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Theorem pgpfac1 20143

Description: Factorization of a finite abelian p-group. There is a direct product decomposition of any abelian group of prime-power order where one of the factors is cyclic and generated by an element of maximal order. (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.ab	⊢ (𝜑 → 𝐴 ∈ 𝐵)

**Assertion**
Ref	Expression
pgpfac1	⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵))

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

Proof of Theorem pgpfac1 Dummy variables 𝑠 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pgpfac1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
2		ablgrp 19846	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3		pgpfac1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgid 19185	. . 3 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
5	1, 2, 4	3syl 19	. 2 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝐺))
6		pgpfac1.ab	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
7		pgpfac1.n	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
8		eleq1 2853	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (𝑠 ∈ (SubGrp‘𝐺) ↔ 𝑢 ∈ (SubGrp‘𝐺)))
9		eleq2 2854	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (𝐴 ∈ 𝑠 ↔ 𝐴 ∈ 𝑢))
10	8, 9	anbi12d 643	. . . . . 6 ⊢ (𝑠 = 𝑢 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) ↔ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢)))
11		eqeq2 2777	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → ((𝑆 ⊕ 𝑡) = 𝑠 ↔ (𝑆 ⊕ 𝑡) = 𝑢))
12	11	anbi2d 641	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢)))
13	12	rexbidv 3189	. . . . . 6 ⊢ (𝑠 = 𝑢 → (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢)))
14	10, 13	imbi12d 347	. . . . 5 ⊢ (𝑠 = 𝑢 → (((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)) ↔ ((𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢))))
15	14	imbi2d 343	. . . 4 ⊢ (𝑠 = 𝑢 → ((𝜑 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))) ↔ (𝜑 → ((𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢)))))
16		eleq1 2853	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠 ∈ (SubGrp‘𝐺) ↔ 𝐵 ∈ (SubGrp‘𝐺)))
17		eleq2 2854	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝐴 ∈ 𝑠 ↔ 𝐴 ∈ 𝐵))
18	16, 17	anbi12d 643	. . . . . 6 ⊢ (𝑠 = 𝐵 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) ↔ (𝐵 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝐵)))
19		eqeq2 2777	. . . . . . . 8 ⊢ (𝑠 = 𝐵 → ((𝑆 ⊕ 𝑡) = 𝑠 ↔ (𝑆 ⊕ 𝑡) = 𝐵))
20	19	anbi2d 641	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵)))
21	20	rexbidv 3189	. . . . . 6 ⊢ (𝑠 = 𝐵 → (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵)))
22	18, 21	imbi12d 347	. . . . 5 ⊢ (𝑠 = 𝐵 → (((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)) ↔ ((𝐵 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝐵) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵))))
23	22	imbi2d 343	. . . 4 ⊢ (𝑠 = 𝐵 → ((𝜑 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))) ↔ (𝜑 → ((𝐵 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝐵) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵)))))
24		bi2.04 391	. . . . . . . . . . 11 ⊢ ((𝑠 ⊊ 𝑢 → (𝑠 ∈ (SubGrp‘𝐺) → (𝐴 ∈ 𝑠 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))) ↔ (𝑠 ∈ (SubGrp‘𝐺) → (𝑠 ⊊ 𝑢 → (𝐴 ∈ 𝑠 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))))
25		impexp 455	. . . . . . . . . . . 12 ⊢ (((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)) ↔ (𝑠 ∈ (SubGrp‘𝐺) → (𝐴 ∈ 𝑠 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))))
26	25	imbi2i 339	. . . . . . . . . . 11 ⊢ ((𝑠 ⊊ 𝑢 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))) ↔ (𝑠 ⊊ 𝑢 → (𝑠 ∈ (SubGrp‘𝐺) → (𝐴 ∈ 𝑠 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))))
27		impexp 455	. . . . . . . . . . . 12 ⊢ (((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)) ↔ (𝑠 ⊊ 𝑢 → (𝐴 ∈ 𝑠 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))))
28	27	imbi2i 339	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ (SubGrp‘𝐺) → ((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))) ↔ (𝑠 ∈ (SubGrp‘𝐺) → (𝑠 ⊊ 𝑢 → (𝐴 ∈ 𝑠 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))))
