Theorem pgpfac 20280

Description: Full factorization of a finite abelian p-group, by iterating pgpfac1 20276. There is a direct product decomposition of any abelian group of prime-power order into cyclic subgroups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)

Hypotheses

Ref	Expression
pgpfac.b	⊢ 𝐵 = (Base‘𝐺)
pgpfac.c	⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
pgpfac.g	⊢ (𝜑 → 𝐺 ∈ Abel)
pgpfac.p	⊢ (𝜑 → 𝑃 pGrp 𝐺)
pgpfac.f	⊢ (𝜑 → 𝐵 ∈ Fin)

Assertion

Ref	Expression
pgpfac	⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))

Distinct variable groups:   𝐶,𝑠   𝑠,𝑟,𝐺   𝐵,𝑠

Allowed substitution hints:   𝜑(𝑠, 𝑟)   𝐵(𝑟)   𝐶(𝑟)   𝑃(𝑠, 𝑟)

Proof of Theorem pgpfac

Dummy variables 𝑡 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pgpfac.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
2		ablgrp 19979	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3		pgpfac.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgid 19318	. . 3 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
5	1, 2, 4	3syl 19	. 2 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝐺))
6		pgpfac.f	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
7		eleq1 2849	. . . . . 6 ⊢ (𝑡 = 𝑢 → (𝑡 ∈ (SubGrp‘𝐺) ↔ 𝑢 ∈ (SubGrp‘𝐺)))
8		eqeq2 2773	. . . . . . . 8 ⊢ (𝑡 = 𝑢 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑢))
9	8	anbi2d 642	. . . . . . 7 ⊢ (𝑡 = 𝑢 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))
10	9	rexbidv 3187	. . . . . 6 ⊢ (𝑡 = 𝑢 → (∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))
11	7, 10	imbi12d 347	. . . . 5 ⊢ (𝑡 = 𝑢 → ((𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ↔ (𝑢 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢))))
12	11	imbi2d 343	. . . 4 ⊢ (𝑡 = 𝑢 → ((𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) ↔ (𝜑 → (𝑢 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))))
13		eleq1 2849	. . . . . 6 ⊢ (𝑡 = 𝐵 → (𝑡 ∈ (SubGrp‘𝐺) ↔ 𝐵 ∈ (SubGrp‘𝐺)))
14		eqeq2 2773	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝐵))
15	14	anbi2d 642	. . . . . . 7 ⊢ (𝑡 = 𝐵 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)))
16	15	rexbidv 3187	. . . . . 6 ⊢ (𝑡 = 𝐵 → (∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)))
17	13, 16	imbi12d 347	. . . . 5 ⊢ (𝑡 = 𝐵 → ((𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ↔ (𝐵 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))))
18	17	imbi2d 343	. . . 4 ⊢ (𝑡 = 𝐵 → ((𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) ↔ (𝜑 → (𝐵 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)))))
19		bi2.04 392	. . . . . . . . 9 ⊢ ((𝑡 ⊊ 𝑢 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) ↔ (𝑡 ∈ (SubGrp‘𝐺) → (𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))))
20	19	imbi2i 339	. . . . . . . 8 ⊢ ((𝜑 → (𝑡 ⊊ 𝑢 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → (𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
21		bi2.04 392	. . . . . . . 8 ⊢ ((𝑡 ⊊ 𝑢 → (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ (𝜑 → (𝑡 ⊊ 𝑢 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
22		bi2.04 392	. . . . . . . 8 ⊢ ((𝑡 ∈ (SubGrp‘𝐺) → (𝜑 → (𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → (𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
23	20, 21, 22	3bitr4i 306	. . . . . . 7 ⊢ ((𝑡 ⊊ 𝑢 → (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ (𝑡 ∈ (SubGrp‘𝐺) → (𝜑 → (𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
24	23	albii 1852	. . . . . 6 ⊢ (∀𝑡(𝑡 ⊊ 𝑢 → (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ ∀𝑡(𝑡 ∈ (SubGrp‘𝐺) → (𝜑 → (𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
25		df-ral 3078	. . . . . 6 ⊢ (∀𝑡 ∈ (SubGrp‘𝐺)(𝜑 → (𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) ↔ ∀𝑡(𝑡 ∈ (SubGrp‘𝐺) → (𝜑 → (𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
26		r19.21v 3188	. . . . . 6 ⊢ (∀𝑡 ∈ (SubGrp‘𝐺)(𝜑 → (𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) ↔ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))))
27	24, 25, 26	3bitr2i 302	. . . . 5 ⊢ (∀𝑡(𝑡 ⊊ 𝑢 → (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))))
28		pgpfac.c	. . . . . . . . 9 ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
29	1	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → 𝐺 ∈ Abel)
30		pgpfac.p	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
31	30	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → 𝑃 pGrp 𝐺)
32	6	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → 𝐵 ∈ Fin)
33		simprr 785	. . . . . . . . 9 ⊢ ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → 𝑢 ∈ (SubGrp‘𝐺))
34		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))
35		psseq1 4038	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (𝑡 ⊊ 𝑢 ↔ 𝑥 ⊊ 𝑢))
36		eqeq2 2773	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑥 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑥))
37	36	anbi2d 642	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥)))
38	37	rexbidv 3187	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥)))
39	35, 38	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑥 → ((𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ↔ (𝑥 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥))))
40	39	cbvralvw 3241	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ↔ ∀𝑥 ∈ (SubGrp‘𝐺)(𝑥 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥)))
41	34, 40	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → ∀𝑥 ∈ (SubGrp‘𝐺)(𝑥 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥)))
42	3, 28, 29, 31, 32, 33, 41	pgpfaclem3 20279	. . . . . . . 8 ⊢ ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢))
43	42	exp32 426	. . . . . . 7 ⊢ (𝜑 → (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) → (𝑢 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢))))
44	43	a1i 11	. . . . . 6 ⊢ (𝑢 ∈ Fin → (𝜑 → (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) → (𝑢 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))))
45	44	a2d 30	. . . . 5 ⊢ (𝑢 ∈ Fin → ((𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) → (𝜑 → (𝑢 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))))
46	27, 45	biimtrid 245	. . . 4 ⊢ (𝑢 ∈ Fin → (∀𝑡(𝑡 ⊊ 𝑢 → (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) → (𝜑 → (𝑢 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))))
47	12, 18, 46	findcard3 9258	. . 3 ⊢ (𝐵 ∈ Fin → (𝜑 → (𝐵 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))))
48	6, 47	mpcom 39	. 2 ⊢ (𝜑 → (𝐵 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)))
49	5, 48	mpd 16	1 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413   ∩ cin 3898   ⊊ wpss 3900   class class class wbr 5103  dom cdm 5651  ran crn 5652  ‘cfv 6531  (class class class)co 7412  Fincfn 8957  Word cword 14638  Basecbs 17367   ↾_s cress 17388  Grpcgrp 19124  SubGrpcsubg 19310   pGrp cpgp 19720  Abelcabl 19975  CycGrpccyg 20071   DProd cdprd 20189

