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Theorem conjsubgen 17865
Description: A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
conjghm.x 𝑋 = (Base‘𝐺)
conjghm.p + = (+g𝐺)
conjghm.m = (-g𝐺)
conjsubg.f 𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))
Assertion
Ref Expression
conjsubgen ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆 ≈ ran 𝐹)
Distinct variable groups:   𝑥,   𝑥, +   𝑥,𝐴   𝑥,𝐺   𝑥,𝑆   𝑥,𝑋
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem conjsubgen
StepHypRef Expression
1 subgrcl 17771 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2 conjghm.x . . . . . . . . 9 𝑋 = (Base‘𝐺)
3 conjghm.p . . . . . . . . 9 + = (+g𝐺)
4 conjghm.m . . . . . . . . 9 = (-g𝐺)
5 eqid 2748 . . . . . . . . 9 (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) = (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴))
62, 3, 4, 5conjghm 17863 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋))
71, 6sylan 489 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋))
87simprd 482 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋)
9 f1of1 6285 . . . . . 6 ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋 → (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1𝑋)
108, 9syl 17 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1𝑋)
112subgss 17767 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1211adantr 472 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆𝑋)
13 f1ssres 6257 . . . . 5 (((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1𝑋𝑆𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋)
1410, 12, 13syl2anc 696 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋)
1512resmptd 5598 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆) = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴)))
16 conjsubg.f . . . . . 6 𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))
1715, 16syl6eqr 2800 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆) = 𝐹)
18 f1eq1 6245 . . . . 5 (((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆) = 𝐹 → (((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋𝐹:𝑆1-1𝑋))
1917, 18syl 17 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → (((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋𝐹:𝑆1-1𝑋))
2014, 19mpbid 222 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝐹:𝑆1-1𝑋)
21 f1f1orn 6297 . . 3 (𝐹:𝑆1-1𝑋𝐹:𝑆1-1-onto→ran 𝐹)
2220, 21syl 17 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝐹:𝑆1-1-onto→ran 𝐹)
23 f1oeng 8128 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐹:𝑆1-1-onto→ran 𝐹) → 𝑆 ≈ ran 𝐹)
2422, 23syldan 488 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆 ≈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  wcel 2127  wss 3703   class class class wbr 4792  cmpt 4869  ran crn 5255  cres 5256  1-1wf1 6034  1-1-ontowf1o 6036  cfv 6037  (class class class)co 6801  cen 8106  Basecbs 16030  +gcplusg 16114  Grpcgrp 17594  -gcsg 17596  SubGrpcsubg 17760   GrpHom cghm 17829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-en 8110  df-0g 16275  df-mgm 17414  df-sgrp 17456  df-mnd 17467  df-grp 17597  df-minusg 17598  df-sbg 17599  df-subg 17763  df-ghm 17830
This theorem is referenced by:  slwhash  18210  sylow2  18212  sylow3lem1  18213
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