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Theorem pgpfac1lem3a 19940

Description: Lemma for pgpfac1 19944. (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 3960	. . 3 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝑆 ⊕ 𝑊))
3		pgpfac1.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
4		pgpprm 19455	. . . . . . . 8 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℙ)
6		pgpfac1.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Abel)
7		ablgrp 19647	. . . . . . . . 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 19451	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∈ ℕ)
13	8, 9, 12	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℕ)
14		pceq0 16800	. . . . . . 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 7413	. . . . . 6 ⊢ ((𝑃 pCnt 𝐸) = 0 → (𝑃↑(𝑃 pCnt 𝐸)) = (𝑃↑0))
17	15, 16	syl6bir 253	. . . . 5 ⊢ (𝜑 → (¬ 𝑃 ∥ 𝐸 → (𝑃↑(𝑃 pCnt 𝐸)) = (𝑃↑0)))
18	10	grpbn0 18847	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
19	8, 18	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ ∅)
20		hashnncl 14322	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
21	9, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
22	19, 21	mpbird 256	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
23	5, 22	pccld 16779	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝐵)) ∈ ℕ₀)
24	10, 11	gexdvds3 19452	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → 𝐸 ∥ (♯‘𝐵))
25	8, 9, 24	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∥ (♯‘𝐵))
26	10	pgphash 19469	. . . . . . . . . . . 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 5173	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵))))
29		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑃 pCnt (♯‘𝐵)) → (𝑃↑𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝐵))))
30	29	breq2d 5159	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑃 pCnt (♯‘𝐵)) → (𝐸 ∥ (𝑃↑𝑘) ↔ 𝐸 ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))))
31	30	rspcev 3612	. . . . . . . . . 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 16811	. . . . . . . . . 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 231	. . . . . . . 8 ⊢ (𝜑 → 𝐸 = (𝑃↑(𝑃 pCnt 𝐸)))
36	35	eqcomd 2738	. . . . . . 7 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝐸)) = 𝐸)
37		prmnn 16607	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
38	5, 37	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
39	38	nncnd 12224	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℂ)
40	39	exp0d 14101	. . . . . . 7 ⊢ (𝜑 → (𝑃↑0) = 1)
41	36, 40	eqeq12d 2748	. . . . . 6 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝐸)) = (𝑃↑0) ↔ 𝐸 = 1))
42	8	grpmndd 18828	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
43	10, 11	gex1 19453	. . . . . . 7 ⊢ (𝐺 ∈ Mnd → (𝐸 = 1 ↔ 𝐵 ≈ 1_o))
44	42, 43	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐸 = 1 ↔ 𝐵 ≈ 1_o))
45	41, 44	bitrd 278	. . . . 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 19035	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
49	8, 48	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
50	49	acsmred 17596	. . . . . . . . . . . 12 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
51		pgpfac1.u	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
52	10	subgss 19001	. . . . . . . . . . . . . 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 3982	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
56		pgpfac1.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
57	56	mrcsncl 17552	. . . . . . . . . . . 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 2837	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
60		pgpfac1.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
61		pgpfac1.l	. . . . . . . . . . 11 ⊢ ⊕ = (LSSum‘𝐺)
62	61	lsmsubg2 19721	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑊 ∈ (SubGrp‘𝐺)) → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
63	6, 59, 60, 62	syl3anc 1371	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
64		pgpfac1.z	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝐺)
65	64	subg0cl 19008	. . . . . . . . 9 ⊢ ((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆 ⊕ 𝑊))
66	63, 65	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (𝑆 ⊕ 𝑊))
67	66	snssd 4811	. . . . . . 7 ⊢ (𝜑 → { 0 } ⊆ (𝑆 ⊕ 𝑊))
68	67	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → { 0 } ⊆ (𝑆 ⊕ 𝑊))
69	1	eldifad 3959	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
70	53, 69	sseldd 3982	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
71	70	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐶 ∈ 𝐵)
72	10, 64	grpidcl 18846	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
73	8, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
74		en1eqsn 9270	. . . . . . . 8 ⊢ (( 0 ∈ 𝐵 ∧ 𝐵 ≈ 1_o) → 𝐵 = { 0 })
75	73, 74	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐵 = { 0 })
76	71, 75	eleqtrd 2835	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐶 ∈ { 0 })
77	68, 76	sseldd 3982	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≈ 1_o) → 𝐶 ∈ (𝑆 ⊕ 𝑊))
78	77	ex 413	. . . 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 12224	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℂ)
83	38	nnne0d 12258	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 0)
84	82, 39, 83	divcan1d 11987	. . . . 5 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑃) = 𝐸)
85	81, 84	eqtr4d 2775	. . . 4 ⊢ (𝜑 → (𝑂‘𝐴) = ((𝐸 / 𝑃) · 𝑃))
86		nndivdvds 16202	. . . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑃) ∈ ℕ)
