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Theorem conjsubgen 18006
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 17912 . . . . . . . 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 2799 . . . . . . . . 9 (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) = (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴))
62, 3, 4, 5conjghm 18004 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋))
71, 6sylan 576 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋))
87simprd 490 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1-onto𝑋)
9 f1of1 6355 . . . . . 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 17908 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1211adantr 473 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆𝑋)
13 f1ssres 6323 . . . . 5 (((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)):𝑋1-1𝑋𝑆𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋)
1410, 12, 13syl2anc 580 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆):𝑆1-1𝑋)
1512resmptd 5664 . . . . . 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 2851 . . . . 5 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴)) ↾ 𝑆) = 𝐹)
18 f1eq1 6311 . . . . 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 224 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝐹:𝑆1-1𝑋)
21 f1f1orn 6367 . . 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 8214 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐹:𝑆1-1-onto→ran 𝐹) → 𝑆 ≈ ran 𝐹)
2422, 23syldan 586 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆 ≈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wss 3769   class class class wbr 4843  cmpt 4922  ran crn 5313  cres 5314  1-1wf1 6098  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6878  cen 8192  Basecbs 16184  +gcplusg 16267  Grpcgrp 17738  -gcsg 17740  SubGrpcsubg 17901   GrpHom cghm 17970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-en 8196  df-0g 16417  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-grp 17741  df-minusg 17742  df-sbg 17743  df-subg 17904  df-ghm 17971
This theorem is referenced by:  slwhash  18352  sylow2  18354  sylow3lem1  18355
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