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Theorem pgpfac1 18746

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 18464	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3		pgpfac1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgid 17860	. . 3 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
5	1, 2, 4	3syl 18	. 2 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝐺))
6		pgpfac1.ab	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
7		pgpfac1.n	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
8		eleq1 2832	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (𝑠 ∈ (SubGrp‘𝐺) ↔ 𝑢 ∈ (SubGrp‘𝐺)))
9		eleq2 2833	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (𝐴 ∈ 𝑠 ↔ 𝐴 ∈ 𝑢))
10	8, 9	anbi12d 624	. . . . . 6 ⊢ (𝑠 = 𝑢 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) ↔ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢)))
11		eqeq2 2776	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → ((𝑆 ⊕ 𝑡) = 𝑠 ↔ (𝑆 ⊕ 𝑡) = 𝑢))
12	11	anbi2d 622	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢)))
13	12	rexbidv 3199	. . . . . 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 335	. . . . 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 331	. . . 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 2832	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠 ∈ (SubGrp‘𝐺) ↔ 𝐵 ∈ (SubGrp‘𝐺)))
17		eleq2 2833	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝐴 ∈ 𝑠 ↔ 𝐴 ∈ 𝐵))
18	16, 17	anbi12d 624	. . . . . 6 ⊢ (𝑠 = 𝐵 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) ↔ (𝐵 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝐵)))
19		eqeq2 2776	. . . . . . . 8 ⊢ (𝑠 = 𝐵 → ((𝑆 ⊕ 𝑡) = 𝑠 ↔ (𝑆 ⊕ 𝑡) = 𝐵))
20	19	anbi2d 622	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵)))
21	20	rexbidv 3199	. . . . . 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 335	. . . . 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 331	. . . 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		bi2.04 377	. . . . . . . . . . 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 441	. . . . . . . . . . . 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 327	. . . . . . . . . . 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 441	. . . . . . . . . . . 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 327	. . . . . . . . . . 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 294	. . . . . . . . . 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 327	. . . . . . . . 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		bi2.04 377	. . . . . . . . 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		bi2.04 377	. . . . . . . . 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 294	. . . . . . . 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 1914	. . . . . . 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 3060	. . . . . . 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 3107	. . . . . . 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 290	. . . . . 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 3855	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (𝑥 ⊊ 𝑢 ↔ 𝑠 ⊊ 𝑢))
39		eleq2 2833	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑠))
40	38, 39	anbi12d 624	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → ((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) ↔ (𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠)))
41		ineq2 3970	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑆 ∩ 𝑦) = (𝑆 ∩ 𝑡))
42	41	eqeq1d 2767	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑆 ∩ 𝑦) = { 0 } ↔ (𝑆 ∩ 𝑡) = { 0 }))
43		oveq2 6850	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑆 ⊕ 𝑦) = (𝑆 ⊕ 𝑡))
44	43	eqeq1d 2767	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑆 ⊕ 𝑦) = 𝑥 ↔ (𝑆 ⊕ 𝑡) = 𝑥))
45	42, 44	anbi12d 624	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥)))
46	45	cbvrexv 3320	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥) ↔ ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥))
47		eqeq2 2776	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑆 ⊕ 𝑡) = 𝑥 ↔ (𝑆 ⊕ 𝑡) = 𝑠))
48	47	anbi2d 622	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))
49	48	rexbidv 3199	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥) ↔ ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))
50	46, 49	syl5bb 274	. . . . . . . . . 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 335	. . . . . . . . 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	cbvralv 3319	. . . . . . . 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 472	. . . . . . . . . 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 472	. . . . . . . . . 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 472	. . . . . . . . . 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 472	. . . . . . . . . 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 799	. . . . . . . . . 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 800	. . . . . . . . . 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 787	. . . . . . . . . . 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 209	. . . . . . . . . 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 18745	. . . . . . . . 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 411	. . . . . . . 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	syl5bir 234	. . . . . . 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 14	. . . . . 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 208	. . . . 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 8410	. . 3 ⊢ (𝐵 ∈ Fin → (𝜑 → ((𝐵 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝐵) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵))))
76	7, 75	mpcom 38	. 2 ⊢ (𝜑 → ((𝐵 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝐵) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵)))
77	5, 6, 76	mp2and 690	1 ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∀wal 1650 = wceq 1652 ∈ wcel 2155 ∀wral 3055 ∃wrex 3056 ∩ cin 3731 ⊊ wpss 3733 {csn 4334 class class class wbr 4809 ‘cfv 6068 (class class class)co 6842 Fincfn 8160 Basecbs 16130 0_gc0g 16366 mrClscmrc 16509 Grpcgrp 17689 SubGrpcsubg 17852 odcod 18208 gExcgex 18209 pGrp cpgp 18210 LSSumclsm 18313 Abelcabl 18460

