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Theorem conjghm 17607
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 473 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐺 ∈ Grp)
43adantr 481 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝐺 ∈ Grp)
51, 2grpcl 17346 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑥𝑋) → (𝐴 + 𝑥) ∈ 𝑋)
653expa 1262 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥𝑋) → (𝐴 + 𝑥) ∈ 𝑋)
7 simplr 791 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥𝑋) → 𝐴𝑋)
8 conjghm.m . . . . . 6 = (-g𝐺)
91, 8grpsubcl 17411 . . . . 5 ((𝐺 ∈ Grp ∧ (𝐴 + 𝑥) ∈ 𝑋𝐴𝑋) → ((𝐴 + 𝑥) 𝐴) ∈ 𝑋)
104, 6, 7, 9syl3anc 1323 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥𝑋) → ((𝐴 + 𝑥) 𝐴) ∈ 𝑋)
11 conjghm.f . . . 4 𝐹 = (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴))
1210, 11fmptd 6341 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹:𝑋𝑋)
133adantr 481 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → 𝐺 ∈ Grp)
14 simplr 791 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → 𝐴𝑋)
15 simprl 793 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → 𝑦𝑋)
161, 2grpcl 17346 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑦𝑋) → (𝐴 + 𝑦) ∈ 𝑋)
1713, 14, 15, 16syl3anc 1323 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝐴 + 𝑦) ∈ 𝑋)
181, 8grpsubcl 17411 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝐴 + 𝑦) ∈ 𝑋𝐴𝑋) → ((𝐴 + 𝑦) 𝐴) ∈ 𝑋)
1913, 17, 14, 18syl3anc 1323 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐴 + 𝑦) 𝐴) ∈ 𝑋)
20 simprr 795 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → 𝑧𝑋)
211, 8grpsubcl 17411 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑧𝑋𝐴𝑋) → (𝑧 𝐴) ∈ 𝑋)
2213, 20, 14, 21syl3anc 1323 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝑧 𝐴) ∈ 𝑋)
231, 2grpass 17347 . . . . . 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 1325 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((((𝐴 + 𝑦) 𝐴) + 𝐴) + (𝑧 𝐴)) = (((𝐴 + 𝑦) 𝐴) + (𝐴 + (𝑧 𝐴))))
251, 2, 8grpnpcan 17423 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝐴 + 𝑦) ∈ 𝑋𝐴𝑋) → (((𝐴 + 𝑦) 𝐴) + 𝐴) = (𝐴 + 𝑦))
2613, 17, 14, 25syl3anc 1323 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐴 + 𝑦) 𝐴) + 𝐴) = (𝐴 + 𝑦))
2726oveq1d 6620 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((((𝐴 + 𝑦) 𝐴) + 𝐴) + (𝑧 𝐴)) = ((𝐴 + 𝑦) + (𝑧 𝐴)))
281, 2, 8grpaddsubass 17421 . . . . . . 7 ((𝐺 ∈ Grp ∧ ((𝐴 + 𝑦) ∈ 𝑋𝑧𝑋𝐴𝑋)) → (((𝐴 + 𝑦) + 𝑧) 𝐴) = ((𝐴 + 𝑦) + (𝑧 𝐴)))
2913, 17, 20, 14, 28syl13anc 1325 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐴 + 𝑦) + 𝑧) 𝐴) = ((𝐴 + 𝑦) + (𝑧 𝐴)))
301, 2grpass 17347 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑦𝑋𝑧𝑋)) → ((𝐴 + 𝑦) + 𝑧) = (𝐴 + (𝑦 + 𝑧)))
3113, 14, 15, 20, 30syl13anc 1325 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐴 + 𝑦) + 𝑧) = (𝐴 + (𝑦 + 𝑧)))
3231oveq1d 6620 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐴 + 𝑦) + 𝑧) 𝐴) = ((𝐴 + (𝑦 + 𝑧)) 𝐴))
3327, 29, 323eqtr2rd 2667 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐴 + (𝑦 + 𝑧)) 𝐴) = ((((𝐴 + 𝑦) 𝐴) + 𝐴) + (𝑧 𝐴)))
341, 2, 8grpaddsubass 17421 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑧𝑋𝐴𝑋)) → ((𝐴 + 𝑧) 𝐴) = (𝐴 + (𝑧 𝐴)))
3513, 14, 20, 14, 34syl13anc 1325 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐴 + 𝑧) 𝐴) = (𝐴 + (𝑧 𝐴)))
3635oveq2d 6621 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐴 + 𝑦) 𝐴) + ((𝐴 + 𝑧) 𝐴)) = (((𝐴 + 𝑦) 𝐴) + (𝐴 + (𝑧 𝐴))))
3724, 33, 363eqtr4d 2670 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐴 + (𝑦 + 𝑧)) 𝐴) = (((𝐴 + 𝑦) 𝐴) + ((𝐴 + 𝑧) 𝐴)))
381, 2grpcl 17346 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → (𝑦 + 𝑧) ∈ 𝑋)
3913, 15, 20, 38syl3anc 1323 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 + 𝑧) ∈ 𝑋)
40 oveq2 6613 . . . . . . 7 (𝑥 = (𝑦 + 𝑧) → (𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝑧)))
4140oveq1d 6620 . . . . . 6 (𝑥 = (𝑦 + 𝑧) → ((𝐴 + 𝑥) 𝐴) = ((𝐴 + (𝑦 + 𝑧)) 𝐴))
42 ovex 6633 . . . . . 6 ((𝐴 + (𝑦 + 𝑧)) 𝐴) ∈ V
4341, 11, 42fvmpt 6240 . . . . 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 6613 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴 + 𝑥) = (𝐴 + 𝑦))
4645oveq1d 6620 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴 + 𝑥) 𝐴) = ((𝐴 + 𝑦) 𝐴))
47 ovex 6633 . . . . . . 7 ((𝐴 + 𝑦) 𝐴) ∈ V
4846, 11, 47fvmpt 6240 . . . . . 6 (𝑦𝑋 → (𝐹𝑦) = ((𝐴 + 𝑦) 𝐴))
4948ad2antrl 763 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹𝑦) = ((𝐴 + 𝑦) 𝐴))
50 oveq2 6613 . . . . . . . 8 (𝑥 = 𝑧 → (𝐴 + 𝑥) = (𝐴 + 𝑧))
