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Theorem pgpfac1 20153

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 19856	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3		pgpfac1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgid 19195	. . 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 2851	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (𝑠 ∈ (SubGrp‘𝐺) ↔ 𝑢 ∈ (SubGrp‘𝐺)))
9		eleq2 2852	. . . . . . 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 2775	. . . . . . . 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 2851	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠 ∈ (SubGrp‘𝐺) ↔ 𝐵 ∈ (SubGrp‘𝐺)))
17		eleq2 2852	. . . . . . 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 2775	. . . . . . . 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 1849	. . . . . . 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 4045	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (𝑥 ⊊ 𝑢 ↔ 𝑠 ⊊ 𝑢))
39		eleq2 2852	. . . . . . . . . . 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 4168	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑆 ∩ 𝑦) = (𝑆 ∩ 𝑡))
42	41	eqeq1d 2765	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑆 ∩ 𝑦) = { 0 } ↔ (𝑆 ∩ 𝑡) = { 0 }))
43		oveq2 7420	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑆 ⊕ 𝑦) = (𝑆 ⊕ 𝑡))
44	43	eqeq1d 2765	. . . . . . . . . . . . 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 2775	. . . . . . . . . . . . 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 20152	. . . . . . . . 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 9244	. . 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 1568 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 ∩ cin 3905 ⊊ wpss 3907 {csn 4590 class class class wbr 5110 ‘cfv 6538 (class class class)co 7412 Fincfn 8944 Basecbs 17270 0_gc0g 17493 mrClscmrc 17636 Grpcgrp 19001 SubGrpcsubg 19187 odcod 19595 gExcgex 19596 pGrp cpgp 19597 LSSumclsm 19705 Abelcabl 19852

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-disj 5078 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-rpss 7722 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-oadd 8458 df-omul 8459 df-er 8695 df-ec 8697 df-qs 8701 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-inf 9404 df-oi 9473 df-dju 9888 df-card 9926 df-acn 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-xnn0 12579 df-z 12593 df-uz 12864 df-q 12974 df-rp 13018 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-dvds 16312 df-gcd 16554 df-prm 16731 df-pc 16898 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-0g 17495 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-grp 19004 df-minusg 19005 df-sbg 19006 df-mulg 19135 df-subg 19190 df-eqg 19192 df-ga 19361 df-cntz 19388 df-od 19599 df-gex 19600 df-pgp 19601 df-lsm 19707 df-cmn 19853 df-abl 19854

This theorem is referenced by: pgpfaclem3 20156

W3C validator