑞 pCnt (♯‘𝐵)) ↔ (𝑞↑𝐶) ∥ (♯‘𝐵)))
67	36, 65, 59, 66	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐶 ≤ (𝑞 pCnt (♯‘𝐵)) ↔ (𝑞↑𝐶) ∥ (♯‘𝐵)))
68	64, 67	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ≤ (𝑞 pCnt (♯‘𝐵)))
69		eluz2 12971	. . . . . . . . . 10 ⊢ ((𝑞 pCnt (♯‘𝐵)) ∈ (ℤ_≥‘𝐶) ↔ (𝐶 ∈ ℤ ∧ (𝑞 pCnt (♯‘𝐵)) ∈ ℤ ∧ 𝐶 ≤ (𝑞 pCnt (♯‘𝐵))))
70	60, 61, 68, 69	syl3anbrc 1362	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ (ℤ_≥‘𝐶))
71		dvdsexp 16498	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℤ ∧ 𝐶 ∈ ℕ₀ ∧ (𝑞 pCnt (♯‘𝐵)) ∈ (ℤ_≥‘𝐶)) → (𝑞↑𝐶) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
72	58, 59, 70, 71	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑𝐶) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
73	56, 72	eqbrtrd 5126	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
74	73	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (♯‘(𝑇‘𝑞)) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
75	30, 35, 50, 55, 74	dvdstrd 16465	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑥 ∈ (𝑇‘𝑞)) → (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵))))
76	26, 75	ssrabdv 3998	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
77		id 23	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → 𝑝 = 𝑞)
78		oveq1 7427	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → (𝑝 pCnt (♯‘𝐵)) = (𝑞 pCnt (♯‘𝐵)))
79	77, 78	oveq12d 7438	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
80	79	breq2d 5114	. . . . . . 7 ⊢ (𝑝 = 𝑞 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))))
81	80	rabbidv 3419	. . . . . 6 ⊢ (𝑝 = 𝑞 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
82	81, 8, 14	fvmpt3i 6999	. . . . 5 ⊢ (𝑞 ∈ 𝐴 → (𝑆‘𝑞) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
83	82	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑞↑(𝑞 pCnt (♯‘𝐵)))})
84	76, 83	sseqtrrd 3945	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ⊆ (𝑆‘𝑞))
85	48	nnnn0d 12667	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∈ ℕ₀)
86		pcdvds 17042	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
87	36, 46, 86	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
88	2	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd 𝑇)
89	3	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom 𝑇 = 𝐴)
90		ablfac1.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
91	90	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
92	88, 89, 91	dprdres 20244	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑇)))
93	92	simpld 500	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd (𝑇 ↾ 𝐷))
94	4	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇:𝐴⟶(SubGrp‘𝐺))
95	94, 91	fssresd 6749	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
96	95	fdmd 6720	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom (𝑇 ↾ 𝐷) = 𝐷)
97		difssd 4073	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐷 ∖ {𝑞}) ⊆ 𝐷)
98	93, 96, 97	dprdres 20244	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})) ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷))))
99	98	simpld 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))
100		dprdsubg 20240	. . . . . . . . . . 11 ⊢ (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺))
101	99, 100	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺))
102	6	lagsubg 19410	. . . . . . . . . 10 ⊢ (((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∥ (♯‘𝐵))
103	101, 19, 102	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∥ (♯‘𝐵))
104		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
105	104	subg0cl 19344	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))
106	101, 105	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (0_g‘𝐺) ∈ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))
107	106	ne0d 4287	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ≠ ∅)
108	6	dprdssv 20232	. . . . . . . . . . . . . 14 ⊢ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ⊆ 𝐵
109		ssfi 9188	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ⊆ 𝐵) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ Fin)
110	19, 108, 109	sylancl 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ∈ Fin)
111		hashnncl 14510	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℕ ↔ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))) ≠ ∅))
113	107, 112	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℕ)
