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

Theorem cntzsubg 19327
Description: Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Hypotheses
Ref Expression
cntzrec.b 𝐵 = (Base‘𝑀)
cntzrec.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntzsubg ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubGrp‘𝑀))

Proof of Theorem cntzsubg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpmnd 18928 . . 3 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2 cntzrec.b . . . 4 𝐵 = (Base‘𝑀)
3 cntzrec.z . . . 4 𝑍 = (Cntz‘𝑀)
42, 3cntzsubm 19326 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
51, 4sylan 578 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
6 simpll 765 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑀 ∈ Grp)
72, 3cntzssv 19316 . . . . . . . . . . . . 13 (𝑍𝑆) ⊆ 𝐵
8 simprl 769 . . . . . . . . . . . . 13 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑥 ∈ (𝑍𝑆))
97, 8sselid 3977 . . . . . . . . . . . 12 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑥𝐵)
10 eqid 2726 . . . . . . . . . . . . 13 (invg𝑀) = (invg𝑀)
112, 10grpinvcl 18975 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → ((invg𝑀)‘𝑥) ∈ 𝐵)
126, 9, 11syl2anc 582 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((invg𝑀)‘𝑥) ∈ 𝐵)
13 ssel2 3974 . . . . . . . . . . . 12 ((𝑆𝐵𝑦𝑆) → 𝑦𝐵)
1413ad2ant2l 744 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑦𝐵)
15 eqid 2726 . . . . . . . . . . . . 13 (+g𝑀) = (+g𝑀)
162, 15grpcl 18929 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ 𝑥𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
176, 9, 12, 16syl3anc 1368 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
182, 15grpass 18930 . . . . . . . . . . 11 ((𝑀 ∈ Grp ∧ (((invg𝑀)‘𝑥) ∈ 𝐵𝑦𝐵 ∧ (𝑥(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)(𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥)))))
196, 12, 14, 17, 18syl13anc 1369 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)(𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥)))))
202, 15grpass 18930 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ (𝑦𝐵𝑥𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))))
216, 14, 9, 12, 20syl13anc 1369 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))))
2221oveq2d 7430 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)(𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥)))))
2319, 22eqtr4d 2769 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥))))
2415, 3cntzi 19317 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
2524adantl 480 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
2625oveq1d 7429 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)))
2726oveq2d 7430 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥))))
2823, 27eqtr4d 2769 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥))))
292, 15grpcl 18929 . . . . . . . . . . 11 ((𝑀 ∈ Grp ∧ 𝑦𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
306, 14, 12, 29syl3anc 1368 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
312, 15grpass 18930 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ (((invg𝑀)‘𝑥) ∈ 𝐵𝑥𝐵 ∧ (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑥)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)(𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥)))))
326, 12, 9, 30, 31syl13anc 1369 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑥)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)(𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥)))))
332, 15grpass 18930 . . . . . . . . . . 11 ((𝑀 ∈ Grp ∧ (𝑥𝐵𝑦𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
346, 9, 14, 12, 33syl13anc 1369 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
3534oveq2d 7430 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)(𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥)))))
3632, 35eqtr4d 2769 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑥)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥))))
3728, 36eqtr4d 2769 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = ((((invg𝑀)‘𝑥)(+g𝑀)𝑥)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
38 eqid 2726 . . . . . . . . . . 11 (0g𝑀) = (0g𝑀)
392, 15, 38, 10grprinv 18978 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) = (0g𝑀))
406, 9, 39syl2anc 582 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) = (0g𝑀))
4140oveq2d 7430 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)))
422, 15grpcl 18929 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ ((invg𝑀)‘𝑥) ∈ 𝐵𝑦𝐵) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵)
436, 12, 14, 42syl3anc 1368 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵)
442, 15, 38grprid 18956 . . . . . . . . 9 ((𝑀 ∈ Grp ∧ (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
456, 43, 44syl2anc 582 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
4641, 45eqtrd 2766 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
472, 15, 38, 10grplinv 18977 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → (((invg𝑀)‘𝑥)(+g𝑀)𝑥) = (0g𝑀))
486, 9, 47syl2anc 582 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑥) = (0g𝑀))
4948oveq1d 7429 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑥)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = ((0g𝑀)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
502, 15, 38grplid 18955 . . . . . . . . 9 ((𝑀 ∈ Grp ∧ (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵) → ((0g𝑀)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
516, 30, 50syl2anc 582 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((0g𝑀)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5249, 51eqtrd 2766 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑥)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5337, 46, 523eqtr3d 2774 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5453anassrs 466 . . . . 5 ((((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) ∧ 𝑦𝑆) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5554ralrimiva 3136 . . . 4 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
56 simplr 767 . . . . 5 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑆𝐵)
57 simpll 765 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑀 ∈ Grp)
58 simpr 483 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑥 ∈ (𝑍𝑆))
597, 58sselid 3977 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑥𝐵)
6057, 59, 11syl2anc 582 . . . . 5 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ((invg𝑀)‘𝑥) ∈ 𝐵)
612, 15, 3cntzel 19311 . . . . 5 ((𝑆𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (((invg𝑀)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥))))
6256, 60, 61syl2anc 582 . . . 4 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → (((invg𝑀)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥))))
6355, 62mpbird 256 . . 3 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ((invg𝑀)‘𝑥) ∈ (𝑍𝑆))
6463ralrimiva 3136 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))
6510issubg3 19132 . . 3 (𝑀 ∈ Grp → ((𝑍𝑆) ∈ (SubGrp‘𝑀) ↔ ((𝑍𝑆) ∈ (SubMnd‘𝑀) ∧ ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))))
6665adantr 479 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → ((𝑍𝑆) ∈ (SubGrp‘𝑀) ↔ ((𝑍𝑆) ∈ (SubMnd‘𝑀) ∧ ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))))
675, 64, 66mpbir2and 711 1 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubGrp‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  wss 3947  cfv 6544  (class class class)co 7414  Basecbs 17206  +gcplusg 17259  0gc0g 17447  Mndcmnd 18720  SubMndcsubmnd 18765  Grpcgrp 18921  invgcminusg 18922  SubGrpcsubg 19108  Cntzccntz 19303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-cnex 11203  ax-resscn 11204  ax-1cn 11205  ax-icn 11206  ax-addcl 11207  ax-addrcl 11208  ax-mulcl 11209  ax-mulrcl 11210  ax-mulcom 11211  ax-addass 11212  ax-mulass 11213  ax-distr 11214  ax-i2m1 11215  ax-1ne0 11216  ax-1rid 11217  ax-rnegex 11218  ax-rrecex 11219  ax-cnre 11220  ax-pre-lttri 11221  ax-pre-lttrn 11222  ax-pre-ltadd 11223  ax-pre-mulgt0 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11289  df-mnf 11290  df-xr 11291  df-ltxr 11292  df-le 11293  df-sub 11485  df-neg 11486  df-nn 12257  df-2 12319  df-sets 17159  df-slot 17177  df-ndx 17189  df-base 17207  df-ress 17236  df-plusg 17272  df-0g 17449  df-mgm 18626  df-sgrp 18705  df-mnd 18721  df-submnd 18767  df-grp 18924  df-minusg 18925  df-subg 19111  df-cntz 19305
This theorem is referenced by:  cntrnsg  19332  lsmcntz  19671  cntrabl  19835  dprdz  20024  dprdcntz2  20032  dmdprdsplit2lem  20039  cntzsdrg  20775
  Copyright terms: Public domain W3C validator