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Theorem pgpfac1 20210

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 19913	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3		pgpfac1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgid 19252	. . 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 2848	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (𝑠 ∈ (SubGrp‘𝐺) ↔ 𝑢 ∈ (SubGrp‘𝐺)))
9		eleq2 2849	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (𝐴 ∈ 𝑠 ↔ 𝐴 ∈ 𝑢))
10	8, 9	anbi12d 644	. . . . . 6 ⊢ (𝑠 = 𝑢 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) ↔ (𝑢 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑢)))
11		eqeq2 2772	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → ((𝑆 ⊕ 𝑡) = 𝑠 ↔ (𝑆 ⊕ 𝑡) = 𝑢))
12	11	anbi2d 642	. . . . . . 7 ⊢ (𝑠 = 𝑢 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑢)))
13	12	rexbidv 3186	. . . . . 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 2848	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠 ∈ (SubGrp‘𝐺) ↔ 𝐵 ∈ (SubGrp‘𝐺)))
17		eleq2 2849	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝐴 ∈ 𝑠 ↔ 𝐴 ∈ 𝐵))
18	16, 17	anbi12d 644	. . . . . 6 ⊢ (𝑠 = 𝐵 → ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) ↔ (𝐵 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝐵)))
19		eqeq2 2772	. . . . . . . 8 ⊢ (𝑠 = 𝐵 → ((𝑆 ⊕ 𝑡) = 𝑠 ↔ (𝑆 ⊕ 𝑡) = 𝐵))
20	19	anbi2d 642	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵)))
21	20	rexbidv 3186	. . . . . 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		bi2.04 392	. . . . . . . . . . 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 456	. . . . . . . . . . . 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 456	. . . . . . . . . . . 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		bi2.04 392	. . . . . . . . 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		bi2.04 392	. . . . . . . . 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 1852	. . . . . . 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 3077	. . . . . . 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 3187	. . . . . . 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 4038	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (𝑥 ⊊ 𝑢 ↔ 𝑠 ⊊ 𝑢))
39		eleq2 2849	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑠))
40	38, 39	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → ((𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥) ↔ (𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠)))
41		ineq2 4160	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑆 ∩ 𝑦) = (𝑆 ∩ 𝑡))
42	41	eqeq1d 2762	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑆 ∩ 𝑦) = { 0 } ↔ (𝑆 ∩ 𝑡) = { 0 }))
43		oveq2 7422	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑆 ⊕ 𝑦) = (𝑆 ⊕ 𝑡))
44	43	eqeq1d 2762	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑆 ⊕ 𝑦) = 𝑥 ↔ (𝑆 ⊕ 𝑡) = 𝑥))
45	42, 44	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥)))
46	45	cbvrexvw 3241	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑦) = { 0 } ∧ (𝑆 ⊕ 𝑦) = 𝑥) ↔ ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥))
47		eqeq2 2772	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑆 ⊕ 𝑡) = 𝑥 ↔ (𝑆 ⊕ 𝑡) = 𝑠))
48	47	anbi2d 642	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑥) ↔ ((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))
49	48	rexbidv 3186	. . . . . . . . . . 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 3240	. . . . . . . 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 486	. . . . . . . . . 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 486	. . . . . . . . . 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 486	. . . . . . . . . 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 486	. . . . . . . . . 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 793	. . . . . . . . . 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 794	. . . . . . . . . 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 783	. . . . . . . . . . 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 20209	. . . . . . . . 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 426	. . . . . . . 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 9254	. . 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 712	1 ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∀wal 1568 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 ∩ cin 3898 ⊊ wpss 3900 {csn 4584 class class class wbr 5103 ‘cfv 6533 (class class class)co 7414 Fincfn 8953 Basecbs 17302 0_gc0g 17525 mrClscmrc 17668 Grpcgrp 19058 SubGrpcsubg 19244 odcod 19652 gExcgex 19653 pGrp cpgp 19654 LSSumclsm 19762 Abelcabl 19909

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9621 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-rpss 7725 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-oadd 8460 df-omul 8461 df-er 8697 df-ec 8699 df-qs 8703 df-map 8829 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9413 df-inf 9414 df-oi 9483 df-dju 9907 df-card 9945 df-acn 9948 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-n0 12530 df-xnn0 12603 df-z 12617 df-uz 12889 df-q 12999 df-rp 13044 df-fz 13563 df-fzo 13711 df-fl 13854 df-mod 13932 df-seq 14067 df-exp 14127 df-fac 14339 df-bc 14368 df-hash 14396 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-clim 15576 df-sum 15775 df-dvds 16344 df-gcd 16586 df-prm 16763 df-pc 16930 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-0g 17527 df-mre 17671 df-mrc 17672 df-acs 17674 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-submnd 18893 df-grp 19061 df-minusg 19062 df-sbg 19063 df-mulg 19192 df-subg 19247 df-eqg 19249 df-ga 19418 df-cntz 19445 df-od 19656 df-gex 19657 df-pgp 19658 df-lsm 19764 df-cmn 19910 df-abl 19911

This theorem is used by: pgpfaclem3 20213

W3C validator