114	113	nnzd 12719	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ∈ ℤ)
115		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑞 → 𝑥 = 𝑞)
116		sneq 4593	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑞 → {𝑥} = {𝑞})
117	116	difeq2d 4063	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑞 → (𝐷 ∖ {𝑥}) = (𝐷 ∖ {𝑞}))
118	117	reseq2d 5969	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑞 → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})) = ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))
119	118	oveq2d 7436	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑞 → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥}))) = (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))
120	119	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑞 → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))) = (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
121	115, 120	breq12d 5115	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))) ↔ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
122	121	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑞 → (¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))) ↔ ¬ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
123		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ (𝑂‘𝑦) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) = (𝑝 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ (𝑂‘𝑦) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
124	9	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐺 ∈ Abel)
125	10	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐵 ∈ Fin)
126		ablfac1c.d	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
127	126	ssrab3 4007	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 ⊆ ℙ
128	127	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐷 ⊆ ℙ)
129		ssidd 3931	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐷 ⊆ 𝐷)
130	2, 3, 90	dprdres 20244	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) ⊆ (𝐺 DProd 𝑇)))
131	130	simpld 500	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ 𝐷))
132		dprdsubg 20240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺dom DProd (𝑇 ↾ 𝐷) → (𝐺 DProd (𝑇 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
133	131, 132	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ 𝐷)) ∈ (SubGrp‘𝐺))
134		difssd 4073	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ 𝐴)
135	2, 3, 134	dprdres 20244	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐺dom DProd (𝑇 ↾ (𝐴 ∖ 𝐷)) ∧ (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ⊆ (𝐺 DProd 𝑇)))
136	135	simpld 500	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))
137		dprdsubg 20240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺dom DProd (𝑇 ↾ (𝐴 ∖ 𝐷)) → (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ∈ (SubGrp‘𝐺))
138	136, 137	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ∈ (SubGrp‘𝐺))
139		difss 4072	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∖ 𝐷) ⊆ 𝐴
140		fssres 6748	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑇:𝐴⟶(SubGrp‘𝐺) ∧ (𝐴 ∖ 𝐷) ⊆ 𝐴) → (𝑇 ↾ (𝐴 ∖ 𝐷)):(𝐴 ∖ 𝐷)⟶(SubGrp‘𝐺))
141	4, 139, 140	sylancl 598	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑇 ↾ (𝐴 ∖ 𝐷)):(𝐴 ∖ 𝐷)⟶(SubGrp‘𝐺))
142	141	fdmd 6720	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom (𝑇 ↾ (𝐴 ∖ 𝐷)) = (𝐴 ∖ 𝐷))
143		fvres 6904	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑞 ∈ (𝐴 ∖ 𝐷) → ((𝑇 ↾ (𝐴 ∖ 𝐷))‘𝑞) = (𝑇‘𝑞))
144	143	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑞 ∈ (𝐴 ∖ 𝐷)) → ((𝑇 ↾ (𝐴 ∖ 𝐷))‘𝑞) = (𝑇‘𝑞))
145		eldif 4049	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑞 ∈ (𝐴 ∖ 𝐷) ↔ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷))
146	31	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑇‘𝑞) ∈ Fin)
147	104	subg0cl 19344	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑇‘𝑞) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝑇‘𝑞))
148	24, 147	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (0_g‘𝐺) ∈ (𝑇‘𝑞))
149	148	snssd 4746	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {(0_g‘𝐺)} ⊆ (𝑇‘𝑞))
150	149	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} ⊆ (𝑇‘𝑞))
151		fvex 6898	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0_g‘𝐺) ∈ V
152		hashsng 14513	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0_g‘𝐺) ∈ V → (♯‘{(0_g‘𝐺)}) = 1)
