Theorem grplactf1o 18594

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 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
grplact.2	⊢ 𝑋 = (Base‘𝐺)
grplact.3	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
grplactf1o	⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴):𝑋–1-1-onto→𝑋)

Distinct variable groups:   𝑔,𝑎,𝐴   𝐺,𝑎,𝑔   + ,𝑎,𝑔   𝑋,𝑎,𝑔

Allowed substitution hints:   𝐹(𝑔,𝑎)

Proof of Theorem grplactf1o

Step	Hyp	Ref	Expression
1		grplact.1	. . 3 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
2		grplact.2	. . 3 ⊢ 𝑋 = (Base‘𝐺)
3		grplact.3	. . 3 ⊢ + = (+g‘𝐺)
4		eqid 2738	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
5	1, 2, 3, 4	grplactcnv 18593	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡(𝐹‘𝐴) = (𝐹‘((invg‘𝐺)‘𝐴))))
6	5	simpld 494	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴):𝑋–1-1-onto→𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ↦ cmpt 5153  ^◡ccnv 5579  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Grpcgrp 18492  inv_gcminusg 18493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496

This theorem is referenced by:  eqgen  18724  dchrsum2  26321  sumdchr2  26323