Theorem pgpfac 19199

Description: Full factorization of a finite abelian p-group, by iterating pgpfac1 19195. 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 18903	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3		pgpfac.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgid 18273	. . 3 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
5	1, 2, 4	3syl 18	. 2 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝐺))
6		pgpfac.f	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
7		eleq1 2877	. . . . . 6 ⊢ (𝑡 = 𝑢 → (𝑡 ∈ (SubGrp‘𝐺) ↔ 𝑢 ∈ (SubGrp‘𝐺)))
8		eqeq2 2810	. . . . . . . 8 ⊢ (𝑡 = 𝑢 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑢))
9	8	anbi2d 631	. . . . . . 7 ⊢ (𝑡 = 𝑢 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))
10	9	rexbidv 3256	. . . . . 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 348	. . . . 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 344	. . . 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 2877	. . . . . 6 ⊢ (𝑡 = 𝐵 → (𝑡 ∈ (SubGrp‘𝐺) ↔ 𝐵 ∈ (SubGrp‘𝐺)))
14		eqeq2 2810	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝐵))
15	14	anbi2d 631	. . . . . . 7 ⊢ (𝑡 = 𝐵 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)))
16	15	rexbidv 3256	. . . . . 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 348	. . . . 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 344	. . . 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 1821	. . . . . 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 3111	. . . . . 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 3142	. . . . . 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 484	. . . . . . . . 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 484	. . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → 𝐵 ∈ Fin)
33		simprr 772	. . . . . . . . 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 770	. . . . . . . . . 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 4015	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (𝑡 ⊊ 𝑢 ↔ 𝑥 ⊊ 𝑢))
36		eqeq2 2810	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑥 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑥))
37	36	anbi2d 631	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥)))
38	37	rexbidv 3256	. . . . . . . . . . . 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 348	. . . . . . . . . . 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 3396	. . . . . . . . . 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 19198	. . . . . . . 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 424	. . . . . . 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 29	. . . . 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	syl5bi 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 8745	. . 3 ⊢ (𝐵 ∈ Fin → (𝜑 → (𝐵 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))))
48	6, 47	mpcom 38	. 2 ⊢ (𝜑 → (𝐵 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)))
49	5, 48	mpd 15	1 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110   ∩ cin 3880   ⊊ wpss 3882   class class class wbr 5030  dom cdm 5519  ran crn 5520  ‘cfv 6324  (class class class)co 7135  Fincfn 8492  Word cword 13857  Basecbs 16475   ↾_s cress 16476  Grpcgrp 18095  SubGrpcsubg 18265   pGrp cpgp 18646  Abelcabl 18899  CycGrpccyg 18989   DProd cdprd 19108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-disj 4996  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-rpss 7429  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-tpos 7875  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-omul 8090  df-er 8272  df-ec 8274  df-qs 8278  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-sup 8890  df-inf 8891  df-oi 8958  df-dju 9314  df-card 9352  df-acn 9355  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-fz 12886  df-fzo 13029  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13426  df-fac 13630  df-bc 13659  df-hash 13687  df-word 13858  df-concat 13914  df-s1 13941  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035  df-dvds 15600  df-gcd 15834  df-prm 16006  df-pc 16164  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-mulg 18217  df-subg 18268  df-eqg 18270  df-ghm 18348  df-gim 18391  df-ga 18412  df-cntz 18439  df-oppg 18466  df-od 18648  df-gex 18649  df-pgp 18650  df-lsm 18753  df-pj1 18754  df-cmn 18900  df-abl 18901  df-cyg 18990  df-dprd 19110

This theorem is referenced by:  ablfaclem3  19202