29	24, 26, 28	3bitr4i 306	. . . . . . . . . 10 ⊢ ((𝑠 ⊊ 𝑢 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))) ↔ (𝑠 ∈ (SubGrp‘𝐺) → ((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))))
30	29	imbi2i 339	. . . . . . . . 9 ⊢ ((𝜑 → (𝑠 ⊊ 𝑢 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))) ↔ (𝜑 → (𝑠 ∈ (SubGrp‘𝐺) → ((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))))
31		bi2.04 391	. . . . . . . . 9 ⊢ ((𝑠 ⊊ 𝑢 → (𝜑 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))) ↔ (𝜑 → (𝑠 ⊊ 𝑢 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))))
32		bi2.04 391	. . . . . . . . 9 ⊢ ((𝑠 ∈ (SubGrp‘𝐺) → (𝜑 → ((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))) ↔ (𝜑 → (𝑠 ∈ (SubGrp‘𝐺) → ((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))))
33	30, 31, 32	3bitr4i 306	. . . . . . . 8 ⊢ ((𝑠 ⊊ 𝑢 → (𝜑 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))) ↔ (𝑠 ∈ (SubGrp‘𝐺) → (𝜑 → ((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))))
34	33	albii 1842	. . . . . . 7 ⊢ (∀𝑠(𝑠 ⊊ 𝑢 → (𝜑 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))) ↔ ∀𝑠(𝑠 ∈ (SubGrp‘𝐺) → (𝜑 → ((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))))
35		df-ral 3080	. . . . . . 7 ⊢ (∀𝑠 ∈ (SubGrp‘𝐺)(𝜑 → ((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))) ↔ ∀𝑠(𝑠 ∈ (SubGrp‘𝐺) → (𝜑 → ((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))))
36		r19.21v 3190	. . . . . . 7 ⊢ (∀𝑠 ∈ (SubGrp‘𝐺)(𝜑 → ((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))) ↔ (𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))))
37	34, 35, 36	3bitr2i 302	. . . . . 6 ⊢ (∀𝑠(𝑠 ⊊ 𝑢 → (𝜑 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))) ↔ (𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))))
38		psseq1 4046	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (𝑥 ⊊ 𝑢 ↔ 𝑠 ⊊ 𝑢))
39		eleq2 2854	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑠))
40	38, 39	anbi12d 643	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → ((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) ↔ (𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠)))
41		ineq2 4169	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑆 ∩ 𝑦) = (𝑆 ∩ 𝑡))
42	41	eqeq1d 2767	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑆 ∩ 𝑦) = { 0 } ↔ (𝑆 ∩ 𝑡) = { 0 }))
43		oveq2 7408	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑆 ⊕ 𝑦) = (𝑆 ⊕ 𝑡))
44	43	eqeq1d 2767	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑆 ⊕ 𝑦) = 𝑥 ↔ (𝑆 ⊕ 𝑡) = 𝑥))
45	42, 44	anbi12d 643	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥)))
46	45	cbvrexvw 3244	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥) ↔ ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥))
47		eqeq2 2777	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑆 ⊕ 𝑡) = 𝑥 ↔ (𝑆 ⊕ 𝑡) = 𝑠))
48	47	anbi2d 641	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))
49	48	rexbidv 3189	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥) ↔ ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))
50	46, 49	bitrid 286	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → (∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥) ↔ ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))
51	40, 50	imbi12d 347	. . . . . . . . 9 ⊢ (𝑥 = 𝑠 → (((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) ↔ ((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))))
52	51	cbvralvw 3243	. . . . . . . 8 ⊢ (∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) ↔ ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))
53		pgpfac1.k	. . . . . . . . . 10 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
54		pgpfac1.s	. . . . . . . . . 10 ⊢ 𝑆 = (𝐾‘{𝐴})
55		pgpfac1.o	. . . . . . . . . 10 ⊢ 𝑂 = (od‘𝐺)
56		pgpfac1.e	. . . . . . . . . 10 ⊢ 𝐸 = (gEx‘𝐺)
57		pgpfac1.z	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝐺)
58		pgpfac1.l	. . . . . . . . . 10 ⊢ ⊕ = (LSSum‘𝐺)
59		pgpfac1.p	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
60	59	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) ∧ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢))) → 𝑃 pGrp 𝐺)
