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Theorem cntzsubg 17976
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 17637 . . 3 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2 cntzrec.b . . . 4 𝐵 = (Base‘𝑀)
3 cntzrec.z . . . 4 𝑍 = (Cntz‘𝑀)
42, 3cntzsubm 17975 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
51, 4sylan 569 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
6 simpll 750 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑀 ∈ Grp)
72, 3cntzssv 17968 . . . . . . . . . . . . 13 (𝑍𝑆) ⊆ 𝐵
8 simprl 754 . . . . . . . . . . . . 13 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑥 ∈ (𝑍𝑆))
97, 8sseldi 3750 . . . . . . . . . . . 12 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑥𝐵)
10 eqid 2771 . . . . . . . . . . . . 13 (invg𝑀) = (invg𝑀)
112, 10grpinvcl 17675 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → ((invg𝑀)‘𝑥) ∈ 𝐵)
126, 9, 11syl2anc 573 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((invg𝑀)‘𝑥) ∈ 𝐵)
13 ssel2 3747 . . . . . . . . . . . 12 ((𝑆𝐵𝑦𝑆) → 𝑦𝐵)
1413ad2ant2l 740 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑦𝐵)
15 eqid 2771 . . . . . . . . . . . . 13 (+g𝑀) = (+g𝑀)
162, 15grpcl 17638 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ 𝑥𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
176, 9, 12, 16syl3anc 1476 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
182, 15grpass 17639 . . . . . . . . . . 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 1478 . . . . . . . . . 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 17639 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ (𝑦𝐵𝑥𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))))
216, 14, 9, 12, 20syl13anc 1478 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))))
2221oveq2d 6809 . . . . . . . . . 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 2808 . . . . . . . . 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 17969 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
2524adantl 467 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
2625oveq1d 6808 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)))
2726oveq2d 6809 . . . . . . . . 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 2808 . . . . . . . 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 17638 . . . . . . . . . . 11 ((𝑀 ∈ Grp ∧ 𝑦𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
306, 14, 12, 29syl3anc 1476 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
312, 15grpass 17639 . . . . . . . . . 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 1478 . . . . . . . . 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 17639 . . . . . . . . . . 11 ((𝑀 ∈ Grp ∧ (𝑥𝐵𝑦𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
346, 9, 14, 12, 33syl13anc 1478 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
3534oveq2d 6809 . . . . . . . . 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 2808 . . . . . . . 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 2808 . . . . . . 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 2771 . . . . . . . . . . 11 (0g𝑀) = (0g𝑀)
392, 15, 38, 10grprinv 17677 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) = (0g𝑀))
406, 9, 39syl2anc 573 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) = (0g𝑀))
4140oveq2d 6809 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)))
422, 15grpcl 17638 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ ((invg𝑀)‘𝑥) ∈ 𝐵𝑦𝐵) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵)
436, 12, 14, 42syl3anc 1476 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵)
442, 15, 38grprid 17661 . . . . . . . . 9 ((𝑀 ∈ Grp ∧ (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
456, 43, 44syl2anc 573 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
4641, 45eqtrd 2805 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
472, 15, 38, 10grplinv 17676 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → (((invg𝑀)‘𝑥)(+g𝑀)𝑥) = (0g𝑀))
486, 9, 47syl2anc 573 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑥) = (0g𝑀))
4948oveq1d 6808 . . . . . . . 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 17660 . . . . . . . . 9 ((𝑀 ∈ Grp ∧ (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵) → ((0g𝑀)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
516, 30, 50syl2anc 573 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((0g𝑀)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5249, 51eqtrd 2805 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑥)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5337, 46, 523eqtr3d 2813 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5453anassrs 458 . . . . 5 ((((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) ∧ 𝑦𝑆) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5554ralrimiva 3115 . . . 4 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
56 simplr 752 . . . . 5 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑆𝐵)
57 simpll 750 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑀 ∈ Grp)
58 simpr 471 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑥 ∈ (𝑍𝑆))
597, 58sseldi 3750 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑥𝐵)
6057, 59, 11syl2anc 573 . . . . 5 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ((invg𝑀)‘𝑥) ∈ 𝐵)
612, 15, 3cntzel 17963 . . . . 5 ((𝑆𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (((invg𝑀)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥))))
6256, 60, 61syl2anc 573 . . . 4 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → (((invg𝑀)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥))))
6355, 62mpbird 247 . . 3 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ((invg𝑀)‘𝑥) ∈ (𝑍𝑆))
6463ralrimiva 3115 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))
6510issubg3 17820 . . 3 (𝑀 ∈ Grp → ((𝑍𝑆) ∈ (SubGrp‘𝑀) ↔ ((𝑍𝑆) ∈ (SubMnd‘𝑀) ∧ ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))))
6665adantr 466 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → ((𝑍𝑆) ∈ (SubGrp‘𝑀) ↔ ((𝑍𝑆) ∈ (SubMnd‘𝑀) ∧ ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))))
675, 64, 66mpbir2and 692 1 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubGrp‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wss 3723  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Mndcmnd 17502  SubMndcsubmnd 17542  Grpcgrp 17630  invgcminusg 17631  SubGrpcsubg 17796  Cntzccntz 17955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-grp 17633  df-minusg 17634  df-subg 17799  df-cntz 17957
This theorem is referenced by:  cntrnsg  17981  lsmcntz  18299  dprdz  18637  dprdcntz2  18645  dmdprdsplit2lem  18652  cntzsdrg  38298
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