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-rpss 7728  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8162  df-tpos 8227  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-oadd 8464  df-omul 8465  df-er 8701  df-ec 8703  df-qs 8707  df-map 8833  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9338  df-sup 9418  df-inf 9419  df-oi 9488  df-dju 9963  df-card 10001  df-acn 10004  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-xnn0 12661  df-z 12675  df-uz 12947  df-q 13057  df-rp 13102  df-fz 13621  df-fzo 13769  df-fl 13912  df-mod 13990  df-seq 14125  df-exp 14185  df-fac 14398  df-bc 14427  df-hash 14455  df-word 14639  df-concat 14696  df-s1 14723  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-clim 15635  df-sum 15834  df-dvds 16403  df-gcd 16645  df-prm 16827  df-pc 16995  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-0g 17592  df-gsum 17593  df-mre 17736  df-mrc 17737  df-acs 17739  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-mhm 18958  df-submnd 18959  df-grp 19127  df-minusg 19128  df-sbg 19129  df-mulg 19258  df-subg 19313  df-eqg 19315  df-ghm 19408  df-gim 19453  df-ga 19484  df-cntz 19511  df-oppg 19540  df-od 19722  df-gex 19723  df-pgp 19724  df-lsm 19830  df-pj1 19831  df-cmn 19976  df-abl 19977  df-cyg 20072  df-dprd 20191

This theorem is used by:  ablfaclem3  20283