61	1	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) ∧ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢))) → 𝐺 ∈ Abel)
62	7	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) ∧ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢))) → 𝐵 ∈ Fin)
63		pgpfac1.oe	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)
64	63	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) ∧ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢))) → (𝑂‘𝐴) = 𝐸)
65		simprrl 792	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) ∧ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢))) → 𝑢 ∈ (SubGrp‘𝐺))
66		simprrr 793	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) ∧ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢))) → 𝐴 ∈ 𝑢)
67		simprl 782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) ∧ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢))) → ∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)))
68	67, 52	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) ∧ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢))) → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))
69	53, 54, 3, 55, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68	pgpfac1lem5 20142	. . . . . . . . 9 ⊢ ((𝜑 ∧ (∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) ∧ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢))) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢))
70	69	exp32 425	. . . . . . . 8 ⊢ (𝜑 → (∀𝑥 ∈ (SubGrp‘𝐺)((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) → ∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥)) → ((𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢))))
71	52, 70	biimtrrid 246	. . . . . . 7 ⊢ (𝜑 → (∀𝑠 ∈ (SubGrp‘𝐺)((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)) → ((𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢))))
72	71	a2i 15	. . . . . 6 ⊢ ((𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))) → (𝜑 → ((𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢))))
73	37, 72	sylbi 220	. . . . 5 ⊢ (∀𝑠(𝑠 ⊊ 𝑢 → (𝜑 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))) → (𝜑 → ((𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢))))
74	73	a1i 11	. . . 4 ⊢ (𝑢 ∈ Fin → (∀𝑠(𝑠 ⊊ 𝑢 → (𝜑 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))) → (𝜑 → ((𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢)))))
75	15, 23, 74	findcard3 9231	. . 3 ⊢ (𝐵 ∈ Fin → (𝜑 → ((𝐵 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝐵) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵))))
76	7, 75	mpcom 39	. 2 ⊢ (𝜑 → ((𝐵 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝐵) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵)))
77	5, 6, 76	mp2and 711	1 ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∀wal 1561 = wceq 1563 ∈ wcel 2145 ∀wral 3079 ∃wrex 3089 ∩ cin 3906 ⊊ wpss 3908 {csn 4585 class class class wbr 5105 ‘cfv 6525 (class class class)co 7400 Fincfn 8931 Basecbs 17259 0_gc0g 17482 mrClscmrc 17625 Grpcgrp 18990 SubGrpcsubg 19177 odcod 19585 gExcgex 19586 pGrp cpgp 19587 LSSumclsm 19695 Abelcabl 19842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-disj 5073 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-rpss 7710 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-oadd 8445 df-omul 8446 df-er 8682 df-ec 8684 df-qs 8688 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-sup 9390 df-inf 9391 df-oi 9460 df-dju 9875 df-card 9913 df-acn 9916 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-n0 12496 df-xnn0 12569 df-z 12583 df-uz 12854 df-q 12964 df-rp 13008 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-fac 14301 df-bc 14330 df-hash 14358 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-clim 15529 df-sum 15728 df-dvds 16301 df-gcd 16543 df-prm 16720 df-pc 16887 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-0g 17484 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-grp 18993 df-minusg 18994 df-sbg 18995 df-mulg 19125 df-subg 19180 df-eqg 19182 df-ga 19351 df-cntz 19378 df-od 19589 df-gex 19590 df-pgp 19591 df-lsm 19697 df-cmn 19843 df-abl 19844

This theorem is referenced by: pgpfaclem3 20146

W3C validator