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2070 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2352 ax-ext 2743 ax-rep 4930 ax-sep 4941 ax-nul 4949 ax-pow 5001 ax-pr 5062 ax-un 7147 ax-inf2 8753 ax-cnex 10245 ax-resscn 10246 ax-1cn 10247 ax-icn 10248 ax-addcl 10249 ax-addrcl 10250 ax-mulcl 10251 ax-mulrcl 10252 ax-mulcom 10253 ax-addass 10254 ax-mulass 10255 ax-distr 10256 ax-i2m1 10257 ax-1ne0 10258 ax-1rid 10259 ax-rnegex 10260 ax-rrecex 10261 ax-cnre 10262 ax-pre-lttri 10263 ax-pre-lttrn 10264 ax-pre-ltadd 10265 ax-pre-mulgt0 10266 ax-pre-sup 10267

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-fal 1666 df-ex 1875 df-nf 1879 df-sb 2063 df-mo 2565 df-eu 2582 df-clab 2752 df-cleq 2758 df-clel 2761 df-nfc 2896 df-ne 2938 df-nel 3041 df-ral 3060 df-rex 3061 df-reu 3062 df-rmo 3063 df-rab 3064 df-v 3352 df-sbc 3597 df-csb 3692 df-dif 3735 df-un 3737 df-in 3739 df-ss 3746 df-pss 3748 df-nul 4080 df-if 4244 df-pw 4317 df-sn 4335 df-pr 4337 df-tp 4339 df-op 4341 df-uni 4595 df-int 4634 df-iun 4678 df-iin 4679 df-disj 4778 df-br 4810 df-opab 4872 df-mpt 4889 df-tr 4912 df-id 5185 df-eprel 5190 df-po 5198 df-so 5199 df-fr 5236 df-se 5237 df-we 5238 df-xp 5283 df-rel 5284 df-cnv 5285 df-co 5286 df-dm 5287 df-rn 5288 df-res 5289 df-ima 5290 df-pred 5865 df-ord 5911 df-on 5912 df-lim 5913 df-suc 5914 df-iota 6031 df-fun 6070 df-fn 6071 df-f 6072 df-f1 6073 df-fo 6074 df-f1o 6075 df-fv 6076 df-isom 6077 df-riota 6803 df-ov 6845 df-oprab 6846 df-mpt2 6847 df-rpss 7135 df-om 7264 df-1st 7366 df-2nd 7367 df-wrecs 7610 df-recs 7672 df-rdg 7710 df-1o 7764 df-2o 7765 df-oadd 7768 df-omul 7769 df-er 7947 df-ec 7949 df-qs 7953 df-map 8062 df-en 8161 df-dom 8162 df-sdom 8163 df-fin 8164 df-sup 8555 df-inf 8556 df-oi 8622 df-card 9016 df-acn 9019 df-cda 9243 df-pnf 10330 df-mnf 10331 df-xr 10332 df-ltxr 10333 df-le 10334 df-sub 10522 df-neg 10523 df-div 10939 df-nn 11275 df-2 11335 df-3 11336 df-n0 11539 df-xnn0 11611 df-z 11625 df-uz 11887 df-q 11990 df-rp 12029 df-fz 12534 df-fzo 12674 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-cj 14124 df-re 14125 df-im 14126 df-sqrt 14260 df-abs 14261 df-clim 14504 df-sum 14702 df-dvds 15266 df-gcd 15498 df-prm 15666 df-pc 15821 df-ndx 16133 df-slot 16134 df-base 16136 df-sets 16137 df-ress 16138 df-plusg 16227 df-0g 16368 df-mre 16512 df-mrc 16513 df-acs 16515 df-mgm 17508 df-sgrp 17550 df-mnd 17561 df-submnd 17602 df-grp 17692 df-minusg 17693 df-sbg 17694 df-mulg 17808 df-subg 17855 df-eqg 17857 df-ga 17986 df-cntz 18013 df-od 18212 df-gex 18213 df-pgp 18214 df-lsm 18315 df-cmn 18461 df-abl 18462

This theorem is referenced by: pgpfaclem3 18749

W3C validator