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Theorem pgpfac1lem3a 20008

Description: Lemma for pgpfac1 20012. (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 3927	. . 3 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝑆 ⊕ 𝑊))
3		pgpfac1.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
4		pgpprm 19523	. . . . . . . 8 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℙ)
6		pgpfac1.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Abel)
7		ablgrp 19715	. . . . . . . . 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 19519	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∈ ℕ)
13	8, 9, 12	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℕ)
14		pceq0 16842	. . . . . . 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 7395	. . . . . 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 18898	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
19	8, 18	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ ∅)
20		hashnncl 14331	. . . . . . . . . . . . 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 16821	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝐵)) ∈ ℕ₀)
24	10, 11	gexdvds3 19520	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∥ (♯‘𝐵))
25	8, 9, 24	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∥ (♯‘𝐵))
26	10	pgphash 19537	. . . . . . . . . . . 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 5133	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵))))
29		oveq2 7395	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑃 pCnt (♯‘𝐵)) → (𝑃↑𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝐵))))
30	29	breq2d 5119	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑃 pCnt (♯‘𝐵)) → (𝐸 ∥ (𝑃↑𝑘) ↔ 𝐸 ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))))
31	30	rspcev 3588	. . . . . . . . . 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 16853	. . . . . . . . . 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 16644	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
38	5, 37	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
39	38	nncnd 12202	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℂ)
40	39	exp0d 14105	. . . . . . 7 ⊢ (𝜑 → (𝑃↑0) = 1)
41	36, 40	eqeq12d 2745	. . . . . 6 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐸)) = (𝑃↑0) ↔ 𝐸 = 1))
42	8	grpmndd 18878	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
43	10, 11	gex1 19521	. . . . . . 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 19093	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
49	8, 48	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
50	49	acsmred 17617	. . . . . . . . . . . 12 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
51		pgpfac1.u	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
52	10	subgss 19059	. . . . . . . . . . . . . 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 3947	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
56		pgpfac1.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
57	56	mrcsncl 17573	. . . . . . . . . . . 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 19789	. . . . . . . . . 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 19066	. . . . . . . . 9 ⊢ ((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆 ⊕ 𝑊))
66	63, 65	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (𝑆 ⊕ 𝑊))
67	66	snssd 4773	. . . . . . 7 ⊢ (𝜑 → { 0 } ⊆ (𝑆 ⊕ 𝑊))
68	67	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → { 0 } ⊆ (𝑆 ⊕ 𝑊))
69	1	eldifad 3926	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
70	53, 69	sseldd 3947	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
71	70	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐶 ∈ 𝐵)
72	10, 64	grpidcl 18897	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
73	8, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
74		en1eqsn 9219	. . . . . . . 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 3947	. . . . 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 12202	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℂ)
83	38	nnne0d 12236	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 0)
84	82, 39, 83	divcan1d 11959	. . . . 5 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑃) = 𝐸)
85	81, 84	eqtr4d 2767	. . . 4 ⊢ (𝜑 → (𝑂‘𝐴) = ((𝐸 / 𝑃) · 𝑃))
86		nndivdvds 16231	. . . . . . . . . . . . 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 12556	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 𝑃) ∈ ℤ)
90		pgpfac1.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
91	89, 90	zmulcld 12644	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑀) ∈ ℤ)
92	55	snssd 4773	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
93	50, 56, 92	mrcssidd 17586	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
94	93, 47	sseqtrrdi 3988	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
95		snssg 4747	. . . . . . . . . . 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 19071	. . . . . . . . 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 16645	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
102	5, 101	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℤ)
103	10, 98	mulgcl 19023	. . . . . . . . . . . 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 19023	. . . . . . . . . . . 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 19756	. . . . . . . . . . 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 7402	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑃) · 𝐶) = (𝐸 · 𝐶))
111	10, 98	mulgass 19043	. . . . . . . . . . . . 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 19511	. . . . . . . . . . . . 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 19043	. . . . . . . . . . . 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 7405	. . . . . . . . . 10 ⊢ (𝜑 → ( 0 (+_g‘𝐺)(((𝐸 / 𝑃) · 𝑀) · 𝐴)) = (((𝐸 / 𝑃) · (𝑃 · 𝐶))(+_g‘𝐺)((𝐸 / 𝑃) · (𝑀 · 𝐴))))
119	10	subgss 19059	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵)
120	59, 119	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
121	120, 100	sseldd 3947	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝐵)
122	10, 107, 64	grplid 18899	. . . . . . . . . . 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 19071	. . . . . . . . . 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 4163	. . . . . . 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 4606	. . . . . 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 19477	. . . . . 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 5131	. . 3 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑃) ∥ ((𝐸 / 𝑃) · 𝑀))
139	88	nnne0d 12236	. . . 4 ⊢ (𝜑 → (𝐸 / 𝑃) ≠ 0)
140		dvdscmulr 16254	. . . 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 3911 ∩ cin 3913 ⊆ wss 3914 ⊊ wpss 3915 ∅c0 4296 {csn 4589 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 1_oc1o 8427 ≈ cen 8915 Fincfn 8918 0cc0 11068 1c1 11069 · cmul 11073 / cdiv 11835 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ↑cexp 14026 ♯chash 14295 ∥ cdvds 16222 ℙcprime 16641 pCnt cpc 16807 Basecbs 17179 +_gcplusg 17220 0_gc0g 17402 Moorecmre 17543 mrClscmrc 17544 ACScacs 17546 Mndcmnd 18661 Grpcgrp 18865 ._gcmg 18999 SubGrpcsubg 19052 odcod 19454 gExcgex 19455 pGrp cpgp 19456 LSSumclsm 19564 Abelcabl 19711

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-dvds 16223 df-gcd 16465 df-prm 16642 df-pc 16808 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-eqg 19057 df-ga 19222 df-cntz 19249 df-od 19458 df-gex 19459 df-pgp 19460 df-lsm 19566 df-cmn 19712 df-abl 19713

This theorem is referenced by: pgpfac1lem3 20009

W3C validator