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Theorem conjsubgen 19270
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 19150 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2 conjghm.x . . . . . . . 8 𝑋 = (Base‘𝐺)
3 conjghm.p . . . . . . . 8 + = (+g𝐺)
4 conjghm.m . . . . . . . 8 = (-g𝐺)
5 eqid 2736 . . . . . . . 8 (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) = (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴))
62, 3, 4, 5conjghm 19268 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋))
71, 6sylan 580 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋))
8 f1of1 6846 . . . . . 6 ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋 → (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1𝑋)
97, 8simpl2im 503 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1𝑋)
102subgss 19146 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1110adantr 480 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆𝑋)
12 f1ssres 6810 . . . . 5 (((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1𝑋𝑆𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋)
139, 11, 12syl2anc 584 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋)
1411resmptd 6057 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆) = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴)))
15 conjsubg.f . . . . . 6 𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))
1614, 15eqtr4di 2794 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆) = 𝐹)
17 f1eq1 6798 . . . . 5 (((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆) = 𝐹 → (((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋𝐹:𝑆1-1𝑋))
1816, 17syl 17 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → (((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋𝐹:𝑆1-1𝑋))
1913, 18mpbid 232 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝐹:𝑆1-1𝑋)
20 f1f1orn 6858 . . 3 (𝐹:𝑆1-1𝑋𝐹:𝑆1-1-onto→ran 𝐹)
2119, 20syl 17 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝐹:𝑆1-1-onto→ran 𝐹)
22 f1oeng 9012 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐹:𝑆1-1-onto→ran 𝐹) → 𝑆 ≈ ran 𝐹)
2321, 22syldan 591 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆 ≈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wss 3950   class class class wbr 5142  cmpt 5224  ran crn 5685  cres 5686  1-1wf1 6557  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  cen 8983  Basecbs 17248  +gcplusg 17298  Grpcgrp 18952  -gcsg 18954  SubGrpcsubg 19139   GrpHom cghm 19231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-en 8987  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956  df-sbg 18957  df-subg 19142  df-ghm 19232
This theorem is referenced by:  slwhash  19643  sylow2  19645  sylow3lem1  19646
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