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Theorem grplmulf1o 18737
Description: Left multiplication by a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.)
Hypotheses
Ref Expression
grplmulf1o.b 𝐵 = (Base‘𝐺)
grplmulf1o.p + = (+g𝐺)
grplmulf1o.n 𝐹 = (𝑥𝐵 ↦ (𝑋 + 𝑥))
Assertion
Ref Expression
grplmulf1o ((𝐺 ∈ Grp ∧ 𝑋𝐵) → 𝐹:𝐵1-1-onto𝐵)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝑥, +   𝑥,𝑋
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem grplmulf1o
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 grplmulf1o.n . 2 𝐹 = (𝑥𝐵 ↦ (𝑋 + 𝑥))
2 grplmulf1o.b . . . 4 𝐵 = (Base‘𝐺)
3 grplmulf1o.p . . . 4 + = (+g𝐺)
42, 3grpcl 18673 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑥𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
543expa 1117 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑥𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
6 eqid 2736 . . . 4 (invg𝐺) = (invg𝐺)
72, 6grpinvcl 18715 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((invg𝐺)‘𝑋) ∈ 𝐵)
82, 3grpcl 18673 . . . 4 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑋) ∈ 𝐵𝑦𝐵) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
983expa 1117 . . 3 (((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑋) ∈ 𝐵) ∧ 𝑦𝐵) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
107, 9syldanl 602 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
11 eqcom 2743 . . 3 (𝑥 = (((invg𝐺)‘𝑋) + 𝑦) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥)
12 simpll 764 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝐺 ∈ Grp)
1310adantrl 713 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
14 simprl 768 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
15 simplr 766 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝑋𝐵)
162, 3grplcan 18725 . . . . 5 ((𝐺 ∈ Grp ∧ ((((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵𝑥𝐵𝑋𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥))
1712, 13, 14, 15, 16syl13anc 1371 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥))
18 eqid 2736 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
192, 3, 18, 6grprinv 18717 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + ((invg𝐺)‘𝑋)) = (0g𝐺))
2019adantr 481 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑋 + ((invg𝐺)‘𝑋)) = (0g𝐺))
2120oveq1d 7344 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = ((0g𝐺) + 𝑦))
227adantr 481 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((invg𝐺)‘𝑋) ∈ 𝐵)
23 simprr 770 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
242, 3grpass 18674 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ ((invg𝐺)‘𝑋) ∈ 𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)))
2512, 15, 22, 23, 24syl13anc 1371 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)))
262, 3, 18grplid 18697 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → ((0g𝐺) + 𝑦) = 𝑦)
2726ad2ant2rl 746 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((0g𝐺) + 𝑦) = 𝑦)
2821, 25, 273eqtr3d 2784 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = 𝑦)
2928eqeq1d 2738 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ 𝑦 = (𝑋 + 𝑥)))
3017, 29bitr3d 280 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((((invg𝐺)‘𝑋) + 𝑦) = 𝑥𝑦 = (𝑋 + 𝑥)))
3111, 30bitrid 282 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 = (((invg𝐺)‘𝑋) + 𝑦) ↔ 𝑦 = (𝑋 + 𝑥)))
321, 5, 10, 31f1o2d 7577 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → 𝐹:𝐵1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  cmpt 5172  1-1-ontowf1o 6472  cfv 6473  (class class class)co 7329  Basecbs 17001  +gcplusg 17051  0gc0g 17239  Grpcgrp 18665  invgcminusg 18666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-0g 17241  df-mgm 18415  df-sgrp 18464  df-mnd 18475  df-grp 18668  df-minusg 18669
This theorem is referenced by:  sylow1lem2  19292  sylow2blem1  19313
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