pgpfac1lem3a - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pgpfac1lem3a		Structured version Visualization version GIF version

Theorem pgpfac1lem3a 19946

Description: Lemma for pgpfac1 19950. (Contributed by Mario Carneiro, 4-Jun-2016.)

**Hypotheses**
Ref	Expression
pgpfac1.k	⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
pgpfac1.s	⊢ 𝑆 = (𝐾‘{𝐴})
pgpfac1.b	⊢ 𝐵 = (Base‘𝐺)
pgpfac1.o	⊢ 𝑂 = (od‘𝐺)
pgpfac1.e	⊢ 𝐸 = (gEx‘𝐺)
pgpfac1.z	⊢ 0 = (0_g‘𝐺)
pgpfac1.l	⊢ ⊕ = (LSSum‘𝐺)
pgpfac1.p	⊢ (𝜑 → 𝑃 pGrp 𝐺)
pgpfac1.g	⊢ (𝜑 → 𝐺 ∈ Abel)
pgpfac1.n	⊢ (𝜑 → 𝐵 ∈ Fin)
pgpfac1.oe	⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)
pgpfac1.u	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
pgpfac1.au	⊢ (𝜑 → 𝐴 ∈ 𝑈)
pgpfac1.w	⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
pgpfac1.i	⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })
pgpfac1.ss	⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈)
pgpfac1.2	⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))
pgpfac1.c	⊢ (𝜑 → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)))
pgpfac1.mg	⊢ · = (._g‘𝐺)
pgpfac1.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
pgpfac1.mw	⊢ (𝜑 → ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴)) ∈ 𝑊)

**Assertion**
Ref	Expression
pgpfac1lem3a	⊢ (𝜑 → (𝑃 ∥ 𝐸 ∧ 𝑃 ∥ 𝑀))

Distinct variable groups: 𝑤,𝐴 𝑤, ⊕ 𝑤,𝑃 𝑤,𝐺 𝑤,𝑈 𝑤,𝐶 𝑤,𝑆 𝑤,𝑊 𝜑,𝑤 𝑤, · 𝑤,𝐾 Allowed substitution hints: 𝐵(𝑤) 𝐸(𝑤) 𝑀(𝑤) 𝑂(𝑤) 0 (𝑤)

