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Theorem grplactcnv 18926
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 2733 . . 3 (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑎𝑋 ↦ (𝐴 + 𝑎))
2 grplact.2 . . . . 5 𝑋 = (Base‘𝐺)
3 grplact.3 . . . . 5 + = (+g𝐺)
42, 3grpcl 18827 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑎𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
543expa 1119 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑎𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
6 grplactcnv.4 . . . . 5 𝐼 = (invg𝐺)
72, 6grpinvcl 18872 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐼𝐴) ∈ 𝑋)
82, 3grpcl 18827 . . . . 5 ((𝐺 ∈ Grp ∧ (𝐼𝐴) ∈ 𝑋𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
983expa 1119 . . . 4 (((𝐺 ∈ Grp ∧ (𝐼𝐴) ∈ 𝑋) ∧ 𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
107, 9syldanl 603 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
11 eqcom 2740 . . . . 5 (𝑎 = ((𝐼𝐴) + 𝑏) ↔ ((𝐼𝐴) + 𝑏) = 𝑎)
12 eqid 2733 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
132, 3, 12, 6grplinv 18874 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐼𝐴) + 𝐴) = (0g𝐺))
1413adantr 482 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐼𝐴) + 𝐴) = (0g𝐺))
1514oveq1d 7424 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((0g𝐺) + 𝑎))
16 simpll 766 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝐺 ∈ Grp)
177adantr 482 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝐼𝐴) ∈ 𝑋)
18 simplr 768 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝐴𝑋)
19 simprl 770 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎𝑋)
202, 3grpass 18828 . . . . . . . 8 ((𝐺 ∈ Grp ∧ ((𝐼𝐴) ∈ 𝑋𝐴𝑋𝑎𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((𝐼𝐴) + (𝐴 + 𝑎)))
2116, 17, 18, 19, 20syl13anc 1373 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((𝐼𝐴) + (𝐴 + 𝑎)))
222, 3, 12grplid 18852 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → ((0g𝐺) + 𝑎) = 𝑎)
2322ad2ant2r 746 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((0g𝐺) + 𝑎) = 𝑎)
2415, 21, 233eqtr3rd 2782 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 = ((𝐼𝐴) + (𝐴 + 𝑎)))
2524eqeq2d 2744 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝑏) = 𝑎 ↔ ((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎))))
2611, 25bitrid 283 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 = ((𝐼𝐴) + 𝑏) ↔ ((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎))))
27 simprr 772 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑏𝑋)
285adantrr 716 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝐴 + 𝑎) ∈ 𝑋)
292, 3grplcan 18885 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑏𝑋 ∧ (𝐴 + 𝑎) ∈ 𝑋 ∧ (𝐼𝐴) ∈ 𝑋)) → (((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
3016, 27, 28, 17, 29syl13anc 1373 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
3126, 30bitrd 279 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 = ((𝐼𝐴) + 𝑏) ↔ 𝑏 = (𝐴 + 𝑎)))
321, 5, 10, 31f1ocnv2d 7659 . 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 18924 . . . . 5 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3534adantl 483 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3635f1oeq1d 6829 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋 ↔ (𝑎𝑋 ↦ (𝐴 + 𝑎)):𝑋1-1-onto𝑋))
3735cnveqd 5876 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3833, 2grplactfval 18924 . . . . . 6 ((𝐼𝐴) ∈ 𝑋 → (𝐹‘(𝐼𝐴)) = (𝑎𝑋 ↦ ((𝐼𝐴) + 𝑎)))
39 oveq2 7417 . . . . . . 7 (𝑎 = 𝑏 → ((𝐼𝐴) + 𝑎) = ((𝐼𝐴) + 𝑏))
4039cbvmptv 5262 . . . . . 6 (𝑎𝑋 ↦ ((𝐼𝐴) + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏))
4138, 40eqtrdi 2789 . . . . 5 ((𝐼𝐴) ∈ 𝑋 → (𝐹‘(𝐼𝐴)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏)))
427, 41syl 17 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘(𝐼𝐴)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏)))
4337, 42eqeq12d 2749 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴) = (𝐹‘(𝐼𝐴)) ↔ (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏))))
4436, 43anbi12d 632 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((𝐹𝐴):𝑋1-1-onto𝑋(𝐹𝐴) = (𝐹‘(𝐼𝐴))) ↔ ((𝑎𝑋 ↦ (𝐴 + 𝑎)):𝑋1-1-onto𝑋(𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏)))))
4532, 44mpbird 257 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋(𝐹𝐴) = (𝐹‘(𝐼𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cmpt 5232  ccnv 5676  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  0gc0g 17385  Grpcgrp 18819  invgcminusg 18820
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823
This theorem is referenced by:  grplactf1o  18927  eqglact  19059  tgplacthmeo  23607  tgpconncompeqg  23616
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