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 19975

Description: Lemma for pgpfac1 19979. (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 3918	. . 3 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝑆 ⊕ 𝑊))
3		pgpfac1.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
4		pgpprm 19490	. . . . . . . 8 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℙ)
6		pgpfac1.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Abel)
7		ablgrp 19682	. . . . . . . . 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 19486	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∈ ℕ)
13	8, 9, 12	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℕ)
14		pceq0 16801	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐸 ∈ ℕ) → ((𝑃 pCnt 𝐸) = 0 ↔ ¬ 𝑃 ∥ 𝐸))
15	5, 13, 14	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((𝑃 pCnt 𝐸) = 0 ↔ ¬ 𝑃 ∥ 𝐸))
16		oveq2 7361	. . . . . 6 ⊢ ((𝑃 pCnt 𝐸) = 0 → (𝑃↑(𝑃 pCnt 𝐸)) = (𝑃↑0))
17	15, 16	biimtrrdi 254	. . . . 5 ⊢ (𝜑 → (¬ 𝑃 ∥ 𝐸 → (𝑃↑(𝑃 pCnt 𝐸)) = (𝑃↑0)))
18	10	grpbn0 18863	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
19	8, 18	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ ∅)
20		hashnncl 14291	. . . . . . . . . . . . 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 16780	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝐵)) ∈ ℕ₀)
24	10, 11	gexdvds3 19487	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∥ (♯‘𝐵))
25	8, 9, 24	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∥ (♯‘𝐵))
26	10	pgphash 19504	. . . . . . . . . . . 12 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) = (𝑃↑(𝑃 pCnt (♯‘𝐵))))
27	3, 9, 26	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐵) = (𝑃↑(𝑃 pCnt (♯‘𝐵))))
28	25, 27	breqtrd 5121	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵))))
29		oveq2 7361	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑃 pCnt (♯‘𝐵)) → (𝑃↑𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝐵))))
30	29	breq2d 5107	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑃 pCnt (♯‘𝐵)) → (𝐸 ∥ (𝑃↑𝑘) ↔ 𝐸 ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))))
31	30	rspcev 3579	. . . . . . . . . 10 ⊢ (((𝑃 pCnt (♯‘𝐵)) ∈ ℕ₀ ∧ 𝐸 ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))) → ∃𝑘 ∈ ℕ₀ 𝐸 ∥ (𝑃↑𝑘))
32	23, 28, 31	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑘 ∈ ℕ₀ 𝐸 ∥ (𝑃↑𝑘))
33		pcprmpw2 16812	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐸 ∈ ℕ) → (∃𝑘 ∈ ℕ₀ 𝐸 ∥ (𝑃↑𝑘) ↔ 𝐸 = (𝑃↑(𝑃 pCnt 𝐸))))
34	5, 13, 33	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑘 ∈ ℕ₀ 𝐸 ∥ (𝑃↑𝑘) ↔ 𝐸 = (𝑃↑(𝑃 pCnt 𝐸))))
35	32, 34	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → 𝐸 = (𝑃↑(𝑃 pCnt 𝐸)))
36	35	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐸)) = 𝐸)
37		prmnn 16603	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
38	5, 37	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
39	38	nncnd 12162	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℂ)
40	39	exp0d 14065	. . . . . . 7 ⊢ (𝜑 → (𝑃↑0) = 1)
41	36, 40	eqeq12d 2745	. . . . . 6 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐸)) = (𝑃↑0) ↔ 𝐸 = 1))
42	8	grpmndd 18843	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
43	10, 11	gex1 19488	. . . . . . 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 239	. . . 4 ⊢ (𝜑 → (¬ 𝑃 ∥ 𝐸 → 𝐵 ≈ 1_o))
47		pgpfac1.s	. . . . . . . . . . 11 ⊢ 𝑆 = (𝐾‘{𝐴})
48	10	subgacs 19058	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
49	8, 48	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
50	49	acsmred 17580	. . . . . . . . . . . 12 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
51		pgpfac1.u	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
52	10	subgss 19024	. . . . . . . . . . . . . 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 3938	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
56		pgpfac1.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
