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Theorem pgpfac1lem4 20034

Description: Lemma for pgpfac1 20036. (Contributed by Mario Carneiro, 27-Apr-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)

**Assertion**
Ref	Expression
pgpfac1lem4	⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))

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

Proof of Theorem pgpfac1lem4 Dummy variables 𝑘 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pgpfac1.k	. . . . . . . 8 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
2		pgpfac1.s	. . . . . . . 8 ⊢ 𝑆 = (𝐾‘{𝐴})
3		pgpfac1.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
4		pgpfac1.o	. . . . . . . 8 ⊢ 𝑂 = (od‘𝐺)
5		pgpfac1.e	. . . . . . . 8 ⊢ 𝐸 = (gEx‘𝐺)
6		pgpfac1.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝐺)
7		pgpfac1.l	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝐺)
8		pgpfac1.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
9		pgpfac1.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Abel)
10		pgpfac1.n	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin)
11		pgpfac1.oe	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)
12		pgpfac1.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
13		pgpfac1.au	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
14		pgpfac1.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
15		pgpfac1.i	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })
16		pgpfac1.ss	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈)
17		pgpfac1.2	. . . . . . . 8 ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))
18		pgpfac1.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)))
19		pgpfac1.mg	. . . . . . . 8 ⊢ · = (._g‘𝐺)
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	pgpfac1lem2 20031	. . . . . . 7 ⊢ (𝜑 → (𝑃 · 𝐶) ∈ (𝑆 ⊕ 𝑊))
21		ablgrp 19739	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
22	9, 21	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Grp)
23	3	subgacs 19115	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
24		acsmre 17626	. . . . . . . . . . 11 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘𝐵) → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
25	22, 23, 24	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
26	3	subgss 19081	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
27	12, 26	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
28	27, 13	sseldd 3974	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
29	1	mrcsncl 17586	. . . . . . . . . 10 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴 ∈ 𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
30	25, 28, 29	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
31	2, 30	eqeltrid 2829	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
32	7	lsmcom 19812	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑊 ∈ (SubGrp‘𝐺)) → (𝑆 ⊕ 𝑊) = (𝑊 ⊕ 𝑆))
33	9, 31, 14, 32	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → (𝑆 ⊕ 𝑊) = (𝑊 ⊕ 𝑆))
34	20, 33	eleqtrd 2827	. . . . . 6 ⊢ (𝜑 → (𝑃 · 𝐶) ∈ (𝑊 ⊕ 𝑆))
35		eqid 2725	. . . . . . 7 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
36	35, 7, 14, 31	lsmelvalm 19605	. . . . . 6 ⊢ (𝜑 → ((𝑃 · 𝐶) ∈ (𝑊 ⊕ 𝑆) ↔ ∃𝑤 ∈ 𝑊 ∃𝑠 ∈ 𝑆 (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)𝑠)))
37	34, 36	mpbid 231	. . . . 5 ⊢ (𝜑 → ∃𝑤 ∈ 𝑊 ∃𝑠 ∈ 𝑆 (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)𝑠))
38		eqid 2725	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤ ↦ (𝑘 · 𝐴)) = (𝑘 ∈ ℤ ↦ (𝑘 · 𝐴))
39	3, 19, 38, 1	cycsubg2 19164	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → (𝐾‘{𝐴}) = ran (𝑘 ∈ ℤ ↦ (𝑘 · 𝐴)))
40	22, 28, 39	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘{𝐴}) = ran (𝑘 ∈ ℤ ↦ (𝑘 · 𝐴)))
41	2, 40	eqtrid 2777	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = ran (𝑘 ∈ ℤ ↦ (𝑘 · 𝐴)))
