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

Theorem conjghm 17781
 Description: Conjugation is an automorphism of the group. (Contributed by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
conjghm.x 𝑋 = (Base‘𝐺)
conjghm.p + = (+g𝐺)
conjghm.m = (-g𝐺)
conjghm.f 𝐹 = (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴))
Assertion
Ref Expression
conjghm ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹 ∈ (𝐺 GrpHom 𝐺) ∧ 𝐹:𝑋1-1-onto𝑋))
Distinct variable groups:   𝑥,   𝑥, +   𝑥,𝐴   𝑥,𝐺   𝑥,𝑋
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem conjghm
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 conjghm.x . . 3 𝑋 = (Base‘𝐺)
2 conjghm.p . . 3 + = (+g𝐺)
3 simpl 474 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐺 ∈ Grp)
43adantr 472 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝐺 ∈ Grp)
51, 2grpcl 17520 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑥𝑋) → (𝐴 + 𝑥) ∈ 𝑋)
653expa 1111 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥𝑋) → (𝐴 + 𝑥) ∈ 𝑋)
7 simplr 809 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝐴𝑋)
8 conjghm.m . . . . . 6 = (-g𝐺)
91, 8grpsubcl 17585 . . . . 5 ((𝐺 ∈ Grp ∧ (𝐴 + 𝑥) ∈ 𝑋𝐴𝑋) → ((𝐴 + 𝑥) 𝐴) ∈ 𝑋)
104, 6, 7, 9syl3anc 1407 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥𝑋) → ((𝐴 + 𝑥) 𝐴) ∈ 𝑋)
11 conjghm.f . . . 4 𝐹 = (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴))
1210, 11fmptd 6468 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹:𝑋𝑋)
133adantr 472 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → 𝐺 ∈ Grp)
14 simplr 809 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → 𝐴𝑋)
15 simprl 811 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → 𝑦𝑋)
161, 2grpcl 17520 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑦𝑋) → (𝐴 + 𝑦) ∈ 𝑋)
1713, 14, 15, 16syl3anc 1407 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝐴 + 𝑦) ∈ 𝑋)
181, 8grpsubcl 17585 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝐴 + 𝑦) ∈ 𝑋𝐴𝑋) → ((𝐴 + 𝑦) 𝐴) ∈ 𝑋)
1913, 17, 14, 18syl3anc 1407 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐴 + 𝑦) 𝐴) ∈ 𝑋)
20 simprr 813 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → 𝑧𝑋)
211, 8grpsubcl 17585 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑧𝑋𝐴𝑋) → (𝑧 𝐴) ∈ 𝑋)
2213, 20, 14, 21syl3anc 1407 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝑧 𝐴) ∈ 𝑋)
231, 2grpass 17521 . . . . . 6 ((𝐺 ∈ Grp ∧ (((𝐴 + 𝑦) 𝐴) ∈ 𝑋𝐴𝑋 ∧ (𝑧 𝐴) ∈ 𝑋)) → ((((𝐴 + 𝑦) 𝐴) + 𝐴) + (𝑧 𝐴)) = (((𝐴 + 𝑦) 𝐴) + (𝐴 + (𝑧 𝐴))))
2413, 19, 14, 22, 23syl13anc 1409 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((((𝐴 + 𝑦) 𝐴) + 𝐴) + (𝑧 𝐴)) = (((𝐴 + 𝑦) 𝐴) + (𝐴 + (𝑧 𝐴))))
251, 2, 8grpnpcan 17597 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝐴 + 𝑦) ∈ 𝑋𝐴𝑋) → (((𝐴 + 𝑦) 𝐴) + 𝐴) = (𝐴 + 𝑦))
2613, 17, 14, 25syl3anc 1407 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐴 + 𝑦) 𝐴) + 𝐴) = (𝐴 + 𝑦))
2726oveq1d 6748 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((((𝐴 + 𝑦) 𝐴) + 𝐴) + (𝑧 𝐴)) = ((𝐴 + 𝑦) + (𝑧 𝐴)))
281, 2, 8grpaddsubass 17595 . . . . . . 7 ((𝐺 ∈ Grp ∧ ((𝐴 + 𝑦) ∈ 𝑋𝑧𝑋𝐴𝑋)) → (((𝐴 + 𝑦) + 𝑧) 𝐴) = ((𝐴 + 𝑦) + (𝑧 𝐴)))
2913, 17, 20, 14, 28syl13anc 1409 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐴 + 𝑦) + 𝑧) 𝐴) = ((𝐴 + 𝑦) + (𝑧 𝐴)))
301, 2grpass 17521 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑦𝑋𝑧𝑋)) → ((𝐴 + 𝑦) + 𝑧) = (𝐴 + (𝑦 + 𝑧)))
3113, 14, 15, 20, 30syl13anc 1409 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐴 + 𝑦) + 𝑧) = (𝐴 + (𝑦 + 𝑧)))
3231oveq1d 6748 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐴 + 𝑦) + 𝑧) 𝐴) = ((𝐴 + (𝑦 + 𝑧)) 𝐴))