Proof of Theorem pgpfac1lem3a Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pgpfac1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)))
2	1	eldifbd 3962	. . 3 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝑆 ⊕ 𝑊))
3		pgpfac1.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
4		pgpprm 19461	. . . . . . . 8 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℙ)
6		pgpfac1.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Abel)
7		ablgrp 19653	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Grp)
9		pgpfac1.n	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin)
10		pgpfac1.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
11		pgpfac1.e	. . . . . . . . 9 ⊢ 𝐸 = (gEx‘𝐺)
12	10, 11	gexcl2 19457	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∈ ℕ)
13	8, 9, 12	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℕ)
14		pceq0 16804	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐸 ∈ ℕ) → ((𝑃 pCnt 𝐸) = 0 ↔ ¬ 𝑃 ∥ 𝐸))
15	5, 13, 14	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((𝑃 pCnt 𝐸) = 0 ↔ ¬ 𝑃 ∥ 𝐸))
16		oveq2 7417	. . . . . 6 ⊢ ((𝑃 pCnt 𝐸) = 0 → (𝑃↑(𝑃 pCnt 𝐸)) = (𝑃↑0))
17	15, 16	syl6bir 254	. . . . 5 ⊢ (𝜑 → (¬ 𝑃 ∥ 𝐸 → (𝑃↑(𝑃 pCnt 𝐸)) = (𝑃↑0)))
18	10	grpbn0 18851	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
19	8, 18	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ ∅)
20		hashnncl 14326	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
21	9, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
22	19, 21	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
23	5, 22	pccld 16783	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝐵)) ∈ ℕ₀)
24	10, 11	gexdvds3 19458	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∥ (♯‘𝐵))
25	8, 9, 24	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∥ (♯‘𝐵))
26	10	pgphash 19475	. . . . . . . . . . . 12 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) = (𝑃↑(𝑃 pCnt (♯‘𝐵))))
27	3, 9, 26	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐵) = (𝑃↑(𝑃 pCnt (♯‘𝐵))))
28	25, 27	breqtrd 5175	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵))))
29		oveq2 7417	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑃 pCnt (♯‘𝐵)) → (𝑃↑𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝐵))))
30	29	breq2d 5161	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑃 pCnt (♯‘𝐵)) → (𝐸 ∥ (𝑃↑𝑘) ↔ 𝐸 ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))))
31	30	rspcev 3613	. . . . . . . . . 10 ⊢ (((𝑃 pCnt (♯‘𝐵)) ∈ ℕ₀ ∧ 𝐸 ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))) → ∃𝑘 ∈ ℕ₀ 𝐸 ∥ (𝑃↑𝑘))
32	23, 28, 31	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑘 ∈ ℕ₀ 𝐸 ∥ (𝑃↑𝑘))
33		pcprmpw2 16815	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐸 ∈ ℕ) → (∃𝑘 ∈ ℕ₀ 𝐸 ∥ (𝑃↑𝑘) ↔ 𝐸 = (𝑃↑(𝑃 pCnt 𝐸))))
34	5, 13, 33	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑘 ∈ ℕ₀ 𝐸 ∥ (𝑃↑𝑘) ↔ 𝐸 = (𝑃↑(𝑃 pCnt 𝐸))))
35	32, 34	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → 𝐸 = (𝑃↑(𝑃 pCnt 𝐸)))
36	35	eqcomd 2739	. . . . . . 7 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐸)) = 𝐸)
37		prmnn 16611	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
38	5, 37	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
39	38	nncnd 12228	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℂ)
40	39	exp0d 14105	. . . . . . 7 ⊢ (𝜑 → (𝑃↑0) = 1)
41	36, 40	eqeq12d 2749	. . . . . 6 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐸)) = (𝑃↑0) ↔ 𝐸 = 1))
42	8	grpmndd 18832	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
43	10, 11	gex1 19459	. . . . . . 7 ⊢ (𝐺 ∈ Mnd → (𝐸 = 1 ↔ 𝐵 ≈ 1_o))
44	42, 43	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐸 = 1 ↔ 𝐵 ≈ 1_o))
45	41, 44	bitrd 279	. . . . 5 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐸)) = (𝑃↑0) ↔ 𝐵 ≈ 1_o))
46	17, 45	sylibd 238	. . . 4 ⊢ (𝜑 → (¬ 𝑃 ∥ 𝐸 → 𝐵 ≈ 1_o))
47		pgpfac1.s	. . . . . . . . . . 11 ⊢ 𝑆 = (𝐾‘{𝐴})
48	10	subgacs 19041	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
49	8, 48	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
50	49	acsmred 17600	. . . . . . . . . . . 12 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
51		pgpfac1.u	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
52	10	subgss 19007	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
53	51, 52	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
54		pgpfac1.au	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
55	53, 54	sseldd 3984	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
56		pgpfac1.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
57	56	mrcsncl 17556	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴 ∈ 𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