57	56	mrcsncl 17536	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴 ∈ 𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
58	50, 55, 57	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
59	47, 58	eqeltrid 2832	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
60		pgpfac1.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
61		pgpfac1.l	. . . . . . . . . . 11 ⊢ ⊕ = (LSSum‘𝐺)
62	61	lsmsubg2 19756	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑊 ∈ (SubGrp‘𝐺)) → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
63	6, 59, 60, 62	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
64		pgpfac1.z	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝐺)
65	64	subg0cl 19031	. . . . . . . . 9 ⊢ ((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆 ⊕ 𝑊))
66	63, 65	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (𝑆 ⊕ 𝑊))
67	66	snssd 4763	. . . . . . 7 ⊢ (𝜑 → { 0 } ⊆ (𝑆 ⊕ 𝑊))
68	67	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → { 0 } ⊆ (𝑆 ⊕ 𝑊))
69	1	eldifad 3917	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
70	53, 69	sseldd 3938	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
71	70	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐶 ∈ 𝐵)
72	10, 64	grpidcl 18862	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
73	8, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
74		en1eqsn 9177	. . . . . . . 8 ⊢ (( 0 ∈ 𝐵 ∧ 𝐵 ≈ 1_o) → 𝐵 = { 0 })
75	73, 74	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐵 = { 0 })
76	71, 75	eleqtrd 2830	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐶 ∈ { 0 })
77	68, 76	sseldd 3938	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐶 ∈ (𝑆 ⊕ 𝑊))
78	77	ex 412	. . . 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 12162	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℂ)
83	38	nnne0d 12196	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 0)
84	82, 39, 83	divcan1d 11919	. . . . 5 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑃) = 𝐸)
85	81, 84	eqtr4d 2767	. . . 4 ⊢ (𝜑 → (𝑂‘𝐴) = ((𝐸 / 𝑃) · 𝑃))
86		nndivdvds 16190	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℕ ∧ 𝑃 ∈ ℕ) → (𝑃 ∥ 𝐸 ↔ (𝐸 / 𝑃) ∈ ℕ))
87	13, 38, 86	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∥ 𝐸 ↔ (𝐸 / 𝑃) ∈ ℕ))
88	80, 87	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑃) ∈ ℕ)
89	88	nnzd 12516	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 𝑃) ∈ ℤ)
90		pgpfac1.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
91	89, 90	zmulcld 12604	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑀) ∈ ℤ)
92	55	snssd 4763	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
93	50, 56, 92	mrcssidd 17549	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
94	93, 47	sseqtrrdi 3979	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
95		snssg 4737	. . . . . . . . . . 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 19036	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ((𝐸 / 𝑃) · 𝑀) ∈ ℤ ∧ 𝐴 ∈ 𝑆) → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝑆)
100	59, 91, 97, 99	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝑆)
101		prmz 16604	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
102	5, 101	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℤ)
103	10, 98	mulgcl 18988	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℤ ∧ 𝐶 ∈ 𝐵) → (𝑃 · 𝐶) ∈ 𝐵)
104	8, 102, 70, 103	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 · 𝐶) ∈ 𝐵)
105	10, 98	mulgcl 18988	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝐵) → (𝑀 · 𝐴) ∈ 𝐵)
106	8, 90, 55, 105	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 · 𝐴) ∈ 𝐵)
107		eqid 2729	. . . . . . . . . . . 12 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
108	10, 98, 107	mulgdi 19723	. . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 / 𝑃) · ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴))) = (((𝐸 / 𝑃) · (𝑃 · 𝐶))(+_g‘𝐺)((𝐸 / 𝑃) · (𝑀 · 𝐴))))
