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Theorem grplactcnv 17711
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 2752 . . 3 (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑎𝑋 ↦ (𝐴 + 𝑎))
2 grplact.2 . . . . 5 𝑋 = (Base‘𝐺)
3 grplact.3 . . . . 5 + = (+g𝐺)
42, 3grpcl 17623 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑎𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
543expa 1111 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑎𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
6 simpl 474 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐺 ∈ Grp)
7 grplactcnv.4 . . . . . 6 𝐼 = (invg𝐺)
82, 7grpinvcl 17660 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐼𝐴) ∈ 𝑋)
96, 8jca 555 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐺 ∈ Grp ∧ (𝐼𝐴) ∈ 𝑋))
102, 3grpcl 17623 . . . . 5 ((𝐺 ∈ Grp ∧ (𝐼𝐴) ∈ 𝑋𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
11103expa 1111 . . . 4 (((𝐺 ∈ Grp ∧ (𝐼𝐴) ∈ 𝑋) ∧ 𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
129, 11sylan 489 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑏𝑋) → ((𝐼𝐴) + 𝑏) ∈ 𝑋)
13 eqcom 2759 . . . . 5 (𝑎 = ((𝐼𝐴) + 𝑏) ↔ ((𝐼𝐴) + 𝑏) = 𝑎)
14 eqid 2752 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
152, 3, 14, 7grplinv 17661 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐼𝐴) + 𝐴) = (0g𝐺))
1615adantr 472 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐼𝐴) + 𝐴) = (0g𝐺))
1716oveq1d 6820 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((0g𝐺) + 𝑎))
18 simpll 807 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝐺 ∈ Grp)
198adantr 472 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝐼𝐴) ∈ 𝑋)
20 simplr 809 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝐴𝑋)
21 simprl 811 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎𝑋)
222, 3grpass 17624 . . . . . . . 8 ((𝐺 ∈ Grp ∧ ((𝐼𝐴) ∈ 𝑋𝐴𝑋𝑎𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((𝐼𝐴) + (𝐴 + 𝑎)))
2318, 19, 20, 21, 22syl13anc 1475 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝐴) + 𝑎) = ((𝐼𝐴) + (𝐴 + 𝑎)))
242, 3, 14grplid 17645 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → ((0g𝐺) + 𝑎) = 𝑎)
2524ad2ant2r 800 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → ((0g𝐺) + 𝑎) = 𝑎)
2617, 23, 253eqtr3rd 2795 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 = ((𝐼𝐴) + (𝐴 + 𝑎)))
2726eqeq2d 2762 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝑏) = 𝑎 ↔ ((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎))))
2813, 27syl5bb 272 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 = ((𝐼𝐴) + 𝑏) ↔ ((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎))))
29 simprr 813 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → 𝑏𝑋)
305adantrr 755 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝐴 + 𝑎) ∈ 𝑋)
312, 3grplcan 17670 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑏𝑋 ∧ (𝐴 + 𝑎) ∈ 𝑋 ∧ (𝐼𝐴) ∈ 𝑋)) → (((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
3218, 29, 30, 19, 31syl13anc 1475 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐼𝐴) + 𝑏) = ((𝐼𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
3328, 32bitrd 268 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎 = ((𝐼𝐴) + 𝑏) ↔ 𝑏 = (𝐴 + 𝑎)))
341, 5, 12, 33f1ocnv2d 7043 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑎𝑋 ↦ (𝐴 + 𝑎)):𝑋1-1-onto𝑋(𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏))))
35 grplact.1 . . . . . 6 𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))
3635, 2grplactfval 17709 . . . . 5 (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
3736adantl 473 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
38 f1oeq1 6280 . . . 4 ((𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)) → ((𝐹𝐴):𝑋1-1-onto𝑋 ↔ (𝑎𝑋 ↦ (𝐴 + 𝑎)):𝑋1-1-onto𝑋))
3937, 38syl 17 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋 ↔ (𝑎𝑋 ↦ (𝐴 + 𝑎)):𝑋1-1-onto𝑋))
4037cnveqd 5445 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
4135, 2grplactfval 17709 . . . . . 6 ((𝐼𝐴) ∈ 𝑋 → (𝐹‘(𝐼𝐴)) = (𝑎𝑋 ↦ ((𝐼𝐴) + 𝑎)))
42 oveq2 6813 . . . . . . 7 (𝑎 = 𝑏 → ((𝐼𝐴) + 𝑎) = ((𝐼𝐴) + 𝑏))
4342cbvmptv 4894 . . . . . 6 (𝑎𝑋 ↦ ((𝐼𝐴) + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏))
4441, 43syl6eq 2802 . . . . 5 ((𝐼𝐴) ∈ 𝑋 → (𝐹‘(𝐼𝐴)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏)))
458, 44syl 17 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘(𝐼𝐴)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏)))
4640, 45eqeq12d 2767 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴) = (𝐹‘(𝐼𝐴)) ↔ (𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏))))
4739, 46anbi12d 749 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((𝐹𝐴):𝑋1-1-onto𝑋(𝐹𝐴) = (𝐹‘(𝐼𝐴))) ↔ ((𝑎𝑋 ↦ (𝐴 + 𝑎)):𝑋1-1-onto𝑋(𝑎𝑋 ↦ (𝐴 + 𝑎)) = (𝑏𝑋 ↦ ((𝐼𝐴) + 𝑏)))))
4834, 47mpbird 247 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋(𝐹𝐴) = (𝐹‘(𝐼𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  cmpt 4873  ccnv 5257  1-1-ontowf1o 6040  cfv 6041  (class class class)co 6805  Basecbs 16051  +gcplusg 16135  0gc0g 16294  Grpcgrp 17615  invgcminusg 17616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-0g 16296  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-grp 17618  df-minusg 17619
This theorem is referenced by:  grplactf1o  17712  eqglact  17838  tgplacthmeo  22100  tgpconncompeqg  22108
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