58	50, 55, 57	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
59	47, 58	eqeltrid 2838	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
60		pgpfac1.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
61		pgpfac1.l	. . . . . . . . . . 11 ⊢ ⊕ = (LSSum‘𝐺)
62	61	lsmsubg2 19727	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑊 ∈ (SubGrp‘𝐺)) → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
63	6, 59, 60, 62	syl3anc 1372	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
64		pgpfac1.z	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝐺)
65	64	subg0cl 19014	. . . . . . . . 9 ⊢ ((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆 ⊕ 𝑊))
66	63, 65	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (𝑆 ⊕ 𝑊))
67	66	snssd 4813	. . . . . . 7 ⊢ (𝜑 → { 0 } ⊆ (𝑆 ⊕ 𝑊))
68	67	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → { 0 } ⊆ (𝑆 ⊕ 𝑊))
69	1	eldifad 3961	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
70	53, 69	sseldd 3984	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
71	70	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐶 ∈ 𝐵)
72	10, 64	grpidcl 18850	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
73	8, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
74		en1eqsn 9274	. . . . . . . 8 ⊢ (( 0 ∈ 𝐵 ∧ 𝐵 ≈ 1_o) → 𝐵 = { 0 })
75	73, 74	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐵 = { 0 })
76	71, 75	eleqtrd 2836	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐶 ∈ { 0 })
77	68, 76	sseldd 3984	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐶 ∈ (𝑆 ⊕ 𝑊))
78	77	ex 414	. . . 4 ⊢ (𝜑 → (𝐵 ≈ 1_o → 𝐶 ∈ (𝑆 ⊕ 𝑊)))
79	46, 78	syld 47	. . 3 ⊢ (𝜑 → (¬ 𝑃 ∥ 𝐸 → 𝐶 ∈ (𝑆 ⊕ 𝑊)))
80	2, 79	mt3d 148	. 2 ⊢ (𝜑 → 𝑃 ∥ 𝐸)
81		pgpfac1.oe	. . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)
82	13	nncnd 12228	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℂ)
83	38	nnne0d 12262	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 0)
84	82, 39, 83	divcan1d 11991	. . . . 5 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑃) = 𝐸)
85	81, 84	eqtr4d 2776	. . . 4 ⊢ (𝜑 → (𝑂‘𝐴) = ((𝐸 / 𝑃) · 𝑃))
86		nndivdvds 16206	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℕ ∧ 𝑃 ∈ ℕ) → (𝑃 ∥ 𝐸 ↔ (𝐸 / 𝑃) ∈ ℕ))
87	13, 38, 86	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∥ 𝐸 ↔ (𝐸 / 𝑃) ∈ ℕ))
88	80, 87	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑃) ∈ ℕ)
89	88	nnzd 12585	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 𝑃) ∈ ℤ)
90		pgpfac1.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
91	89, 90	zmulcld 12672	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑀) ∈ ℤ)
92	55	snssd 4813	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
93	50, 56, 92	mrcssidd 17569	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
94	93, 47	sseqtrrdi 4034	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
95		snssg 4788	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑈 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
96	54, 95	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
97	94, 96	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
98		pgpfac1.mg	. . . . . . . . . 10 ⊢ · = (._g‘𝐺)
99	98	subgmulgcl 19019	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ((𝐸 / 𝑃) · 𝑀) ∈ ℤ ∧ 𝐴 ∈ 𝑆) → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝑆)
100	59, 91, 97, 99	syl3anc 1372	. . . . . . . 8 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝑆)
101		prmz 16612	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
102	5, 101	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℤ)
103	10, 98	mulgcl 18971	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℤ ∧ 𝐶 ∈ 𝐵) → (𝑃 · 𝐶) ∈ 𝐵)
104	8, 102, 70, 103	syl3anc 1372	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 · 𝐶) ∈ 𝐵)
105	10, 98	mulgcl 18971	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝐵) → (𝑀 · 𝐴) ∈ 𝐵)
106	8, 90, 55, 105	syl3anc 1372	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 · 𝐴) ∈ 𝐵)
107		eqid 2733	. . . . . . . . . . . 12 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
108	10, 98, 107	mulgdi 19694	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Abel ∧ ((𝐸 / 𝑃) ∈ ℤ ∧ (𝑃 · 𝐶) ∈ 𝐵 ∧ (𝑀 · 𝐴) ∈ 𝐵)) → ((𝐸 / 𝑃) · ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴))) = (((𝐸 / 𝑃) · (𝑃 · 𝐶))(+_g‘𝐺)((𝐸 / 𝑃) · (𝑀 · 𝐴))))
109	6, 89, 104, 106, 108	syl13anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 / 𝑃) · ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴))) = (((𝐸 / 𝑃) · (𝑃 · 𝐶))(+_g‘𝐺)((𝐸 / 𝑃) · (𝑀 · 𝐴))))
110	84	oveq1d 7424	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑃) · 𝐶) = (𝐸 · 𝐶))