3327, 29, 323eqtr2rd 2733 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐴 + (𝑦 + 𝑧)) 𝐴) = ((((𝐴 + 𝑦) 𝐴) + 𝐴) + (𝑧 𝐴)))
341, 2, 8grpaddsubass 17595 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑧𝑋𝐴𝑋)) → ((𝐴 + 𝑧) 𝐴) = (𝐴 + (𝑧 𝐴)))
3513, 14, 20, 14, 34syl13anc 1409 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐴 + 𝑧) 𝐴) = (𝐴 + (𝑧 𝐴)))
3635oveq2d 6749 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐴 + 𝑦) 𝐴) + ((𝐴 + 𝑧) 𝐴)) = (((𝐴 + 𝑦) 𝐴) + (𝐴 + (𝑧 𝐴))))
3724, 33, 363eqtr4d 2736 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐴 + (𝑦 + 𝑧)) 𝐴) = (((𝐴 + 𝑦) 𝐴) + ((𝐴 + 𝑧) 𝐴)))
381, 2grpcl 17520 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → (𝑦 + 𝑧) ∈ 𝑋)
3913, 15, 20, 38syl3anc 1407 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 + 𝑧) ∈ 𝑋)
40 oveq2 6741 . . . . . . 7 (𝑥 = (𝑦 + 𝑧) → (𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝑧)))
4140oveq1d 6748 . . . . . 6 (𝑥 = (𝑦 + 𝑧) → ((𝐴 + 𝑥) 𝐴) = ((𝐴 + (𝑦 + 𝑧)) 𝐴))
42 ovex 6761 . . . . . 6 ((𝐴 + (𝑦 + 𝑧)) 𝐴) ∈ V
4341, 11, 42fvmpt 6364 . . . . 5 ((𝑦 + 𝑧) ∈ 𝑋 → (𝐹‘(𝑦 + 𝑧)) = ((𝐴 + (𝑦 + 𝑧)) 𝐴))
4439, 43syl 17 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹‘(𝑦 + 𝑧)) = ((𝐴 + (𝑦 + 𝑧)) 𝐴))
45 oveq2 6741 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴 + 𝑥) = (𝐴 + 𝑦))
4645oveq1d 6748 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴 + 𝑥) 𝐴) = ((𝐴 + 𝑦) 𝐴))
47 ovex 6761 . . . . . . 7 ((𝐴 + 𝑦) 𝐴) ∈ V
4846, 11, 47fvmpt 6364 . . . . . 6 (𝑦𝑋 → (𝐹𝑦) = ((𝐴 + 𝑦) 𝐴))
4948ad2antrl 766 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹𝑦) = ((𝐴 + 𝑦) 𝐴))
50 oveq2 6741 . . . . . . . 8 (𝑥 = 𝑧 → (𝐴 + 𝑥) = (𝐴 + 𝑧))
5150oveq1d 6748 . . . . . . 7 (𝑥 = 𝑧 → ((𝐴 + 𝑥) 𝐴) = ((𝐴 + 𝑧) 𝐴))
52 ovex 6761 . . . . . . 7 ((𝐴 + 𝑧) 𝐴) ∈ V
5351, 11, 52fvmpt 6364 . . . . . 6 (𝑧𝑋 → (𝐹𝑧) = ((𝐴 + 𝑧) 𝐴))
5453ad2antll 767 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹𝑧) = ((𝐴 + 𝑧) 𝐴))
5549, 54oveq12d 6751 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐹𝑦) + (𝐹𝑧)) = (((𝐴 + 𝑦) 𝐴) + ((𝐴 + 𝑧) 𝐴)))
5637, 44, 553eqtr4d 2736 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹‘(𝑦 + 𝑧)) = ((𝐹𝑦) + (𝐹𝑧)))
571, 1, 2, 2, 3, 3, 12, 56isghmd 17759 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐺))
583adantr 472 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝐺 ∈ Grp)
59 eqid 2692 . . . . . 6 (invg𝐺) = (invg𝐺)
601, 59grpinvcl 17557 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
6160adantr 472 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
62 simpr 479 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
63 simplr 809 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝐴𝑋)
641, 2grpcl 17520 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑦𝑋𝐴𝑋) → (𝑦 + 𝐴) ∈ 𝑋)
6558, 62, 63, 64syl3anc 1407 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → (𝑦 + 𝐴) ∈ 𝑋)
661, 2grpcl 17520 . . . 4 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝐴) ∈ 𝑋 ∧ (𝑦 + 𝐴) ∈ 𝑋) → (((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) ∈ 𝑋)
6758, 61, 65, 66syl3anc 1407 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → (((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) ∈ 𝑋)
683adantr 472 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → 𝐺 ∈ Grp)
6965adantrl 754 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑦 + 𝐴) ∈ 𝑋)
706adantrr 755 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝐴 + 𝑥) ∈ 𝑋)
7160adantr 472 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((invg𝐺)‘𝐴) ∈ 𝑋)
