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Theorem conjsubgen 18463
 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 18356 . . . . . . 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 2758 . . . . . . . 8 (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) = (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴))
62, 3, 4, 5conjghm 18461 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋))
71, 6sylan 583 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋))
8 f1of1 6605 . . . . . 6 ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋 → (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1𝑋)
97, 8simpl2im 507 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1𝑋)
102subgss 18352 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1110adantr 484 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆𝑋)
12 f1ssres 6572 . . . . 5 (((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1𝑋𝑆𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋)
139, 11, 12syl2anc 587 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋)
1411resmptd 5884 . . . . . 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 2811 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆) = 𝐹)
17 f1eq1 6559 . . . . 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 235 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝐹:𝑆1-1𝑋)
20 f1f1orn 6617 . . 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 8551 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐹:𝑆1-1-onto→ran 𝐹) → 𝑆 ≈ ran 𝐹)
2321, 22syldan 594 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆 ≈ ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3860   class class class wbr 5035   ↦ cmpt 5115  ran crn 5528   ↾ cres 5529  –1-1→wf1 6336  –1-1-onto→wf1o 6338  ‘cfv 6339  (class class class)co 7155   ≈ cen 8529  Basecbs 16546  +gcplusg 16628  Grpcgrp 18174  -gcsg 18176  SubGrpcsubg 18345   GrpHom cghm 18427 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7698  df-2nd 7699  df-en 8533  df-0g 16778  df-mgm 17923  df-sgrp 17972  df-mnd 17983  df-grp 18177  df-minusg 18178  df-sbg 18179  df-subg 18348  df-ghm 18428 This theorem is referenced by:  slwhash  18821  sylow2  18823  sylow3lem1  18824
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