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Theorem pgpfac 19128
Description: Full factorization of a finite abelian p-group, by iterating pgpfac1 19124. 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 18833 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3 pgpfac.b . . . 4 𝐵 = (Base‘𝐺)
43subgid 18213 . . 3 (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
51, 2, 43syl 18 . 2 (𝜑𝐵 ∈ (SubGrp‘𝐺))
6 pgpfac.f . . 3 (𝜑𝐵 ∈ Fin)
7 eleq1 2904 . . . . . 6 (𝑡 = 𝑢 → (𝑡 ∈ (SubGrp‘𝐺) ↔ 𝑢 ∈ (SubGrp‘𝐺)))
8 eqeq2 2837 . . . . . . . 8 (𝑡 = 𝑢 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑢))
98anbi2d 628 . . . . . . 7 (𝑡 = 𝑢 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑢)))
109rexbidv 3301 . . . . . 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 346 . . . . 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 342 . . . 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 2904 . . . . . 6 (𝑡 = 𝐵 → (𝑡 ∈ (SubGrp‘𝐺) ↔ 𝐵 ∈ (SubGrp‘𝐺)))
14 eqeq2 2837 . . . . . . . 8 (𝑡 = 𝐵 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝐵))
1514anbi2d 628 . . . . . . 7 (𝑡 = 𝐵 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)))
1615rexbidv 3301 . . . . . 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 346 . . . . 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 342 . . . 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 bi2.04 389 . . . . . . . . 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 337 . . . . . . . 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 bi2.04 389 . . . . . . . 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 bi2.04 389 . . . . . . . 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 304 . . . . . . 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 1813 . . . . . 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 3147 . . . . . 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 3179 . . . . . 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 300 . . . . 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 769 . . . . . . . . 9 ((𝜑 ∧ (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑢 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ∧ 𝑢 ∈ (SubGrp‘𝐺))) → 𝑢 ∈ (SubGrp‘𝐺))
34 simprl 767 . . . . . . . . . 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 4067 . . . . . . . . . . . 12 (𝑡 = 𝑥 → (𝑡𝑢𝑥𝑢))
36 eqeq2 2837 . . . . . . . . . . . . . 14 (𝑡 = 𝑥 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑥))
3736anbi2d 628 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑥)))
3837rexbidv 3301 . . . . . . . . . . . 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 346 . . . . . . . . . . 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 3457 . . . . . . . . . 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 219 . . . . . . . . 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 19127 . . . . . . . 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 421 . . . . . . 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 243 . . . 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 8753 . . 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 396  wal 1528   = wceq 1530  wcel 2107  wral 3142  wrex 3143  {crab 3146  cin 3938  wpss 3940   class class class wbr 5062  dom cdm 5553  ran crn 5554  cfv 6351  (class class class)co 7151  Fincfn 8501  Word cword 13854  Basecbs 16475  s cress 16476  Grpcgrp 18035  SubGrpcsubg 18205   pGrp cpgp 18576  Abelcabl 18829  CycGrpccyg 18918   DProd cdprd 19037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-disj 5028  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-rpss 7442  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-tpos 7886  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-omul 8101  df-er 8282  df-ec 8284  df-qs 8288  df-map 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-acn 9363  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-q 12341  df-rp 12383  df-fz 12886  df-fzo 13027  df-fl 13155  df-mod 13231  df-seq 13363  df-exp 13423  df-fac 13627  df-bc 13656  df-hash 13684  df-word 13855  df-concat 13916  df-s1 13943  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-clim 14838  df-sum 15036  df-dvds 15600  df-gcd 15836  df-prm 16008  df-pc 16166  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 17892  df-mnd 17903  df-mhm 17946  df-submnd 17947  df-grp 18038  df-minusg 18039  df-sbg 18040  df-mulg 18157  df-subg 18208  df-eqg 18210  df-ghm 18288  df-gim 18331  df-ga 18352  df-cntz 18379  df-oppg 18406  df-od 18578  df-gex 18579  df-pgp 18580  df-lsm 18683  df-pj1 18684  df-cmn 18830  df-abl 18831  df-cyg 18919  df-dprd 19039
This theorem is referenced by:  ablfaclem3  19131
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