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Theorem cntzsubg 17592
 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 17252 . . 3 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2 cntzrec.b . . . 4 𝐵 = (Base‘𝑀)
3 cntzrec.z . . . 4 𝑍 = (Cntz‘𝑀)
42, 3cntzsubm 17591 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
51, 4sylan 487 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
6 simpll 786 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑀 ∈ Grp)
72, 3cntzssv 17584 . . . . . . . . . . . . 13 (𝑍𝑆) ⊆ 𝐵
8 simprl 790 . . . . . . . . . . . . 13 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑥 ∈ (𝑍𝑆))
97, 8sseldi 3566 . . . . . . . . . . . 12 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑥𝐵)
10 eqid 2610 . . . . . . . . . . . . 13 (invg𝑀) = (invg𝑀)
112, 10grpinvcl 17290 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → ((invg𝑀)‘𝑥) ∈ 𝐵)
126, 9, 11syl2anc 691 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((invg𝑀)‘𝑥) ∈ 𝐵)
13 ssel2 3563 . . . . . . . . . . . 12 ((𝑆𝐵𝑦𝑆) → 𝑦𝐵)
1413ad2ant2l 778 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑦𝐵)
15 eqid 2610 . . . . . . . . . . . . 13 (+g𝑀) = (+g𝑀)
162, 15grpcl 17253 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ 𝑥𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
176, 9, 12, 16syl3anc 1318 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
182, 15grpass 17254 . . . . . . . . . . 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 1320 . . . . . . . . . 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 17254 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ (𝑦𝐵𝑥𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))))
216, 14, 9, 12, 20syl13anc 1320 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))))
2221oveq2d 6565 . . . . . . . . . 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 2647 . . . . . . . . 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 17585 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
2524adantl 481 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
2625oveq1d 6564 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)))
2726oveq2d 6565 . . . . . . . . 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 2647 . . . . . . . 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 17253 . . . . . . . . . . 11 ((𝑀 ∈ Grp ∧ 𝑦𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
306, 14, 12, 29syl3anc 1318 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
312, 15grpass 17254 . . . . . . . . . 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 1320 . . . . . . . . 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 17254 . . . . . . . . . . 11 ((𝑀 ∈ Grp ∧ (𝑥𝐵𝑦𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
346, 9, 14, 12, 33syl13anc 1320 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
3534oveq2d 6565 . . . . . . . . 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 2647 . . . . . . . 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 2647 . . . . . . 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 2610 . . . . . . . . . . 11 (0g𝑀) = (0g𝑀)
392, 15, 38, 10grprinv 17292 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) = (0g𝑀))
406, 9, 39syl2anc 691 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) = (0g𝑀))
4140oveq2d 6565 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)))
422, 15grpcl 17253 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ ((invg𝑀)‘𝑥) ∈ 𝐵𝑦𝐵) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵)
436, 12, 14, 42syl3anc 1318 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵)
442, 15, 38grprid 17276 . . . . . . . . 9 ((𝑀 ∈ Grp ∧ (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
456, 43, 44syl2anc 691 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
4641, 45eqtrd 2644 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
472, 15, 38, 10grplinv 17291 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → (((invg𝑀)‘𝑥)(+g𝑀)𝑥) = (0g𝑀))
486, 9, 47syl2anc 691 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑥) = (0g𝑀))
4948oveq1d 6564 . . . . . . . 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 17275 . . . . . . . . 9 ((𝑀 ∈ Grp ∧ (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵) → ((0g𝑀)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
516, 30, 50syl2anc 691 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((0g𝑀)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5249, 51eqtrd 2644 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑥)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5337, 46, 523eqtr3d 2652 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5453anassrs 678 . . . . 5 ((((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) ∧ 𝑦𝑆) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5554ralrimiva 2949 . . . 4 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
56 simplr 788 . . . . 5 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑆𝐵)
57 simpll 786 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑀 ∈ Grp)
58 simpr 476 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑥 ∈ (𝑍𝑆))
597, 58sseldi 3566 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑥𝐵)
6057, 59, 11syl2anc 691 . . . . 5 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ((invg𝑀)‘𝑥) ∈ 𝐵)
612, 15, 3cntzel 17579 . . . . 5 ((𝑆𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (((invg𝑀)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥))))
6256, 60, 61syl2anc 691 . . . 4 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → (((invg𝑀)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥))))
6355, 62mpbird 246 . . 3 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ((invg𝑀)‘𝑥) ∈ (𝑍𝑆))
6463ralrimiva 2949 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))
6510issubg3 17435 . . 3 (𝑀 ∈ Grp → ((𝑍𝑆) ∈ (SubGrp‘𝑀) ↔ ((𝑍𝑆) ∈ (SubMnd‘𝑀) ∧ ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))))
6665adantr 480 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → ((𝑍𝑆) ∈ (SubGrp‘𝑀) ↔ ((𝑍𝑆) ∈ (SubMnd‘𝑀) ∧ ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))))
675, 64, 66mpbir2and 959 1 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubGrp‘𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Mndcmnd 17117  SubMndcsubmnd 17157  Grpcgrp 17245  invgcminusg 17246  SubGrpcsubg 17411  Cntzccntz 17571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-subg 17414  df-cntz 17573 This theorem is referenced by:  cntrnsg  17597  lsmcntz  17915  dprdz  18252  dprdcntz2  18260  dmdprdsplit2lem  18267  cntzsdrg  36791
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