Theorem pgpfac 20301

Description: Full factorization of a finite abelian p-group, by iterating pgpfac1 20297. 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 20000	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3		pgpfac.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgid 19339	. . 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 20300	. . . . . . . 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 9275	. . 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 4034   class class class wbr 5103  dom cdm 5651  ran crn 5652  ‘cfv 6539  (class class class)co 7421  Fincfn 8974  Word cword 14659  Basecbs 17388   ↾_s cress 17409  Grpcgrp 19145  SubGrpcsubg 19331   pGrp cpgp 19741  Abelcabl 19996  CycGrpccyg 20092   DProd cdprd 20210

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 7752  ax-inf2 9643  ax-cnex 11257  ax-resscn 11258  ax-1cn 11259  ax-icn 11260  ax-addcl 11261  ax-addrcl 11262  ax-mulcl 11263  ax-mulrcl 11264  ax-mulcom 11265  ax-addass 11266  ax-mulass 11267  ax-distr 11268  ax-i2m1 11269  ax-1ne0 11270  ax-1rid 11271  ax-rnegex 11272  ax-rrecex 11273  ax-cnre 11274  ax-pre-lttri 11275  ax-pre-lttrn 11276  ax-pre-ltadd 11277  ax-pre-mulgt0 11278  ax-pre-sup 11279

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 3900  df-un 3902  df-in 3904  df-ss 3915  df-pss 4035  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 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-of 7694  df-rpss 7740  df-om 7879  df-1st 8002  df-2nd 8003  df-supp 8179  df-tpos 8244  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-2o 8478  df-oadd 8481  df-omul 8482  df-er 8718  df-ec 8720  df-qs 8724  df-map 8850  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9355  df-sup 9435  df-inf 9436  df-oi 9505  df-dju 9983  df-card 10021  df-acn 10024  df-pnf 11346  df-mnf 11347  df-xr 11348  df-ltxr 11349  df-le 11350  df-sub 11544  df-neg 11545  df-div 11975  df-nn 12337  df-2 12406  df-3 12407  df-n0 12608  df-xnn0 12681  df-z 12695  df-uz 12967  df-q 13077  df-rp 13122  df-fz 13641  df-fzo 13790  df-fl 13933  df-mod 14011  df-seq 14146  df-exp 14206  df-fac 14419  df-bc 14448  df-hash 14476  df-word 14660  df-concat 14717  df-s1 14744  df-cj 15267  df-re 15268  df-im 15269  df-sqrt 15403  df-abs 15404  df-clim 15656  df-sum 15855  df-dvds 16424  df-gcd 16666  df-prm 16848  df-pc 17016  df-sets 17343  df-slot 17361  df-ndx 17373  df-base 17389  df-ress 17410  df-plusg 17442  df-0g 17613  df-gsum 17614  df-mre 17757  df-mrc 17758  df-acs 17760  df-mgm 18817  df-sgrp 18909  df-mnd 18925  df-mhm 18979  df-submnd 18980  df-grp 19148  df-minusg 19149  df-sbg 19150  df-mulg 19279  df-subg 19334  df-eqg 19336  df-ghm 19429  df-gim 19474  df-ga 19505  df-cntz 19532  df-oppg 19561  df-od 19743  df-gex 19744  df-pgp 19745  df-lsm 19851  df-pj1 19852  df-cmn 19997  df-abl 19998  df-cyg 20093  df-dprd 20212

This theorem is used by:  ablfaclem3  20304