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Theorem grplactcnv 18313
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 2738 . . 3 (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑎𝑋 ↦ (𝐴 + 𝑎))
2 grplact.2 . . . . 5 𝑋 = (Base‘𝐺)
3 grplact.3 . . . . 5 + = (+g𝐺)
42, 3grpcl 18220 . . . 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 18262 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐼𝐴) ∈ 𝑋)
82, 3grpcl 18220 . . . . 5 ((𝐺 ∈ Grp ∧ (𝐼𝐴) ∈ 𝑋𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
983expa 1119 . . . 4 (((𝐺 ∈ Grp ∧ (𝐼𝐴) ∈ 𝑋) ∧ 𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
107, 9syldanl 605 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
11 eqcom 2745 . . . . 5 (𝑎 = ((𝐼𝐴) + 𝑏) ↔ ((𝐼𝐴) + 𝑏) = 𝑎)
12 eqid 2738 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
132, 3, 12, 6grplinv 18263 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐼𝐴) + 𝐴) = (0g𝐺))
1413adantr 484 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐼𝐴) + 𝐴) = (0g𝐺))
1514oveq1d 7179 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((0g𝐺) + 𝑎))
16 simpll 767 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝐺 ∈ Grp)
177adantr 484 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝐼𝐴) ∈ 𝑋)
18 simplr 769 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝐴𝑋)
19 simprl 771 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎𝑋)
202, 3grpass 18221 . . . . . . . 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 18244 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → ((0g𝐺) + 𝑎) = 𝑎)
2322ad2ant2r 747 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((0g𝐺) + 𝑎) = 𝑎)
2415, 21, 233eqtr3rd 2782 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 = ((𝐼𝐴) + (𝐴 + 𝑎)))
2524eqeq2d 2749 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝑏) = 𝑎 ↔ ((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎))))
2611, 25syl5bb 286 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 = ((𝐼𝐴) + 𝑏) ↔ ((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎))))
27 simprr 773 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑏𝑋)
285adantrr 717 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝐴 + 𝑎) ∈ 𝑋)
292, 3grplcan 18272 . . . . 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 282 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 = ((𝐼𝐴) + 𝑏) ↔ 𝑏 = (𝐴 + 𝑎)))
321, 5, 10, 31f1ocnv2d 7408 . 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 18311 . . . . 5 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3534adantl 485 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3635f1oeq1d 6607 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋 ↔ (𝑎𝑋 ↦ (𝐴 + 𝑎)):𝑋1-1-onto𝑋))
3735cnveqd 5712 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3833, 2grplactfval 18311 . . . . . 6 ((𝐼𝐴) ∈ 𝑋 → (𝐹‘(𝐼𝐴)) = (𝑎𝑋 ↦ ((𝐼𝐴) + 𝑎)))
39 oveq2 7172 . . . . . . 7 (𝑎 = 𝑏 → ((𝐼𝐴) + 𝑎) = ((𝐼𝐴) + 𝑏))
4039cbvmptv 5130 . . . . . 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 2754 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴) = (𝐹‘(𝐼𝐴)) ↔ (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏))))
4436, 43anbi12d 634 . 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 260 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋(𝐹𝐴) = (𝐹‘(𝐼𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  cmpt 5107  ccnv 5518  1-1-ontowf1o 6332  cfv 6333  (class class class)co 7164  Basecbs 16579  +gcplusg 16661  0gc0g 16809  Grpcgrp 18212  invgcminusg 18213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-0g 16811  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-grp 18215  df-minusg 18216
This theorem is referenced by:  grplactf1o  18314  eqglact  18442  tgplacthmeo  22847  tgpconncompeqg  22856
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