110	84	oveq1d 7368	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑃) · 𝐶) = (𝐸 · 𝐶))
111	10, 98	mulgass 19008	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑃) · 𝐶) = ((𝐸 / 𝑃) · (𝑃 · 𝐶)))
113	10, 11, 98, 64	gexid 19478	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝐵 → (𝐸 · 𝐶) = 0 )
114	70, 113	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 · 𝐶) = 0 )
115	110, 112, 114	3eqtr3rd 2773	. . . . . . . . . . 11 ⊢ (𝜑 → 0 = ((𝐸 / 𝑃) · (𝑃 · 𝐶)))
116	10, 98	mulgass 19008	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) = ((𝐸 / 𝑃) · (𝑀 · 𝐴)))
118	115, 117	oveq12d 7371	. . . . . . . . . 10 ⊢ (𝜑 → ( 0 (+_g‘𝐺)(((𝐸 / 𝑃) · 𝑀) · 𝐴)) = (((𝐸 / 𝑃) · (𝑃 · 𝐶))(+_g‘𝐺)((𝐸 / 𝑃) · (𝑀 · 𝐴))))
119	10	subgss 19024	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵)
120	59, 119	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
121	120, 100	sseldd 3938	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝐵)
122	10, 107, 64	grplid 18864	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝐵) → ( 0 (+_g‘𝐺)(((𝐸 / 𝑃) · 𝑀) · 𝐴)) = (((𝐸 / 𝑃) · 𝑀) · 𝐴))
123	8, 121, 122	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ( 0 (+_g‘𝐺)(((𝐸 / 𝑃) · 𝑀) · 𝐴)) = (((𝐸 / 𝑃) · 𝑀) · 𝐴))
124	109, 118, 123	3eqtr2d 2770	. . . . . . . . 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 19036	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 / 𝑃) · ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴))) ∈ 𝑊)
128	124, 127	eqeltrrd 2829	. . . . . . . 8 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝑊)
129	100, 128	elind 4153	. . . . . . 7 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ (𝑆 ∩ 𝑊))
130		pgpfac1.i	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })
131	129, 130	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ { 0 })
132		elsni 4596	. . . . . 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 19444	. . . . . 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 1373	. . . . 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 5119	. . 3 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑃) ∥ ((𝐸 / 𝑃) · 𝑀))
139	88	nnne0d 12196	. . . 4 ⊢ (𝜑 → (𝐸 / 𝑃) ≠ 0)
140		dvdscmulr 16213	. . . 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 1376	. . 3 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑃) ∥ ((𝐸 / 𝑃) · 𝑀) ↔ 𝑃 ∥ 𝑀))
142	138, 141	mpbid 232	. 2 ⊢ (𝜑 → 𝑃 ∥ 𝑀)
143	80, 142	jca 511	1 ⊢ (𝜑 → (𝑃 ∥ 𝐸 ∧ 𝑃 ∥ 𝑀))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ∖ cdif 3902 ∩ cin 3904 ⊆ wss 3905 ⊊ wpss 3906 ∅c0 4286 {csn 4579 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 1_oc1o 8388 ≈ cen 8876 Fincfn 8879 0cc0 11028 1c1 11029 · cmul 11033 / cdiv 11795 ℕcn 12146 ℕ₀cn0 12402 ℤcz 12489 ↑cexp 13986 ♯chash 14255 ∥ cdvds 16181 ℙcprime 16600 pCnt cpc 16766 Basecbs 17138 +_gcplusg 17179 0_gc0g 17361 Moorecmre 17502 mrClscmrc 17503 ACScacs 17505 Mndcmnd 18626 Grpcgrp 18830 ._gcmg 18964 SubGrpcsubg 19017 odcod 19421 gExcgex 19422 pGrp cpgp 19423 LSSumclsm 19531 Abelcabl 19678

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-disj 5063 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8632 df-ec 8634 df-qs 8638 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-oi 9421 df-dju 9816 df-card 9854 df-acn 9857 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 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 12754 df-q 12868 df-rp 12912 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612 df-dvds 16182 df-gcd 16424 df-prm 16601 df-pc 16767 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-0g 17363 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-mulg 18965 df-subg 19020 df-eqg 19022 df-ga 19187 df-cntz 19214 df-od 19425 df-gex 19426 df-pgp 19427 df-lsm 19533 df-cmn 19679 df-abl 19680

This theorem is referenced by: pgpfac1lem3 19976

W3C validator