721, 2grplcan 17567 . . . . . 6 ((𝐺 ∈ Grp ∧ ((𝑦 + 𝐴) ∈ 𝑋 ∧ (𝐴 + 𝑥) ∈ 𝑋 ∧ ((invg𝐺)‘𝐴) ∈ 𝑋)) → ((((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) = (((invg𝐺)‘𝐴) + (𝐴 + 𝑥)) ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
7368, 69, 70, 71, 72syl13anc 1409 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) = (((invg𝐺)‘𝐴) + (𝐴 + 𝑥)) ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
74 eqid 2692 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
751, 2, 74, 59grplinv 17558 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴) + 𝐴) = (0g𝐺))
7675adantr 472 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (((invg𝐺)‘𝐴) + 𝐴) = (0g𝐺))
7776oveq1d 6748 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((((invg𝐺)‘𝐴) + 𝐴) + 𝑥) = ((0g𝐺) + 𝑥))
78 simplr 809 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑋)
79 simprl 811 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
801, 2grpass 17521 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝐴𝑋𝑥𝑋)) → ((((invg𝐺)‘𝐴) + 𝐴) + 𝑥) = (((invg𝐺)‘𝐴) + (𝐴 + 𝑥)))
8168, 71, 78, 79, 80syl13anc 1409 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((((invg𝐺)‘𝐴) + 𝐴) + 𝑥) = (((invg𝐺)‘𝐴) + (𝐴 + 𝑥)))
821, 2, 74grplid 17542 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((0g𝐺) + 𝑥) = 𝑥)
8382ad2ant2r 800 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((0g𝐺) + 𝑥) = 𝑥)
8477, 81, 833eqtr3rd 2735 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥 = (((invg𝐺)‘𝐴) + (𝐴 + 𝑥)))
8584eqeq2d 2702 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) = 𝑥 ↔ (((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) = (((invg𝐺)‘𝐴) + (𝐴 + 𝑥))))
86 simprr 813 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
871, 2, 8grpsubadd 17593 . . . . . 6 ((𝐺 ∈ Grp ∧ ((𝐴 + 𝑥) ∈ 𝑋𝐴𝑋𝑦𝑋)) → (((𝐴 + 𝑥) 𝐴) = 𝑦 ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
8868, 70, 78, 86, 87syl13anc 1409 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (((𝐴 + 𝑥) 𝐴) = 𝑦 ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
8973, 85, 883bitr4d 300 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) = 𝑥 ↔ ((𝐴 + 𝑥) 𝐴) = 𝑦))
90 eqcom 2699 . . . 4 (𝑥 = (((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) ↔ (((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) = 𝑥)
91 eqcom 2699 . . . 4 (𝑦 = ((𝐴 + 𝑥) 𝐴) ↔ ((𝐴 + 𝑥) 𝐴) = 𝑦)
9289, 90, 913bitr4g 303 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥 = (((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) ↔ 𝑦 = ((𝐴 + 𝑥) 𝐴)))
9311, 10, 67, 92f1o2d 6972 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹:𝑋1-1-onto𝑋)
9457, 93jca 555 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹 ∈ (𝐺 GrpHom 𝐺) ∧ 𝐹:𝑋1-1-onto𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1564   ∈ wcel 2071   ↦ cmpt 4805  –1-1-onto→wf1o 5968  ‘cfv 5969  (class class class)co 6733  Basecbs 15948  +gcplusg 16032  0gc0g 16191  Grpcgrp 17512  invgcminusg 17513  -gcsg 17514   GrpHom cghm 17747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-rep 4847  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-reu 2989  df-rmo 2990  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-iun 4598  df-br 4729  df-opab 4789  df-mpt 4806  df-id 5096  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-f1 5974  df-fo 5975  df-f1o 5976  df-fv 5977  df-riota 6694  df-ov 6736  df-oprab 6737  df-mpt2 6738  df-1st 7253  df-2nd 7254  df-0g 16193  df-mgm 17332  df-sgrp 17374  df-mnd 17385  df-grp 17515  df-minusg 17516  df-sbg 17517  df-ghm 17748 This theorem is referenced by:  conjsubg  17782  conjsubgen  17783
 Copyright terms: Public domain W3C validator