153	151, 152	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (♯‘{(0_g‘𝐺)}) = 1
154	56	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))
155	36	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∈ ℙ)
156		iddvdsexp 16449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑞 ∈ ℤ ∧ 𝐶 ∈ ℕ) → 𝑞 ∥ (𝑞↑𝐶))
157	58, 156	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∥ (𝑞↑𝐶))
158	64	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → (𝑞↑𝐶) ∥ (♯‘𝐵))
159	56, 34	eqeltrrd 2861	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑𝐶) ∈ ℤ)
160		dvdstr 16464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑞 ∈ ℤ ∧ (𝑞↑𝐶) ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → ((𝑞 ∥ (𝑞↑𝐶) ∧ (𝑞↑𝐶) ∥ (♯‘𝐵)) → 𝑞 ∥ (♯‘𝐵)))
161	58, 159, 65, 160	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞 ∥ (𝑞↑𝐶) ∧ (𝑞↑𝐶) ∥ (♯‘𝐵)) → 𝑞 ∥ (♯‘𝐵)))
162	161	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → ((𝑞 ∥ (𝑞↑𝐶) ∧ (𝑞↑𝐶) ∥ (♯‘𝐵)) → 𝑞 ∥ (♯‘𝐵)))
163	157, 158, 162	mp2and 712	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∥ (♯‘𝐵))
164		breq1 5105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑤 = 𝑞 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑞 ∥ (♯‘𝐵)))
165	164, 126	elrab2 3648	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑞 ∈ 𝐷 ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ (♯‘𝐵)))
166	155, 163, 165	sylanbrc 595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝐶 ∈ ℕ) → 𝑞 ∈ 𝐷)
167	166	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐶 ∈ ℕ → 𝑞 ∈ 𝐷))
168	167	con3d 153	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞 ∈ 𝐷 → ¬ 𝐶 ∈ ℕ))
169	168	impr 460	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ¬ 𝐶 ∈ ℕ)
170	59	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝐶 ∈ ℕ₀)
171		elnn0 12608	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐶 ∈ ℕ₀ ↔ (𝐶 ∈ ℕ ∨ 𝐶 = 0))
172	170, 171	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐶 ∈ ℕ ∨ 𝐶 = 0))
173	172	ord 878	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (¬ 𝐶 ∈ ℕ → 𝐶 = 0))
174	169, 173	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝐶 = 0)
175	174	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑞↑𝐶) = (𝑞↑0))
176	38	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝑞 ∈ ℕ)
177	176	nncnd 12351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → 𝑞 ∈ ℂ)
178	177	exp0d 14283	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑞↑0) = 1)
179	154, 175, 178	3eqtrd 2799	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (♯‘(𝑇‘𝑞)) = 1)
180	153, 179	eqtr4id 2814	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (♯‘{(0_g‘𝐺)}) = (♯‘(𝑇‘𝑞)))
181		snfi 9071	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ {(0_g‘𝐺)} ∈ Fin
182		hashen 14491	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (({(0_g‘𝐺)} ∈ Fin ∧ (𝑇‘𝑞) ∈ Fin) → ((♯‘{(0_g‘𝐺)}) = (♯‘(𝑇‘𝑞)) ↔ {(0_g‘𝐺)} ≈ (𝑇‘𝑞)))
183	181, 146, 182	sylancr 599	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ((♯‘{(0_g‘𝐺)}) = (♯‘(𝑇‘𝑞)) ↔ {(0_g‘𝐺)} ≈ (𝑇‘𝑞)))
184	180, 183	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} ≈ (𝑇‘𝑞))
185		fisseneq 9254	. . . . . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} = (𝑇‘𝑞))
187	104	subg0cl 19344	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐺 DProd (𝑇 ↾ 𝐷)) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd (𝑇 ↾ 𝐷)))
188	133, 187	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (0_g‘𝐺) ∈ (𝐺 DProd (𝑇 ↾ 𝐷)))
189	188	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (0_g‘𝐺) ∈ (𝐺 DProd (𝑇 ↾ 𝐷)))
190	189	snssd 4746	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → {(0_g‘𝐺)} ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
191	186, 190	eqsstrrd 3943	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝑇‘𝑞) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
192	145, 191	sylan2b 606	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑞 ∈ (𝐴 ∖ 𝐷)) → (𝑇‘𝑞) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
193	144, 192	eqsstrd 3942	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑞 ∈ (𝐴 ∖ 𝐷)) → ((𝑇 ↾ (𝐴 ∖ 𝐷))‘𝑞) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
194	136, 142, 133, 193	dprdlub 20242	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷))) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷)))
195		eqid 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