42	41	rexeqdv 3316	. . . . . . 7 ⊢ (𝜑 → (∃𝑠 ∈ 𝑆 (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)𝑠) ↔ ∃𝑠 ∈ ran (𝑘 ∈ ℤ ↦ (𝑘 · 𝐴))(𝑃 · 𝐶) = (𝑤(-_g‘𝐺)𝑠)))
43		ovex 7446	. . . . . . . . 9 ⊢ (𝑘 · 𝐴) ∈ V
44	43	rgenw 3055	. . . . . . . 8 ⊢ ∀𝑘 ∈ ℤ (𝑘 · 𝐴) ∈ V
45		oveq2 7421	. . . . . . . . . 10 ⊢ (𝑠 = (𝑘 · 𝐴) → (𝑤(-_g‘𝐺)𝑠) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴)))
46	45	eqeq2d 2736	. . . . . . . . 9 ⊢ (𝑠 = (𝑘 · 𝐴) → ((𝑃 · 𝐶) = (𝑤(-_g‘𝐺)𝑠) ↔ (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴))))
47	38, 46	rexrnmptw 7098	. . . . . . . 8 ⊢ (∀𝑘 ∈ ℤ (𝑘 · 𝐴) ∈ V → (∃𝑠 ∈ ran (𝑘 ∈ ℤ ↦ (𝑘 · 𝐴))(𝑃 · 𝐶) = (𝑤(-_g‘𝐺)𝑠) ↔ ∃𝑘 ∈ ℤ (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴))))
48	44, 47	ax-mp 5	. . . . . . 7 ⊢ (∃𝑠 ∈ ran (𝑘 ∈ ℤ ↦ (𝑘 · 𝐴))(𝑃 · 𝐶) = (𝑤(-_g‘𝐺)𝑠) ↔ ∃𝑘 ∈ ℤ (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴)))
49	42, 48	bitrdi 286	. . . . . 6 ⊢ (𝜑 → (∃𝑠 ∈ 𝑆 (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)𝑠) ↔ ∃𝑘 ∈ ℤ (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴))))
50	49	rexbidv 3169	. . . . 5 ⊢ (𝜑 → (∃𝑤 ∈ 𝑊 ∃𝑠 ∈ 𝑆 (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)𝑠) ↔ ∃𝑤 ∈ 𝑊 ∃𝑘 ∈ ℤ (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴))))
51	37, 50	mpbid 231	. . . 4 ⊢ (𝜑 → ∃𝑤 ∈ 𝑊 ∃𝑘 ∈ ℤ (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴)))
52		rexcom 3278	. . . 4 ⊢ (∃𝑤 ∈ 𝑊 ∃𝑘 ∈ ℤ (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴)) ↔ ∃𝑘 ∈ ℤ ∃𝑤 ∈ 𝑊 (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴)))
53	51, 52	sylib 217	. . 3 ⊢ (𝜑 → ∃𝑘 ∈ ℤ ∃𝑤 ∈ 𝑊 (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴)))
54	22	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑤 ∈ 𝑊) → 𝐺 ∈ Grp)
55	3	subgss 19081	. . . . . . . . . . 11 ⊢ (𝑊 ∈ (SubGrp‘𝐺) → 𝑊 ⊆ 𝐵)
56	14, 55	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ⊆ 𝐵)
57	56	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → 𝑊 ⊆ 𝐵)
58	57	sselda 3973	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑤 ∈ 𝑊) → 𝑤 ∈ 𝐵)
59		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑤 ∈ 𝑊) → 𝑘 ∈ ℤ)
60	28	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑤 ∈ 𝑊) → 𝐴 ∈ 𝐵)
61	3, 19	mulgcl 19045	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑘 ∈ ℤ ∧ 𝐴 ∈ 𝐵) → (𝑘 · 𝐴) ∈ 𝐵)
62	54, 59, 60, 61	syl3anc 1368	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑤 ∈ 𝑊) → (𝑘 · 𝐴) ∈ 𝐵)
63		pgpprm 19547	. . . . . . . . . . 11 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
64		prmz 16640	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
65	8, 63, 64	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
66	18	eldifad 3953	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
67	27, 66	sseldd 3974	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
68	3, 19	mulgcl 19045	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℤ ∧ 𝐶 ∈ 𝐵) → (𝑃 · 𝐶) ∈ 𝐵)
69	22, 65, 67, 68	syl3anc 1368	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 · 𝐶) ∈ 𝐵)
70	69	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑤 ∈ 𝑊) → (𝑃 · 𝐶) ∈ 𝐵)
71		eqid 2725	. . . . . . . . 9 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
72	3, 71, 35	grpsubadd 18983	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑤 ∈ 𝐵 ∧ (𝑘 · 𝐴) ∈ 𝐵 ∧ (𝑃 · 𝐶) ∈ 𝐵)) → ((𝑤(-_g‘𝐺)(𝑘 · 𝐴)) = (𝑃 · 𝐶) ↔ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) = 𝑤))
73	54, 58, 62, 70, 72	syl13anc 1369	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑤 ∈ 𝑊) → ((𝑤(-_g‘𝐺)(𝑘 · 𝐴)) = (𝑃 · 𝐶) ↔ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) = 𝑤))
74		eqcom 2732	. . . . . . 7 ⊢ ((𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴)) ↔ (𝑤(-_g‘𝐺)(𝑘 · 𝐴)) = (𝑃 · 𝐶))
