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Theorem grplactcnv 18140
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 2818 . . 3 (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑎𝑋 ↦ (𝐴 + 𝑎))
2 grplact.2 . . . . 5 𝑋 = (Base‘𝐺)
3 grplact.3 . . . . 5 + = (+g𝐺)
42, 3grpcl 18049 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑎𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
543expa 1110 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑎𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
6 grplactcnv.4 . . . . 5 𝐼 = (invg𝐺)
72, 6grpinvcl 18089 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐼𝐴) ∈ 𝑋)
82, 3grpcl 18049 . . . . 5 ((𝐺 ∈ Grp ∧ (𝐼𝐴) ∈ 𝑋𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
983expa 1110 . . . 4 (((𝐺 ∈ Grp ∧ (𝐼𝐴) ∈ 𝑋) ∧ 𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
107, 9syldanl 601 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
11 eqcom 2825 . . . . 5 (𝑎 = ((𝐼𝐴) + 𝑏) ↔ ((𝐼𝐴) + 𝑏) = 𝑎)
12 eqid 2818 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
132, 3, 12, 6grplinv 18090 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐼𝐴) + 𝐴) = (0g𝐺))
1413adantr 481 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐼𝐴) + 𝐴) = (0g𝐺))
1514oveq1d 7160 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((0g𝐺) + 𝑎))
16 simpll 763 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝐺 ∈ Grp)
177adantr 481 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝐼𝐴) ∈ 𝑋)
18 simplr 765 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝐴𝑋)
19 simprl 767 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎𝑋)
202, 3grpass 18050 . . . . . . . 8 ((𝐺 ∈ Grp ∧ ((𝐼𝐴) ∈ 𝑋𝐴𝑋𝑎𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((𝐼𝐴) + (𝐴 + 𝑎)))
2116, 17, 18, 19, 20syl13anc 1364 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((𝐼𝐴) + (𝐴 + 𝑎)))
222, 3, 12grplid 18071 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → ((0g𝐺) + 𝑎) = 𝑎)
2322ad2ant2r 743 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((0g𝐺) + 𝑎) = 𝑎)
2415, 21, 233eqtr3rd 2862 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 = ((𝐼𝐴) + (𝐴 + 𝑎)))
2524eqeq2d 2829 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝑏) = 𝑎 ↔ ((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎))))
2611, 25syl5bb 284 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 = ((𝐼𝐴) + 𝑏) ↔ ((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎))))
27 simprr 769 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑏𝑋)
285adantrr 713 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝐴 + 𝑎) ∈ 𝑋)
292, 3grplcan 18099 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑏𝑋 ∧ (𝐴 + 𝑎) ∈ 𝑋 ∧ (𝐼𝐴) ∈ 𝑋)) → (((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
3016, 27, 28, 17, 29syl13anc 1364 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
3126, 30bitrd 280 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 = ((𝐼𝐴) + 𝑏) ↔ 𝑏 = (𝐴 + 𝑎)))
321, 5, 10, 31f1ocnv2d 7387 . 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 18138 . . . . 5 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3534adantl 482 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
36 f1oeq1 6597 . . . 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 5739 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3933, 2grplactfval 18138 . . . . . 6 ((𝐼𝐴) ∈ 𝑋 → (𝐹‘(𝐼𝐴)) = (𝑎𝑋 ↦ ((𝐼𝐴) + 𝑎)))
40 oveq2 7153 . . . . . . 7 (𝑎 = 𝑏 → ((𝐼𝐴) + 𝑎) = ((𝐼𝐴) + 𝑏))
4140cbvmptv 5160 . . . . . 6 (𝑎𝑋 ↦ ((𝐼𝐴) + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏))
4239, 41syl6eq 2869 . . . . 5 ((𝐼𝐴) ∈ 𝑋 → (𝐹‘(𝐼𝐴)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏)))
437, 42syl 17 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘(𝐼𝐴)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏)))
4438, 43eqeq12d 2834 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴) = (𝐹‘(𝐼𝐴)) ↔ (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏))))
4537, 44anbi12d 630 . 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 258 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋(𝐹𝐴) = (𝐹‘(𝐼𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  cmpt 5137  ccnv 5547  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Grpcgrp 18041  invgcminusg 18042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045
This theorem is referenced by:  grplactf1o  18141  eqglact  18269  tgplacthmeo  22639  tgpconncompeqg  22647
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