111	10, 98	mulgass 18991	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ ((𝐸 / 𝑃) ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝐶 ∈ 𝐵)) → (((𝐸 / 𝑃) · 𝑃) · 𝐶) = ((𝐸 / 𝑃) · (𝑃 · 𝐶)))
112	8, 89, 102, 70, 111	syl13anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑃) · 𝐶) = ((𝐸 / 𝑃) · (𝑃 · 𝐶)))
113	10, 11, 98, 64	gexid 19449	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝐵 → (𝐸 · 𝐶) = 0 )
114	70, 113	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 · 𝐶) = 0 )
115	110, 112, 114	3eqtr3rd 2782	. . . . . . . . . . 11 ⊢ (𝜑 → 0 = ((𝐸 / 𝑃) · (𝑃 · 𝐶)))
116	10, 98	mulgass 18991	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ ((𝐸 / 𝑃) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝐵)) → (((𝐸 / 𝑃) · 𝑀) · 𝐴) = ((𝐸 / 𝑃) · (𝑀 · 𝐴)))
117	8, 89, 90, 55, 116	syl13anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) = ((𝐸 / 𝑃) · (𝑀 · 𝐴)))
118	115, 117	oveq12d 7427	. . . . . . . . . 10 ⊢ (𝜑 → ( 0 (+_g‘𝐺)(((𝐸 / 𝑃) · 𝑀) · 𝐴)) = (((𝐸 / 𝑃) · (𝑃 · 𝐶))(+_g‘𝐺)((𝐸 / 𝑃) · (𝑀 · 𝐴))))
119	10	subgss 19007	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵)
120	59, 119	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
121	120, 100	sseldd 3984	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝐵)
122	10, 107, 64	grplid 18852	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝐵) → ( 0 (+_g‘𝐺)(((𝐸 / 𝑃) · 𝑀) · 𝐴)) = (((𝐸 / 𝑃) · 𝑀) · 𝐴))
123	8, 121, 122	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ( 0 (+_g‘𝐺)(((𝐸 / 𝑃) · 𝑀) · 𝐴)) = (((𝐸 / 𝑃) · 𝑀) · 𝐴))
124	109, 118, 123	3eqtr2d 2779	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 / 𝑃) · ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴))) = (((𝐸 / 𝑃) · 𝑀) · 𝐴))
125		pgpfac1.mw	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴)) ∈ 𝑊)
126	98	subgmulgcl 19019	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (SubGrp‘𝐺) ∧ (𝐸 / 𝑃) ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴)) ∈ 𝑊) → ((𝐸 / 𝑃) · ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴))) ∈ 𝑊)
127	60, 89, 125, 126	syl3anc 1372	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 / 𝑃) · ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴))) ∈ 𝑊)
128	124, 127	eqeltrrd 2835	. . . . . . . 8 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝑊)
129	100, 128	elind 4195	. . . . . . 7 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ (𝑆 ∩ 𝑊))
130		pgpfac1.i	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })
131	129, 130	eleqtrd 2836	. . . . . 6 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ { 0 })
132		elsni 4646	. . . . . 6 ⊢ ((((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ { 0 } → (((𝐸 / 𝑃) · 𝑀) · 𝐴) = 0 )
133	131, 132	syl 17	. . . . 5 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) = 0 )
134		pgpfac1.o	. . . . . . 7 ⊢ 𝑂 = (od‘𝐺)
135	10, 134, 98, 64	oddvds 19415	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵 ∧ ((𝐸 / 𝑃) · 𝑀) ∈ ℤ) → ((𝑂‘𝐴) ∥ ((𝐸 / 𝑃) · 𝑀) ↔ (((𝐸 / 𝑃) · 𝑀) · 𝐴) = 0 ))
136	8, 55, 91, 135	syl3anc 1372	. . . . 5 ⊢ (𝜑 → ((𝑂‘𝐴) ∥ ((𝐸 / 𝑃) · 𝑀) ↔ (((𝐸 / 𝑃) · 𝑀) · 𝐴) = 0 ))
137	133, 136	mpbird 257	. . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∥ ((𝐸 / 𝑃) · 𝑀))
138	85, 137	eqbrtrrd 5173	. . 3 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑃) ∥ ((𝐸 / 𝑃) · 𝑀))
139	88	nnne0d 12262	. . . 4 ⊢ (𝜑 → (𝐸 / 𝑃) ≠ 0)
140		dvdscmulr 16228	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ ((𝐸 / 𝑃) ∈ ℤ ∧ (𝐸 / 𝑃) ≠ 0)) → (((𝐸 / 𝑃) · 𝑃) ∥ ((𝐸 / 𝑃) · 𝑀) ↔ 𝑃 ∥ 𝑀))
141	102, 90, 89, 139, 140	syl112anc 1375	. . 3 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑃) ∥ ((𝐸 / 𝑃) · 𝑀) ↔ 𝑃 ∥ 𝑀))
142	138, 141	mpbid 231	. 2 ⊢ (𝜑 → 𝑃 ∥ 𝑀)
143	80, 142	jca 513	1 ⊢ (𝜑 → (𝑃 ∥ 𝐸 ∧ 𝑃 ∥ 𝑀))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 ⊊ wpss 3950 ∅c0 4323 {csn 4629 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 1_oc1o 8459 ≈ cen 8936 Fincfn 8939 0cc0 11110 1c1 11111 · cmul 11115 / cdiv 11871 ℕcn 12212 ℕ₀cn0 12472 ℤcz 12558 ↑cexp 14027 ♯chash 14290 ∥ cdvds 16197 ℙcprime 16608 pCnt cpc 16769 Basecbs 17144 +_gcplusg 17197 0_gc0g 17385 Moorecmre 17526 mrClscmrc 17527 ACScacs 17529 Mndcmnd 18625 Grpcgrp 18819 ._gcmg 18950 SubGrpcsubg 19000 odcod 19392 gExcgex 19393 pGrp cpgp 19394 LSSumclsm 19502 Abelcabl 19649

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-ec 8705 df-qs 8709 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-dvds 16198 df-gcd 16436 df-prm 16609 df-pc 16770 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-0g 17387 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-eqg 19005 df-ga 19154 df-cntz 19181 df-od 19396 df-gex 19397 df-pgp 19398 df-lsm 19504 df-cmn 19650 df-abl 19651

This theorem is referenced by: pgpfac1lem3 19947

W3C validator