75		eqcom 2732	. . . . . . 7 ⊢ (𝑤 = ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ↔ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) = 𝑤)
76	73, 74, 75	3bitr4g 313	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑤 ∈ 𝑊) → ((𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴)) ↔ 𝑤 = ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴))))
77	76	rexbidva 3167	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∃𝑤 ∈ 𝑊 (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴)) ↔ ∃𝑤 ∈ 𝑊 𝑤 = ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴))))
78		risset 3221	. . . . 5 ⊢ (((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊 ↔ ∃𝑤 ∈ 𝑊 𝑤 = ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)))
79	77, 78	bitr4di 288	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∃𝑤 ∈ 𝑊 (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴)) ↔ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊))
80	79	rexbidva 3167	. . 3 ⊢ (𝜑 → (∃𝑘 ∈ ℤ ∃𝑤 ∈ 𝑊 (𝑃 · 𝐶) = (𝑤(-_g‘𝐺)(𝑘 · 𝐴)) ↔ ∃𝑘 ∈ ℤ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊))
81	53, 80	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑘 ∈ ℤ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)
82	8	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → 𝑃 pGrp 𝐺)
83	9	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → 𝐺 ∈ Abel)
84	10	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → 𝐵 ∈ Fin)
85	11	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → (𝑂‘𝐴) = 𝐸)
86	12	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → 𝑈 ∈ (SubGrp‘𝐺))
87	13	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → 𝐴 ∈ 𝑈)
88	14	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → 𝑊 ∈ (SubGrp‘𝐺))
89	15	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → (𝑆 ∩ 𝑊) = { 0 })
90	16	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → (𝑆 ⊕ 𝑊) ⊆ 𝑈)
91	17	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))
92	18	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)))
93		simprl 769	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → 𝑘 ∈ ℤ)
94		simprr 771	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)
95		eqid 2725	. . 3 ⊢ (𝐶(+_g‘𝐺)((𝑘 / 𝑃) · 𝐴)) = (𝐶(+_g‘𝐺)((𝑘 / 𝑃) · 𝐴))
96	1, 2, 3, 4, 5, 6, 7, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 19, 93, 94, 95	pgpfac1lem3 20033	. 2 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ ((𝑃 · 𝐶)(+_g‘𝐺)(𝑘 · 𝐴)) ∈ 𝑊)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))
97	81, 96	rexlimddv 3151	1 ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 Vcvv 3463 ∖ cdif 3938 ∩ cin 3940 ⊆ wss 3941 ⊊ wpss 3942 {csn 4625 class class class wbr 5144 ↦ cmpt 5227 ran crn 5674 ‘cfv 6543 (class class class)co 7413 Fincfn 8957 / cdiv 11896 ℤcz 12583 ℙcprime 16636 Basecbs 17174 +_gcplusg 17227 0_gc0g 17415 Moorecmre 17556 mrClscmrc 17557 ACScacs 17559 Grpcgrp 18889 -_gcsg 18891 ._gcmg 19022 SubGrpcsubg 19074 odcod 19478 gExcgex 19479 pGrp cpgp 19480 LSSumclsm 19588 Abelcabl 19735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-disj 5110 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8718 df-ec 8720 df-qs 8724 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-oi 9528 df-dju 9919 df-card 9957 df-acn 9960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-xnn0 12570 df-z 12584 df-uz 12848 df-q 12958 df-rp 13002 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-fac 14260 df-bc 14289 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-sum 15660 df-dvds 16226 df-gcd 16464 df-prm 16637 df-pc 16800 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-0g 17417 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-eqg 19079 df-ga 19240 df-cntz 19267 df-od 19482 df-gex 19483 df-pgp 19484 df-lsm 19590 df-cmn 19736 df-abl 19737

This theorem is referenced by: pgpfac1lem5 20035

W3C validator