5150oveq1d 6620 . . . . . . 7 (𝑥 = 𝑧 → ((𝐴 + 𝑥) 𝐴) = ((𝐴 + 𝑧) 𝐴))
52 ovex 6633 . . . . . . 7 ((𝐴 + 𝑧) 𝐴) ∈ V
5351, 11, 52fvmpt 6240 . . . . . 6 (𝑧𝑋 → (𝐹𝑧) = ((𝐴 + 𝑧) 𝐴))
5453ad2antll 764 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹𝑧) = ((𝐴 + 𝑧) 𝐴))
5549, 54oveq12d 6623 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐹𝑦) + (𝐹𝑧)) = (((𝐴 + 𝑦) 𝐴) + ((𝐴 + 𝑧) 𝐴)))
5637, 44, 553eqtr4d 2670 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹‘(𝑦 + 𝑧)) = ((𝐹𝑦) + (𝐹𝑧)))
571, 1, 2, 2, 3, 3, 12, 56isghmd 17585 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐺))
583adantr 481 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝐺 ∈ Grp)
59 eqid 2626 . . . . . 6 (invg𝐺) = (invg𝐺)
601, 59grpinvcl 17383 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
6160adantr 481 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
62 simpr 477 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
63 simplr 791 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝐴𝑋)
641, 2grpcl 17346 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑦𝑋𝐴𝑋) → (𝑦 + 𝐴) ∈ 𝑋)
6558, 62, 63, 64syl3anc 1323 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → (𝑦 + 𝐴) ∈ 𝑋)
661, 2grpcl 17346 . . . 4 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝐴) ∈ 𝑋 ∧ (𝑦 + 𝐴) ∈ 𝑋) → (((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) ∈ 𝑋)
6758, 61, 65, 66syl3anc 1323 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → (((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) ∈ 𝑋)
683adantr 481 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → 𝐺 ∈ Grp)
6965adantrl 751 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑦 + 𝐴) ∈ 𝑋)
706adantrr 752 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝐴 + 𝑥) ∈ 𝑋)
7160adantr 481 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((invg𝐺)‘𝐴) ∈ 𝑋)
721, 2grplcan 17393 . . . . . 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 1325 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) = (((invg𝐺)‘𝐴) + (𝐴 + 𝑥)) ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
74 eqid 2626 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
751, 2, 74, 59grplinv 17384 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴) + 𝐴) = (0g𝐺))
7675adantr 481 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (((invg𝐺)‘𝐴) + 𝐴) = (0g𝐺))
7776oveq1d 6620 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((((invg𝐺)‘𝐴) + 𝐴) + 𝑥) = ((0g𝐺) + 𝑥))
78 simplr 791 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → 𝐴𝑋)
79 simprl 793 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
801, 2grpass 17347 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝐴𝑋𝑥𝑋)) → ((((invg𝐺)‘𝐴) + 𝐴) + 𝑥) = (((invg𝐺)‘𝐴) + (𝐴 + 𝑥)))
8168, 71, 78, 79, 80syl13anc 1325 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((((invg𝐺)‘𝐴) + 𝐴) + 𝑥) = (((invg𝐺)‘𝐴) + (𝐴 + 𝑥)))
821, 2, 74grplid 17368 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((0g𝐺) + 𝑥) = 𝑥)
8382ad2ant2r 782 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((0g𝐺) + 𝑥) = 𝑥)
8477, 81, 833eqtr3rd 2669 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥 = (((invg𝐺)‘𝐴) + (𝐴 + 𝑥)))
8584eqeq2d 2636 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → ((((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) = 𝑥 ↔ (((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) = (((invg𝐺)‘𝐴) + (𝐴 + 𝑥))))
86 simprr 795 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
871, 2, 8grpsubadd 17419 . . . . . 6 ((𝐺 ∈ Grp ∧ ((𝐴 + 𝑥) ∈ 𝑋𝐴𝑋𝑦𝑋)) → (((𝐴 + 𝑥) 𝐴) = 𝑦 ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
8868, 70, 78, 86, 87syl13anc 1325 . . . . 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 2633 . . . 4 (𝑥 = (((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) ↔ (((invg𝐺)‘𝐴) + (𝑦 + 𝐴)) = 𝑥)
91 eqcom 2633 . . . 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 6841 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹:𝑋1-1-onto𝑋)
9457, 93jca 554 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 384   = wceq 1480  wcel 1992  cmpt 4678  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  0gc0g 16016  Grpcgrp 17338  invgcminusg 17339  -gcsg 17340   GrpHom cghm 17573
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  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-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-sbg 17343  df-ghm 17574
This theorem is referenced by:  conjsubg  17608  conjsubgen  17609
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