MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pgpfac Structured version   Visualization version   GIF version

Theorem pgpfac 18415
Description: Full factorization of a finite abelian p-group, by iterating pgpfac1 18411. 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) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
pgpfac.g (𝜑𝐺 ∈ Abel)
pgpfac.p (𝜑𝑃 pGrp 𝐺)
pgpfac.f (𝜑𝐵 ∈ 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.
StepHypRef Expression
1 pgpfac.g . . 3 (𝜑𝐺 ∈ Abel)
2 ablgrp 18130 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3 pgpfac.b . . . 4 𝐵 = (Base‘𝐺)
43subgid 17528 . . 3 (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
51, 2, 43syl 18 . 2 (𝜑𝐵 ∈ (SubGrp‘𝐺))
6 pgpfac.f . . 3 (𝜑𝐵 ∈ Fin)
7 eleq1 2686 . . . . . 6 (𝑡 = 𝑢 → (𝑡 ∈ (SubGrp‘𝐺) ↔ 𝑢 ∈ (SubGrp‘𝐺)))
8 eqeq2 2632 . . . . . . . 8 (𝑡 = 𝑢 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑢))
98anbi2d 739 . . . . . . 7 (𝑡 = 𝑢 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))
109rexbidv 3046 . . . . . 6 (𝑡 = 𝑢 → (∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))
117, 10imbi12d 334 . . . . 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))))
1211imbi2d 330 . . . 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 2686 . . . . . 6 (𝑡 = 𝐵 → (𝑡 ∈ (SubGrp‘𝐺) ↔ 𝐵 ∈ (SubGrp‘𝐺)))
14 eqeq2 2632 . . . . . . . 8 (𝑡 = 𝐵 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝐵))
1514anbi2d 739 . . . . . . 7 (𝑡 = 𝐵 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)))
1615rexbidv 3046 . . . . . 6 (𝑡 = 𝐵 → (∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)))
1713, 16imbi12d 334 . . . . 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))))
1817imbi2d 330 . . . 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 bi2.04 376 . . . . . . . . 9 ((𝑡𝑢 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) ↔ (𝑡 ∈ (SubGrp‘𝐺) → (𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))))
2019imbi2i 326 . . . . . . . 8 ((𝜑 → (𝑡𝑢 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → (𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
21 bi2.04 376 . . . . . . . 8 ((𝑡𝑢 → (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ (𝜑 → (𝑡𝑢 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
22 bi2.04 376 . . . . . . . 8 ((𝑡 ∈ (SubGrp‘𝐺) → (𝜑 → (𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → (𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
2320, 21, 223bitr4i 292 . . . . . . 7 ((𝑡𝑢 → (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ (𝑡 ∈ (SubGrp‘𝐺) → (𝜑 → (𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
2423albii 1744 . . . . . 6 (∀𝑡(𝑡𝑢 → (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ ∀𝑡(𝑡 ∈ (SubGrp‘𝐺) → (𝜑 → (𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
25 df-ral 2912 . . . . . 6 (∀𝑡 ∈ (SubGrp‘𝐺)(𝜑 → (𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) ↔ ∀𝑡(𝑡 ∈ (SubGrp‘𝐺) → (𝜑 → (𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))))
26 r19.21v 2955 . . . . . 6 (∀𝑡 ∈ (SubGrp‘𝐺)(𝜑 → (𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) ↔ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))))
2724, 25, 263bitr2i 288 . . . . 5 (∀𝑡(𝑡𝑢 → (𝜑 → (𝑡 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))) ↔ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))))
28 pgpfac.c . . . . . . . . 9 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
291adantr 481 . . . . . . . . 9 ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → 𝐺 ∈ Abel)
30 pgpfac.p . . . . . . . . . 10 (𝜑𝑃 pGrp 𝐺)
3130adantr 481 . . . . . . . . 9 ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → 𝑃 pGrp 𝐺)
326adantr 481 . . . . . . . . 9 ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → 𝐵 ∈ Fin)
33 simprr 795 . . . . . . . . 9 ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → 𝑢 ∈ (SubGrp‘𝐺))
34 simprl 793 . . . . . . . . . 10 ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))
35 psseq1 3677 . . . . . . . . . . . 12 (𝑡 = 𝑥 → (𝑡𝑢𝑥𝑢))
36 eqeq2 2632 . . . . . . . . . . . . . 14 (𝑡 = 𝑥 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑥))
3736anbi2d 739 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥)))
3837rexbidv 3046 . . . . . . . . . . . 12 (𝑡 = 𝑥 → (∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥)))
3935, 38imbi12d 334 . . . . . . . . . . 11 (𝑡 = 𝑥 → ((𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ↔ (𝑥𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥))))
4039cbvralv 3162 . . . . . . . . . 10 (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ↔ ∀𝑥 ∈ (SubGrp‘𝐺)(𝑥𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥)))
4134, 40sylib 208 . . . . . . . . 9 ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → ∀𝑥 ∈ (SubGrp‘𝐺)(𝑥𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥)))
423, 28, 29, 31, 32, 33, 41pgpfaclem3 18414 . . . . . . . 8 ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢))
4342exp32 630 . . . . . . 7 (𝜑 → (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) → (𝑢 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢))))
4443a1i 11 . . . . . 6 (𝑢 ∈ Fin → (𝜑 → (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) → (𝑢 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))))
4544a2d 29 . . . . 5 (𝑢 ∈ Fin → ((𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) → (𝜑 → (𝑢 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))))
4627, 45syl5bi 232 . . . 4 (𝑢 ∈ Fin → (∀𝑡(𝑡𝑢 → (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)))))
4712, 18, 46findcard3 8155 . . 3 (𝐵 ∈ Fin → (𝜑 → (𝐵 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))))
486, 47mpcom 38 . 2 (𝜑 → (𝐵 ∈ (SubGrp‘𝐺) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)))
495, 48mpd 15 1 (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  cin 3558  wpss 3560   class class class wbr 4618  dom cdm 5079  ran crn 5080  cfv 5852  (class class class)co 6610  Fincfn 7907  Word cword 13238  Basecbs 15792  s cress 15793  Grpcgrp 17354  SubGrpcsubg 17520   pGrp cpgp 17878  Abelcabl 18126  CycGrpccyg 18211   DProd cdprd 18324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-rpss 6897  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-tpos 7304  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-omul 7517  df-er 7694  df-ec 7696  df-qs 7700  df-map 7811  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-sup 8300  df-inf 8301  df-oi 8367  df-card 8717  df-acn 8720  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-xnn0 11316  df-z 11330  df-uz 11640  df-q 11741  df-rp 11785  df-fz 12277  df-fzo 12415  df-fl 12541  df-mod 12617  df-seq 12750  df-exp 12809  df-fac 13009  df-bc 13038  df-hash 13066  df-word 13246  df-concat 13248  df-s1 13249  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-clim 14161  df-sum 14359  df-dvds 14919  df-gcd 15152  df-prm 15321  df-pc 15477  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-0g 16034  df-gsum 16035  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mhm 17267  df-submnd 17268  df-grp 17357  df-minusg 17358  df-sbg 17359  df-mulg 17473  df-subg 17523  df-eqg 17525  df-ghm 17590  df-gim 17633  df-ga 17655  df-cntz 17682  df-oppg 17708  df-od 17880  df-gex 17881  df-pgp 17882  df-lsm 17983  df-pj1 17984  df-cmn 18127  df-abl 18128  df-cyg 18212  df-dprd 18326
This theorem is referenced by:  ablfaclem3  18418
  Copyright terms: Public domain W3C validator