196	195	lsmss2 19881	. . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ 𝐷))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))) = (𝐺 DProd (𝑇 ↾ 𝐷)))
198		disjdif 4425	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∩ (𝐴 ∖ 𝐷)) = ∅
199	198	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐷 ∩ (𝐴 ∖ 𝐷)) = ∅)
200		undif2 4430	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∪ (𝐴 ∖ 𝐷)) = (𝐷 ∪ 𝐴)
201		ssequn1 4184	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐷 ⊆ 𝐴 ↔ (𝐷 ∪ 𝐴) = 𝐴)
202	90, 201	sylib 221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐷 ∪ 𝐴) = 𝐴)
203	200, 202	eqtr2id 2808	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 = (𝐷 ∪ (𝐴 ∖ 𝐷)))
204	4, 199, 203, 195, 2	dprdsplit 20264	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐺 DProd (𝑇 ↾ 𝐷))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))))
205	1	simprd 501	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 DProd 𝑇) = 𝐵)
206	204, 205	eqtr3d 2797	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ 𝐷))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ (𝐴 ∖ 𝐷)))) = 𝐵)
207	197, 206	eqtr3d 2797	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵)
208	131, 207	jca 521	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵))
209	208	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → (𝐺dom DProd (𝑇 ↾ 𝐷) ∧ (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵))
210	4, 90	fssresd 6749	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑇 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺))
211	210	fdmd 6720	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom (𝑇 ↾ 𝐷) = 𝐷)
212	211	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → dom (𝑇 ↾ 𝐷) = 𝐷)
213	90	sselda 3908	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → 𝑞 ∈ 𝐴)
214	213, 59	syldan 603	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → 𝐶 ∈ ℕ₀)
215	214	adantlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℙ) ∧ 𝑞 ∈ 𝐷) → 𝐶 ∈ ℕ₀)
216		fvres 6904	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 ∈ 𝐷 → ((𝑇 ↾ 𝐷)‘𝑞) = (𝑇‘𝑞))
217	216	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → ((𝑇 ↾ 𝐷)‘𝑞) = (𝑇‘𝑞))
218	217	fveq2d 6889	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → (♯‘((𝑇 ↾ 𝐷)‘𝑞)) = (♯‘(𝑇‘𝑞)))
219	213, 56	syldan 603	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶))
220	218, 219	eqtrd 2795	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐷) → (♯‘((𝑇 ↾ 𝐷)‘𝑞)) = (𝑞↑𝐶))
221	220	adantlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℙ) ∧ 𝑞 ∈ 𝐷) → (♯‘((𝑇 ↾ 𝐷)‘𝑞)) = (𝑞↑𝐶))
222		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝑥 ∈ ℙ)
223		fzfid 14116	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...(♯‘𝐵)) ∈ Fin)
224		prmnn 16849	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℕ)
225	224	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ ℕ)
226		prmz 16850	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℤ)
227		dvdsle 16480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℤ ∧ (♯‘𝐵) ∈ ℕ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
228	226, 45, 227	syl2anr 609	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
229	228	3impia 1135	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ≤ (♯‘𝐵))
230	45	nnzd 12719	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
231	230	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (♯‘𝐵) ∈ ℤ)
232		fznn 13726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐵) ∈ ℤ → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
233	231, 232	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
234	225, 229, 233	mpbir2and 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ (1...(♯‘𝐵)))
235	234	rabssdv 3999	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ⊆ (1...(♯‘𝐵)))
236	126, 235	eqsstrid 3946	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 ⊆ (1...(♯‘𝐵)))
237	223, 236	ssfid 9260	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐷 ∈ Fin)
238	237	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → 𝐷 ∈ Fin)
239	6, 7, 123, 124, 125, 128, 126, 129, 209, 212, 215, 221, 222, 238	ablfac1eulem 20288	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℙ) → ¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))))
240	239	ralrimiva 3154	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ ℙ ¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))))
