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Theorem grplactcnv 18196
Description: The left group action of element 𝐴 of group 𝐺 maps the underlying set 𝑋 of 𝐺 one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
grplact.1 𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))
grplact.2 𝑋 = (Base‘𝐺)
grplact.3 + = (+g𝐺)
grplactcnv.4 𝐼 = (invg𝐺)
Assertion
Ref Expression
grplactcnv ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋(𝐹𝐴) = (𝐹‘(𝐼𝐴))))
Distinct variable groups:   𝑔,𝑎,𝐴   𝐺,𝑎,𝑔   𝐼,𝑎,𝑔   + ,𝑎,𝑔   𝑋,𝑎,𝑔
Allowed substitution hints:   𝐹(𝑔,𝑎)

Proof of Theorem grplactcnv
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 eqid 2821 . . 3 (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑎𝑋 ↦ (𝐴 + 𝑎))
2 grplact.2 . . . . 5 𝑋 = (Base‘𝐺)
3 grplact.3 . . . . 5 + = (+g𝐺)
42, 3grpcl 18105 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑎𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
543expa 1114 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑎𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
6 grplactcnv.4 . . . . 5 𝐼 = (invg𝐺)
72, 6grpinvcl 18145 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐼𝐴) ∈ 𝑋)
82, 3grpcl 18105 . . . . 5 ((𝐺 ∈ Grp ∧ (𝐼𝐴) ∈ 𝑋𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
983expa 1114 . . . 4 (((𝐺 ∈ Grp ∧ (𝐼𝐴) ∈ 𝑋) ∧ 𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
107, 9syldanl 603 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
11 eqcom 2828 . . . . 5 (𝑎 = ((𝐼𝐴) + 𝑏) ↔ ((𝐼𝐴) + 𝑏) = 𝑎)
12 eqid 2821 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
132, 3, 12, 6grplinv 18146 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐼𝐴) + 𝐴) = (0g𝐺))
1413adantr 483 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐼𝐴) + 𝐴) = (0g𝐺))
1514oveq1d 7165 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((0g𝐺) + 𝑎))
16 simpll 765 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝐺 ∈ Grp)
177adantr 483 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝐼𝐴) ∈ 𝑋)
18 simplr 767 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝐴𝑋)
19 simprl 769 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎𝑋)
202, 3grpass 18106 . . . . . . . 8 ((𝐺 ∈ Grp ∧ ((𝐼𝐴) ∈ 𝑋𝐴𝑋𝑎𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((𝐼𝐴) + (𝐴 + 𝑎)))
2116, 17, 18, 19, 20syl13anc 1368 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((𝐼𝐴) + (𝐴 + 𝑎)))
222, 3, 12grplid 18127 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → ((0g𝐺) + 𝑎) = 𝑎)
2322ad2ant2r 745 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((0g𝐺) + 𝑎) = 𝑎)
2415, 21, 233eqtr3rd 2865 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 = ((𝐼𝐴) + (𝐴 + 𝑎)))
2524eqeq2d 2832 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝑏) = 𝑎 ↔ ((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎))))
2611, 25syl5bb 285 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 = ((𝐼𝐴) + 𝑏) ↔ ((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎))))
27 simprr 771 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑏𝑋)
285adantrr 715 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝐴 + 𝑎) ∈ 𝑋)
292, 3grplcan 18155 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑏𝑋 ∧ (𝐴 + 𝑎) ∈ 𝑋 ∧ (𝐼𝐴) ∈ 𝑋)) → (((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
3016, 27, 28, 17, 29syl13anc 1368 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
3126, 30bitrd 281 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 = ((𝐼𝐴) + 𝑏) ↔ 𝑏 = (𝐴 + 𝑎)))
321, 5, 10, 31f1ocnv2d 7392 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑎𝑋 ↦ (𝐴 + 𝑎)):𝑋1-1-onto𝑋(𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏))))
33 grplact.1 . . . . . 6 𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))
3433, 2grplactfval 18194 . . . . 5 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3534adantl 484 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
36 f1oeq1 6599 . . . 4 ((𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)) → ((𝐹𝐴):𝑋1-1-onto𝑋 ↔ (𝑎𝑋 ↦ (𝐴 + 𝑎)):𝑋1-1-onto𝑋))
3735, 36syl 17 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋 ↔ (𝑎𝑋 ↦ (𝐴 + 𝑎)):𝑋1-1-onto𝑋))
3835cnveqd 5741 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3933, 2grplactfval 18194 . . . . . 6 ((𝐼𝐴) ∈ 𝑋 → (𝐹‘(𝐼𝐴)) = (𝑎𝑋 ↦ ((𝐼𝐴) + 𝑎)))
40 oveq2 7158 . . . . . . 7 (𝑎 = 𝑏 → ((𝐼𝐴) + 𝑎) = ((𝐼𝐴) + 𝑏))
4140cbvmptv 5162 . . . . . 6 (𝑎𝑋 ↦ ((𝐼𝐴) + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏))
4239, 41syl6eq 2872 . . . . 5 ((𝐼𝐴) ∈ 𝑋 → (𝐹‘(𝐼𝐴)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏)))
437, 42syl 17 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘(𝐼𝐴)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏)))
4438, 43eqeq12d 2837 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴) = (𝐹‘(𝐼𝐴)) ↔ (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏))))
4537, 44anbi12d 632 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((𝐹𝐴):𝑋1-1-onto𝑋(𝐹𝐴) = (𝐹‘(𝐼𝐴))) ↔ ((𝑎𝑋 ↦ (𝐴 + 𝑎)):𝑋1-1-onto𝑋(𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏)))))
4632, 45mpbird 259 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋(𝐹𝐴) = (𝐹‘(𝐼𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  cmpt 5139  ccnv 5549  1-1-ontowf1o 6349  cfv 6350  (class class class)co 7150  Basecbs 16477  +gcplusg 16559  0gc0g 16707  Grpcgrp 18097  invgcminusg 18098
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101
This theorem is referenced by:  grplactf1o  18197  eqglact  18325  tgplacthmeo  22705  tgpconncompeqg  22714
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