89	88	nnzd 12581	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 𝑃) ∈ ℤ)
90		pgpfac1.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
91	89, 90	zmulcld 12668	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑀) ∈ ℤ)
92	55	snssd 4811	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
93	50, 56, 92	mrcssidd 17565	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
94	93, 47	sseqtrrdi 4032	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
95		snssg 4786	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑈 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
96	54, 95	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
97	94, 96	mpbird 256	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
98		pgpfac1.mg	. . . . . . . . . 10 ⊢ · = (._g‘𝐺)
99	98	subgmulgcl 19013	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ((𝐸 / 𝑃) · 𝑀) ∈ ℤ ∧ 𝐴 ∈ 𝑆) → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝑆)
100	59, 91, 97, 99	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝑆)
101		prmz 16608	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
102	5, 101	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℤ)
103	10, 98	mulgcl 18965	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℤ ∧ 𝐶 ∈ 𝐵) → (𝑃 · 𝐶) ∈ 𝐵)
104	8, 102, 70, 103	syl3anc 1371	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 · 𝐶) ∈ 𝐵)
105	10, 98	mulgcl 18965	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝐵) → (𝑀 · 𝐴) ∈ 𝐵)
106	8, 90, 55, 105	syl3anc 1371	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 · 𝐴) ∈ 𝐵)
107		eqid 2732	. . . . . . . . . . . 12 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
108	10, 98, 107	mulgdi 19688	. . . . . . . . . . 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 1372	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 / 𝑃) · ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴))) = (((𝐸 / 𝑃) · (𝑃 · 𝐶))(+_g‘𝐺)((𝐸 / 𝑃) · (𝑀 · 𝐴))))
110	84	oveq1d 7420	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑃) · 𝐶) = (𝐸 · 𝐶))
111	10, 98	mulgass 18985	. . . . . . . . . . . . 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 1372	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑃) · 𝐶) = ((𝐸 / 𝑃) · (𝑃 · 𝐶)))
113	10, 11, 98, 64	gexid 19443	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝐵 → (𝐸 · 𝐶) = 0 )
114	70, 113	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 · 𝐶) = 0 )
115	110, 112, 114	3eqtr3rd 2781	. . . . . . . . . . 11 ⊢ (𝜑 → 0 = ((𝐸 / 𝑃) · (𝑃 · 𝐶)))
116	10, 98	mulgass 18985	. . . . . . . . . . . 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 1372	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) = ((𝐸 / 𝑃) · (𝑀 · 𝐴)))
118	115, 117	oveq12d 7423	. . . . . . . . . 10 ⊢ (𝜑 → ( 0 (+_g‘𝐺)(((𝐸 / 𝑃) · 𝑀) · 𝐴)) = (((𝐸 / 𝑃) · (𝑃 · 𝐶))(+_g‘𝐺)((𝐸 / 𝑃) · (𝑀 · 𝐴))))
119	10	subgss 19001	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵)
120	59, 119	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
121	120, 100	sseldd 3982	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝐵)
122	10, 107, 64	grplid 18848	. . . . . . . . . . 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 2778	. . . . . . . . 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 19013	. . . . . . . . . 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 1371	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 / 𝑃) · ((𝑃 · 𝐶)(+_g‘𝐺)(𝑀 · 𝐴))) ∈ 𝑊)
128	124, 127	eqeltrrd 2834	. . . . . . . 8 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ 𝑊)
129	100, 128	elind 4193	. . . . . . 7 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ (𝑆 ∩ 𝑊))
130		pgpfac1.i	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })
131	129, 130	eleqtrd 2835	. . . . . 6 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑀) · 𝐴) ∈ { 0 })
132		elsni 4644	. . . . . 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 19409	. . . . . 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 1371	. . . . 5 ⊢ (𝜑 → ((𝑂‘𝐴) ∥ ((𝐸 / 𝑃) · 𝑀) ↔ (((𝐸 / 𝑃) · 𝑀) · 𝐴) = 0 ))
137	133, 136	mpbird 256	. . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∥ ((𝐸 / 𝑃) · 𝑀))
138	85, 137	eqbrtrrd 5171	. . 3 ⊢ (𝜑 → ((𝐸 / 𝑃) · 𝑃) ∥ ((𝐸 / 𝑃) · 𝑀))
139	88	nnne0d 12258	. . . 4 ⊢ (𝜑 → (𝐸 / 𝑃) ≠ 0)
140		dvdscmulr 16224	. . . 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 1374	. . 3 ⊢ (𝜑 → (((𝐸 / 𝑃) · 𝑃) ∥ ((𝐸 / 𝑃) · 𝑀) ↔ 𝑃 ∥ 𝑀))
142	138, 141	mpbid 231	. 2 ⊢ (𝜑 → 𝑃 ∥ 𝑀)
143	80, 142	jca 512	1 ⊢ (𝜑 → (𝑃 ∥ 𝐸 ∧ 𝑃 ∥ 𝑀))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∖ cdif 3944 ∩ cin 3946 ⊆ wss 3947 ⊊ wpss 3948 ∅c0 4321 {csn 4627 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 1_oc1o 8455 ≈ cen 8932 Fincfn 8935 0cc0 11106 1c1 11107 · cmul 11111 / cdiv 11867 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ↑cexp 14023 ♯chash 14286 ∥ cdvds 16193 ℙcprime 16604 pCnt cpc 16765 Basecbs 17140 +_gcplusg 17193 0_gc0g 17381 Moorecmre 17522 mrClscmrc 17523 ACScacs 17525 Mndcmnd 18621 Grpcgrp 18815 ._gcmg 18944 SubGrpcsubg 18994 odcod 19386 gExcgex 19387 pGrp cpgp 19388 LSSumclsm 19496 Abelcabl 19643

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-dvds 16194 df-gcd 16432 df-prm 16605 df-pc 16766 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-0g 17383 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-eqg 18999 df-ga 19148 df-cntz 19175 df-od 19390 df-gex 19391 df-pgp 19392 df-lsm 19498 df-cmn 19644 df-abl 19645

This theorem is referenced by: pgpfac1lem3 19941

W3C validator