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Theorem conjsubgen 19176
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 19058 . . . . . . 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 2726 . . . . . . . 8 (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴))
62, 3, 4, 5conjghm 19174 . . . . . . 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 579 . . . . . 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 6826 . . . . . 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 503 . . . . 5 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)):𝑋–1-1→𝑋)
102subgss 19054 . . . . . 6 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝑆 βŠ† 𝑋)
1110adantr 480 . . . . 5 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† 𝑋)
12 f1ssres 6789 . . . . 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 583 . . . 4 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) β†Ύ 𝑆):𝑆–1-1→𝑋)
1411resmptd 6034 . . . . . 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 2784 . . . . 5 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) β†Ύ 𝑆) = 𝐹)
17 f1eq1 6776 . . . . 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 6838 . . 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 8969 . 2 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐹:𝑆–1-1-ontoβ†’ran 𝐹) β†’ 𝑆 β‰ˆ ran 𝐹)
2321, 22syldan 590 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 395   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943   class class class wbr 5141   ↦ cmpt 5224  ran crn 5670   β†Ύ cres 5671  β€“1-1β†’wf1 6534  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  (class class class)co 7405   β‰ˆ cen 8938  Basecbs 17153  +gcplusg 17206  Grpcgrp 18863  -gcsg 18865  SubGrpcsubg 19047   GrpHom cghm 19138
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-en 8942  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19050  df-ghm 19139
This theorem is referenced by:  slwhash  19544  sylow2  19546  sylow3lem1  19547
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