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

Theorem cntzsubg 18399
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 18042 . . 3 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2 cntzrec.b . . . 4 𝐵 = (Base‘𝑀)
3 cntzrec.z . . . 4 𝑍 = (Cntz‘𝑀)
42, 3cntzsubm 18398 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
51, 4sylan 580 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
6 simpll 763 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑀 ∈ Grp)
72, 3cntzssv 18390 . . . . . . . . . . . . 13 (𝑍𝑆) ⊆ 𝐵
8 simprl 767 . . . . . . . . . . . . 13 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑥 ∈ (𝑍𝑆))
97, 8sseldi 3968 . . . . . . . . . . . 12 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑥𝐵)
10 eqid 2825 . . . . . . . . . . . . 13 (invg𝑀) = (invg𝑀)
112, 10grpinvcl 18083 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → ((invg𝑀)‘𝑥) ∈ 𝐵)
126, 9, 11syl2anc 584 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((invg𝑀)‘𝑥) ∈ 𝐵)
13 ssel2 3965 . . . . . . . . . . . 12 ((𝑆𝐵𝑦𝑆) → 𝑦𝐵)
1413ad2ant2l 742 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑦𝐵)
15 eqid 2825 . . . . . . . . . . . . 13 (+g𝑀) = (+g𝑀)
162, 15grpcl 18043 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ 𝑥𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
176, 9, 12, 16syl3anc 1365 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
182, 15grpass 18044 . . . . . . . . . . 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 1366 . . . . . . . . . 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 18044 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ (𝑦𝐵𝑥𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))))
216, 14, 9, 12, 20syl13anc 1366 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))))
2221oveq2d 7167 . . . . . . . . . 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 2863 . . . . . . . . 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 18391 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
2524adantl 482 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
2625oveq1d 7166 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)))
2726oveq2d 7167 . . . . . . . . 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 2863 . . . . . . . 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 18043 . . . . . . . . . . 11 ((𝑀 ∈ Grp ∧ 𝑦𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
306, 14, 12, 29syl3anc 1365 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
312, 15grpass 18044 . . . . . . . . . 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 1366 . . . . . . . . 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 18044 . . . . . . . . . . 11 ((𝑀 ∈ Grp ∧ (𝑥𝐵𝑦𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
346, 9, 14, 12, 33syl13anc 1366 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
3534oveq2d 7167 . . . . . . . . 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 2863 . . . . . . . 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 2863 . . . . . . 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 2825 . . . . . . . . . . 11 (0g𝑀) = (0g𝑀)
392, 15, 38, 10grprinv 18085 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) = (0g𝑀))
406, 9, 39syl2anc 584 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) = (0g𝑀))
4140oveq2d 7167 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)))
422, 15grpcl 18043 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ ((invg𝑀)‘𝑥) ∈ 𝐵𝑦𝐵) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵)
436, 12, 14, 42syl3anc 1365 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵)
442, 15, 38grprid 18066 . . . . . . . . 9 ((𝑀 ∈ Grp ∧ (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
456, 43, 44syl2anc 584 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
4641, 45eqtrd 2860 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
472, 15, 38, 10grplinv 18084 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → (((invg𝑀)‘𝑥)(+g𝑀)𝑥) = (0g𝑀))
486, 9, 47syl2anc 584 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑥) = (0g𝑀))
4948oveq1d 7166 . . . . . . . 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 18065 . . . . . . . . 9 ((𝑀 ∈ Grp ∧ (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵) → ((0g𝑀)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
516, 30, 50syl2anc 584 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((0g𝑀)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5249, 51eqtrd 2860 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑥)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5337, 46, 523eqtr3d 2868 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5453anassrs 468 . . . . 5 ((((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) ∧ 𝑦𝑆) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5554ralrimiva 3186 . . . 4 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
56 simplr 765 . . . . 5 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑆𝐵)
57 simpll 763 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑀 ∈ Grp)
58 simpr 485 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑥 ∈ (𝑍𝑆))
597, 58sseldi 3968 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑥𝐵)
6057, 59, 11syl2anc 584 . . . . 5 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ((invg𝑀)‘𝑥) ∈ 𝐵)
612, 15, 3cntzel 18385 . . . . 5 ((𝑆𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (((invg𝑀)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥))))
6256, 60, 61syl2anc 584 . . . 4 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → (((invg𝑀)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥))))
6355, 62mpbird 258 . . 3 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ((invg𝑀)‘𝑥) ∈ (𝑍𝑆))
6463ralrimiva 3186 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))
6510issubg3 18229 . . 3 (𝑀 ∈ Grp → ((𝑍𝑆) ∈ (SubGrp‘𝑀) ↔ ((𝑍𝑆) ∈ (SubMnd‘𝑀) ∧ ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))))
6665adantr 481 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → ((𝑍𝑆) ∈ (SubGrp‘𝑀) ↔ ((𝑍𝑆) ∈ (SubMnd‘𝑀) ∧ ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))))
675, 64, 66mpbir2and 709 1 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubGrp‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3142  wss 3939  cfv 6351  (class class class)co 7151  Basecbs 16475  +gcplusg 16557  0gc0g 16705  Mndcmnd 17902  SubMndcsubmnd 17945  Grpcgrp 18035  invgcminusg 18036  SubGrpcsubg 18205  Cntzccntz 18377
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-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  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
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  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-iun 4918  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-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-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-submnd 17947  df-grp 18038  df-minusg 18039  df-subg 18208  df-cntz 18379
This theorem is referenced by:  cntrnsg  18404  lsmcntz  18727  cntrabl  18885  dprdz  19074  dprdcntz2  19082  dmdprdsplit2lem  19089  cntzsdrg  19503
  Copyright terms: Public domain W3C validator