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Theorem pgpfac1 19196

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 18905	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3		pgpfac1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgid 18275	. . 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 2900	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (𝑠 ∈ (SubGrp‘𝐺) ↔ 𝑢 ∈ (SubGrp‘𝐺)))
9		eleq2 2901	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (𝐴 ∈ 𝑠 ↔ 𝐴 ∈ 𝑢))
10	8, 9	anbi12d 632	. . . . . 6 ⊢ (𝑠 = 𝑢 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) ↔ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢)))
11		eqeq2 2833	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → ((𝑆 ⊕ 𝑡) = 𝑠 ↔ (𝑆 ⊕ 𝑡) = 𝑢))
12	11	anbi2d 630	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢)))
13	12	rexbidv 3297	. . . . . 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 2900	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠 ∈ (SubGrp‘𝐺) ↔ 𝐵 ∈ (SubGrp‘𝐺)))
17		eleq2 2901	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝐴 ∈ 𝑠 ↔ 𝐴 ∈ 𝐵))
18	16, 17	anbi12d 632	. . . . . 6 ⊢ (𝑠 = 𝐵 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) ↔ (𝐵 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝐵)))
19		eqeq2 2833	. . . . . . . 8 ⊢ (𝑠 = 𝐵 → ((𝑆 ⊕ 𝑡) = 𝑠 ↔ (𝑆 ⊕ 𝑡) = 𝐵))
20	19	anbi2d 630	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵)))
21	20	rexbidv 3297	. . . . . 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 453	. . . . . . . . . . . 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 338	. . . . . . . . . . 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 453	. . . . . . . . . . . 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 338	. . . . . . . . . . 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 305	. . . . . . . . . 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 338	. . . . . . . . 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 305	. . . . . . . 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 1816	. . . . . . 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 3143	. . . . . . 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 3175	. . . . . . 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 301	. . . . . 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 4063	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (𝑥 ⊊ 𝑢 ↔ 𝑠 ⊊ 𝑢))
39		eleq2 2901	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑠))
40	38, 39	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → ((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) ↔ (𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠)))
41		ineq2 4182	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑆 ∩ 𝑦) = (𝑆 ∩ 𝑡))
42	41	eqeq1d 2823	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑆 ∩ 𝑦) = { 0 } ↔ (𝑆 ∩ 𝑡) = { 0 }))
43		oveq2 7158	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑆 ⊕ 𝑦) = (𝑆 ⊕ 𝑡))
44	43	eqeq1d 2823	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑆 ⊕ 𝑦) = 𝑥 ↔ (𝑆 ⊕ 𝑡) = 𝑥))
45	42, 44	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥)))
46	45	cbvrexvw 3450	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥) ↔ ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥))
47		eqeq2 2833	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑆 ⊕ 𝑡) = 𝑥 ↔ (𝑆 ⊕ 𝑡) = 𝑠))
48	47	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))
49	48	rexbidv 3297	. . . . . . . . . . 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 285	. . . . . . . . . 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 3449	. . . . . . . 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 483	. . . . . . . . . 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 483	. . . . . . . . . 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 483	. . . . . . . . . 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 483	. . . . . . . . . 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 779	. . . . . . . . . 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 780	. . . . . . . . . 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 769	. . . . . . . . . . 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 220	. . . . . . . . . 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 19195	. . . . . . . . 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 423	. . . . . . . 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 245	. . . . . . 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 219	. . . . 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 8755	. . 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 697	1 ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ∩ cin 3934 ⊊ wpss 3936 {csn 4560 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 Basecbs 16477 0_gc0g 16707 mrClscmrc 16848 Grpcgrp 18097 SubGrpcsubg 18267 odcod 18646 gExcgex 18647 pGrp cpgp 18648 LSSumclsm 18753 Abelcabl 18901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-rpss 7443 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-dvds 15602 df-gcd 15838 df-prm 16010 df-pc 16168 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-eqg 18272 df-ga 18414 df-cntz 18441 df-od 18650 df-gex 18651 df-pgp 18652 df-lsm 18755 df-cmn 18902 df-abl 18903

This theorem is referenced by: pgpfaclem3 19199

W3C validator