241	240	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ∀𝑥 ∈ ℙ ¬ 𝑥 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑥})))))
242	122, 241, 36	rspcdva 3577	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ¬ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
243		coprm 16887	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞 ∥ (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ↔ (𝑞 gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1))
245	242, 244	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1)
246		rpexp1i 16899	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 16	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) gcd (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = 1)
249		coprmdvds2 16829	. . . . . . . . . 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 1400	. . . . . . . . 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 712	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞↑(𝑞 pCnt (♯‘𝐵))) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) ∥ (♯‘𝐵))
252		eqid 2760	. . . . . . . . . 10 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
253		inss1 4133	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∩ {𝑞}) ⊆ 𝐷
254	253	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐷 ∩ {𝑞}) ⊆ 𝐷)
255	93, 96, 254	dprdres 20244	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ (𝐺 DProd (𝑇 ↾ 𝐷))))
256	255	simpld 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))
257		dprdsubg 20240	. . . . . . . . . . 11 ⊢ (𝐺dom DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ (SubGrp‘𝐺))
258	256, 257	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ (SubGrp‘𝐺))
259		inass 4124	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∩ {𝑞}) ∩ (𝐷 ∖ {𝑞})) = (𝐷 ∩ ({𝑞} ∩ (𝐷 ∖ {𝑞})))
260		disjdif 4425	. . . . . . . . . . . . . 14 ⊢ ({𝑞} ∩ (𝐷 ∖ {𝑞})) = ∅
261	260	ineq2i 4114	. . . . . . . . . . . . 13 ⊢ (𝐷 ∩ ({𝑞} ∩ (𝐷 ∖ {𝑞}))) = (𝐷 ∩ ∅)
262		in0 4344	. . . . . . . . . . . . 13 ⊢ (𝐷 ∩ ∅) = ∅
263	259, 261, 262	3eqtri 2787	. . . . . . . . . . . 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 20255	. . . . . . . . . 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 20254	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))))
267	6	dprdssv 20232	. . . . . . . . . . 11 ⊢ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ 𝐵
268		ssfi 9188	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ⊆ 𝐵) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ Fin)
269	19, 267, 268	sylancl 598	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) ∈ Fin)
270	195, 104, 252, 258, 101, 265, 266, 269, 110	lsmhash 19919	. . . . . . . . 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 4434	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∩ {𝑞}) ∪ (𝐷 ∖ {𝑞})) = 𝐷
272	271	eqcomi 2769	. . . . . . . . . . . . 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 20264	. . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd (𝑇 ↾ 𝐷)) = 𝐵)
276	274, 275	eqtr3d 2797	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) = 𝐵)
277	276	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘((𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))) = (♯‘𝐵))
278		snssi 4745	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ 𝐷 → {𝑞} ⊆ 𝐷)
279	278	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → {𝑞} ⊆ 𝐷)
280		sseqin2 4120	. . . . . . . . . . . . . . . 16 ⊢ ({𝑞} ⊆ 𝐷 ↔ (𝐷 ∩ {𝑞}) = {𝑞})
281	279, 280	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐷 ∩ {𝑞}) = {𝑞})
282	281	reseq2d 5969	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ((𝑇 ↾ 𝐷) ↾ {𝑞}))
283	282	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ {𝑞})))
284	93	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → 𝐺dom DProd (𝑇 ↾ 𝐷))
285	211	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → dom (𝑇 ↾ 𝐷) = 𝐷)
286		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → 𝑞 ∈ 𝐷)
287	284, 285, 286	dpjlem 20267	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ {𝑞})) = ((𝑇 ↾ 𝐷)‘𝑞))
288	216	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → ((𝑇 ↾ 𝐷)‘𝑞) = (𝑇‘𝑞))
289	283, 287, 288	3eqtrd 2799	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
290		simprr 785	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ¬ 𝑞 ∈ 𝐷)
291		disjsn 4671	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∩ {𝑞}) = ∅ ↔ ¬ 𝑞 ∈ 𝐷)
