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Theorem conjsubgen 19042
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 18934 . . . . . . 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 2737 . . . . . . . 8 (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴))
62, 3, 4, 5conjghm 19040 . . . . . . 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 581 . . . . . 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 6784 . . . . . 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 505 . . . . 5 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)):𝑋–1-1→𝑋)
102subgss 18930 . . . . . 6 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝑆 βŠ† 𝑋)
1110adantr 482 . . . . 5 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† 𝑋)
12 f1ssres 6747 . . . . 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 585 . . . 4 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) β†Ύ 𝑆):𝑆–1-1→𝑋)
1411resmptd 5995 . . . . . 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 2795 . . . . 5 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ ((𝐴 + π‘₯) βˆ’ 𝐴)) β†Ύ 𝑆) = 𝐹)
17 f1eq1 6734 . . . . 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 6796 . . 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 8912 . 2 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐹:𝑆–1-1-ontoβ†’ran 𝐹) β†’ 𝑆 β‰ˆ ran 𝐹)
2321, 22syldan 592 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 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911   class class class wbr 5106   ↦ cmpt 5189  ran crn 5635   β†Ύ cres 5636  β€“1-1β†’wf1 6494  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358   β‰ˆ cen 8881  Basecbs 17084  +gcplusg 17134  Grpcgrp 18749  -gcsg 18751  SubGrpcsubg 18923   GrpHom cghm 19006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-en 8885  df-0g 17324  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-grp 18752  df-minusg 18753  df-sbg 18754  df-subg 18926  df-ghm 19007
This theorem is referenced by:  slwhash  19407  sylow2  19409  sylow3lem1  19410
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