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Theorem cntzsubg 17701
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 17361 . . 3 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2 cntzrec.b . . . 4 𝐵 = (Base‘𝑀)
3 cntzrec.z . . . 4 𝑍 = (Cntz‘𝑀)
42, 3cntzsubm 17700 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
51, 4sylan 488 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
6 simpll 789 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑀 ∈ Grp)
72, 3cntzssv 17693 . . . . . . . . . . . . 13 (𝑍𝑆) ⊆ 𝐵
8 simprl 793 . . . . . . . . . . . . 13 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑥 ∈ (𝑍𝑆))
97, 8sseldi 3585 . . . . . . . . . . . 12 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑥𝐵)
10 eqid 2621 . . . . . . . . . . . . 13 (invg𝑀) = (invg𝑀)
112, 10grpinvcl 17399 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → ((invg𝑀)‘𝑥) ∈ 𝐵)
126, 9, 11syl2anc 692 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((invg𝑀)‘𝑥) ∈ 𝐵)
13 ssel2 3582 . . . . . . . . . . . 12 ((𝑆𝐵𝑦𝑆) → 𝑦𝐵)
1413ad2ant2l 781 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → 𝑦𝐵)
15 eqid 2621 . . . . . . . . . . . . 13 (+g𝑀) = (+g𝑀)
162, 15grpcl 17362 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ 𝑥𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
176, 9, 12, 16syl3anc 1323 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
182, 15grpass 17363 . . . . . . . . . . 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 1325 . . . . . . . . . 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 17363 . . . . . . . . . . . 12 ((𝑀 ∈ Grp ∧ (𝑦𝐵𝑥𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))))
216, 14, 9, 12, 20syl13anc 1325 . . . . . . . . . . 11 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))))
2221oveq2d 6626 . . . . . . . . . 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 2658 . . . . . . . . 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 17694 . . . . . . . . . . . 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 6625 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = ((𝑦(+g𝑀)𝑥)(+g𝑀)((invg𝑀)‘𝑥)))
2726oveq2d 6626 . . . . . . . . 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 2658 . . . . . . . 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 17362 . . . . . . . . . . 11 ((𝑀 ∈ Grp ∧ 𝑦𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
306, 14, 12, 29syl3anc 1323 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵)
312, 15grpass 17363 . . . . . . . . . 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 1325 . . . . . . . . 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 17363 . . . . . . . . . . 11 ((𝑀 ∈ Grp ∧ (𝑥𝐵𝑦𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
346, 9, 14, 12, 33syl13anc 1325 . . . . . . . . . 10 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)((invg𝑀)‘𝑥)) = (𝑥(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))))
3534oveq2d 6626 . . . . . . . . 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 2658 . . . . . . . 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 2658 . . . . . . 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 2621 . . . . . . . . . . 11 (0g𝑀) = (0g𝑀)
392, 15, 38, 10grprinv 17401 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) = (0g𝑀))
406, 9, 39syl2anc 692 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (𝑥(+g𝑀)((invg𝑀)‘𝑥)) = (0g𝑀))
4140oveq2d 6626 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)))
422, 15grpcl 17362 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ ((invg𝑀)‘𝑥) ∈ 𝐵𝑦𝐵) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵)
436, 12, 14, 42syl3anc 1323 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵)
442, 15, 38grprid 17385 . . . . . . . . 9 ((𝑀 ∈ Grp ∧ (((invg𝑀)‘𝑥)(+g𝑀)𝑦) ∈ 𝐵) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
456, 43, 44syl2anc 692 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(0g𝑀)) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
4641, 45eqtrd 2655 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑦)(+g𝑀)(𝑥(+g𝑀)((invg𝑀)‘𝑥))) = (((invg𝑀)‘𝑥)(+g𝑀)𝑦))
472, 15, 38, 10grplinv 17400 . . . . . . . . . 10 ((𝑀 ∈ Grp ∧ 𝑥𝐵) → (((invg𝑀)‘𝑥)(+g𝑀)𝑥) = (0g𝑀))
486, 9, 47syl2anc 692 . . . . . . . . 9 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑥) = (0g𝑀))
4948oveq1d 6625 . . . . . . . 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 17384 . . . . . . . . 9 ((𝑀 ∈ Grp ∧ (𝑦(+g𝑀)((invg𝑀)‘𝑥)) ∈ 𝐵) → ((0g𝑀)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
516, 30, 50syl2anc 692 . . . . . . . 8 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((0g𝑀)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5249, 51eqtrd 2655 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → ((((invg𝑀)‘𝑥)(+g𝑀)𝑥)(+g𝑀)(𝑦(+g𝑀)((invg𝑀)‘𝑥))) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5337, 46, 523eqtr3d 2663 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ (𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆)) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5453anassrs 679 . . . . 5 ((((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) ∧ 𝑦𝑆) → (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
5554ralrimiva 2961 . . . 4 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥)))
56 simplr 791 . . . . 5 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑆𝐵)
57 simpll 789 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑀 ∈ Grp)
58 simpr 477 . . . . . . 7 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑥 ∈ (𝑍𝑆))
597, 58sseldi 3585 . . . . . 6 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → 𝑥𝐵)
6057, 59, 11syl2anc 692 . . . . 5 (((𝑀 ∈ Grp ∧ 𝑆𝐵) ∧ 𝑥 ∈ (𝑍𝑆)) → ((invg𝑀)‘𝑥) ∈ 𝐵)
612, 15, 3cntzel 17688 . . . . 5 ((𝑆𝐵 ∧ ((invg𝑀)‘𝑥) ∈ 𝐵) → (((invg𝑀)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invg𝑀)‘𝑥)(+g𝑀)𝑦) = (𝑦(+g𝑀)((invg𝑀)‘𝑥))))
6256, 60, 61syl2anc 692 . . . 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 2961 . 2 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → ∀𝑥 ∈ (𝑍𝑆)((invg𝑀)‘𝑥) ∈ (𝑍𝑆))
6510issubg3 17544 . . 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 956 1 ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubGrp‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3559  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  0gc0g 16032  Mndcmnd 17226  SubMndcsubmnd 17266  Grpcgrp 17354  invgcminusg 17355  SubGrpcsubg 17520  Cntzccntz 17680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-0g 16034  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-submnd 17268  df-grp 17357  df-minusg 17358  df-subg 17523  df-cntz 17682
This theorem is referenced by:  cntrnsg  17706  lsmcntz  18024  dprdz  18361  dprdcntz2  18369  dmdprdsplit2lem  18376  cntzsdrg  37288
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