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Theorem conjsubgen 19282
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 19162 . . . . . . 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 2735 . . . . . . . 8 (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) = (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴))
62, 3, 4, 5conjghm 19280 . . . . . . 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 6848 . . . . . 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 19158 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1110adantr 480 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆𝑋)
12 f1ssres 6812 . . . . 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 6060 . . . . . 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 2793 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆) = 𝐹)
17 f1eq1 6800 . . . . 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 6860 . . 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 9010 . 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 1537  wcel 2106  wss 3963   class class class wbr 5148  cmpt 5231  ran crn 5690  cres 5691  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  cen 8981  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  -gcsg 18966  SubGrpcsubg 19151   GrpHom cghm 19243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-en 8985  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-ghm 19244
This theorem is referenced by:  slwhash  19657  sylow2  19659  sylow3lem1  19660
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