292	290, 291	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐷 ∩ {𝑞}) = ∅)
293	292	reseq2d 5969	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ((𝑇 ↾ 𝐷) ↾ ∅))
294		res0 5973	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ↾ 𝐷) ↾ ∅) = ∅
295	293, 294	eqtrdi 2811	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})) = ∅)
296	295	oveq2d 7436	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝐺 DProd ∅))
297	104	dprd0 20247	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0_g‘𝐺)}))
298	40, 297	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0_g‘𝐺)}))
299	298	simprd 501	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺 DProd ∅) = {(0_g‘𝐺)})
300	299	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐺 DProd ∅) = {(0_g‘𝐺)})
301	296, 300, 186	3eqtrd 2799	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ∈ 𝐷)) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
302	301	anassrs 473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ ¬ 𝑞 ∈ 𝐷) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
303	289, 302	pm2.61dan 825	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞}))) = (𝑇‘𝑞))
304	303	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∩ {𝑞})))) = (♯‘(𝑇‘𝑞)))
305	304	oveq1d 7435	. . . . . . . . 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 2803	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) = ((♯‘(𝑇‘𝑞)) · (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞}))))))
307	251, 306	breqtrd 5130	. . . . . . 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 12388	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝐺 DProd ((𝑇 ↾ 𝐷) ↾ (𝐷 ∖ {𝑞})))) ≠ 0)
309		dvdsmulcr 16455	. . . . . . . 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 1401	. . . . . . 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 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘𝐵))) ∥ (♯‘(𝑇‘𝑞)))
312		dvdseq 16484	. . . . . 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 852	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
314	6, 7, 8, 9, 10, 11	ablfac1a 20285	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
315	313, 314	eqtr4d 2798	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (♯‘(𝑆‘𝑞)))
316		hashen 14491	. . . . 5 ⊢ (((𝑇‘𝑞) ∈ Fin ∧ (𝑆‘𝑞) ∈ Fin) → ((♯‘(𝑇‘𝑞)) = (♯‘(𝑆‘𝑞)) ↔ (𝑇‘𝑞) ≈ (𝑆‘𝑞)))
317	31, 23, 316	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((♯‘(𝑇‘𝑞)) = (♯‘(𝑆‘𝑞)) ↔ (𝑇‘𝑞) ≈ (𝑆‘𝑞)))
318	315, 317	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) ≈ (𝑆‘𝑞))
319		fisseneq 9254	. . 3 ⊢ (((𝑆‘𝑞) ∈ Fin ∧ (𝑇‘𝑞) ⊆ (𝑆‘𝑞) ∧ (𝑇‘𝑞) ≈ (𝑆‘𝑞)) → (𝑇‘𝑞) = (𝑆‘𝑞))
320	23, 84, 318, 319	syl3anc 1398	. 2 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑇‘𝑞) = (𝑆‘𝑞))
321	5, 18, 320	eqfnfvd 7032	1 ⊢ (𝜑 → 𝑇 = 𝑆)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 {crab 3412 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ∩ cin 4089 ∪ cun 4149 ∅c0 4278 {csn 4583 class class class wbr 5102 ↦ cmpt 5185 dom cdm 5650 ↾ cres 5652 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ≈ cen 8970 Fincfn 8973 0cc0 11200 1c1 11201 · cmul 11205 ≤ cle 11344 ℕcn 12335 ℕ₀cn0 12606 ℤcz 12693 ℤ_≥cuz 12965 ...cfz 13639 ↑cexp 14204 ♯chash 14474 ∥ cdvds 16422 gcd cgcd 16664 ℙcprime 16846 pCnt cpc 17014 Basecbs 17387 0_gc0g 17610 Grpcgrp 19144 SubGrpcsubg 19330 Cntzccntz 19529 odcod 19738 LSSumclsm 19848 Abelcabl 19995 DProd cdprd 20209

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-tpos 8243 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-omul 8481 df-er 8717 df-ec 8719 df-qs 8723 df-map 8849 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-sup 9434 df-inf 9435 df-oi 9504 df-dju 9982 df-card 10020 df-acn 10023 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-xnn0 12680 df-z 12694 df-uz 12966 df-q 13076 df-rp 13121 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-sum 15854 df-dvds 16423 df-gcd 16665 df-prm 16847 df-pc 17015 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-0g 17612 df-gsum 17613 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-submnd 18979 df-grp 19147 df-minusg 19148 df-sbg 19149 df-mulg 19278 df-subg 19333 df-eqg 19335 df-ghm 19428 df-gim 19473 df-ga 19504 df-cntz 19531 df-oppg 19560 df-od 19742 df-lsm 19850 df-pj1 19851 df-cmn 19996 df-abl 19997 df-dprd 20211

This theorem is used by: (None)

W3C validator