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Theorem conjsubgen 19207
Description: A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
conjghm.x 𝑋 = (Baseβ€˜πΊ)
conjghm.p + = (+gβ€˜πΊ)
conjghm.m βˆ’ = (-gβ€˜πΊ)
conjsubg.f 𝐹 = (π‘₯ ∈ 𝑆 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴))
Assertion
Ref Expression
conjsubgen ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 β‰ˆ ran 𝐹)
Distinct variable groups:   π‘₯, βˆ’   π‘₯, +   π‘₯,𝐴   π‘₯,𝐺   π‘₯,𝑆   π‘₯,𝑋
Allowed substitution hint:   𝐹(π‘₯)
Proof of Theorem conjsubgen
StepHypRef Expression
1 subgrcl 19088 . . . . . . 7 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
2 conjghm.x . . . . . . . 8 𝑋 = (Baseβ€˜πΊ)
3 conjghm.p . . . . . . . 8 + = (+gβ€˜πΊ)
4 conjghm.m . . . . . . . 8 βˆ’ = (-gβ€˜πΊ)
5 eqid 2725 . . . . . . . 8 (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴))
62, 3, 4, 5conjghm 19205 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)):𝑋–1-1-onto→𝑋))
71, 6sylan 578 . . . . . 6 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)):𝑋–1-1-onto→𝑋))
8 f1of1 6832 . . . . . 6 ((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)):𝑋–1-1-onto→𝑋 β†’ (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)):𝑋–1-1→𝑋)
97, 8simpl2im 502 . . . . 5 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)):𝑋–1-1→𝑋)
102subgss 19084 . . . . . 6 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝑆 βŠ† 𝑋)
1110adantr 479 . . . . 5 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† 𝑋)
12 f1ssres 6795 . . . . 5 (((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)):𝑋–1-1→𝑋 ∧ 𝑆 βŠ† 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) β†Ύ 𝑆):𝑆–1-1→𝑋)
139, 11, 12syl2anc 582 . . . 4 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) β†Ύ 𝑆):𝑆–1-1→𝑋)
1411resmptd 6039 . . . . . 6 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) β†Ύ 𝑆) = (π‘₯ ∈ 𝑆 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)))
15 conjsubg.f . . . . . 6 𝐹 = (π‘₯ ∈ 𝑆 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴))
1614, 15eqtr4di 2783 . . . . 5 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) β†Ύ 𝑆) = 𝐹)
17 f1eq1 6782 . . . . 5 (((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) β†Ύ 𝑆) = 𝐹 β†’ (((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) β†Ύ 𝑆):𝑆–1-1→𝑋 ↔ 𝐹:𝑆–1-1→𝑋))
1816, 17syl 17 . . . 4 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ (((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) β†Ύ 𝑆):𝑆–1-1→𝑋 ↔ 𝐹:𝑆–1-1→𝑋))
1913, 18mpbid 231 . . 3 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ 𝐹:𝑆–1-1→𝑋)
20 f1f1orn 6844 . . 3 (𝐹:𝑆–1-1→𝑋 β†’ 𝐹:𝑆–1-1-ontoβ†’ran 𝐹)
2119, 20syl 17 . 2 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ 𝐹:𝑆–1-1-ontoβ†’ran 𝐹)
22 f1oeng 8988 . 2 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐹:𝑆–1-1-ontoβ†’ran 𝐹) β†’ 𝑆 β‰ˆ ran 𝐹)
2321, 22syldan 589 1 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 β‰ˆ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   βŠ† wss 3940   class class class wbr 5143   ↦ cmpt 5226  ran crn 5673   β†Ύ cres 5674  β€“1-1β†’wf1 6539  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7415   β‰ˆ cen 8957  Basecbs 17177  +gcplusg 17230  Grpcgrp 18892  -gcsg 18894  SubGrpcsubg 19077   GrpHom cghm 19169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-en 8961  df-0g 17420  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18895  df-minusg 18896  df-sbg 18897  df-subg 19080  df-ghm 19170
This theorem is referenced by:  slwhash  19581  sylow2  19583  sylow3lem1  19584
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