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Theorem grplmulf1o 18826
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 18761 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑥𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
543expa 1119 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑥𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
6 eqid 2733 . . . 4 (invg𝐺) = (invg𝐺)
72, 6grpinvcl 18803 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((invg𝐺)‘𝑋) ∈ 𝐵)
82, 3grpcl 18761 . . . 4 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑋) ∈ 𝐵𝑦𝐵) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
983expa 1119 . . 3 (((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑋) ∈ 𝐵) ∧ 𝑦𝐵) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
107, 9syldanl 603 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
11 eqcom 2740 . . 3 (𝑥 = (((invg𝐺)‘𝑋) + 𝑦) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥)
12 simpll 766 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝐺 ∈ Grp)
1310adantrl 715 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
14 simprl 770 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
15 simplr 768 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝑋𝐵)
162, 3grplcan 18814 . . . . 5 ((𝐺 ∈ Grp ∧ ((((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵𝑥𝐵𝑋𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥))
1712, 13, 14, 15, 16syl13anc 1373 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥))
18 eqid 2733 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
192, 3, 18, 6grprinv 18806 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + ((invg𝐺)‘𝑋)) = (0g𝐺))
2019adantr 482 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑋 + ((invg𝐺)‘𝑋)) = (0g𝐺))
2120oveq1d 7373 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = ((0g𝐺) + 𝑦))
227adantr 482 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((invg𝐺)‘𝑋) ∈ 𝐵)
23 simprr 772 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
242, 3grpass 18762 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ ((invg𝐺)‘𝑋) ∈ 𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)))
2512, 15, 22, 23, 24syl13anc 1373 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)))
262, 3, 18grplid 18785 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → ((0g𝐺) + 𝑦) = 𝑦)
2726ad2ant2rl 748 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((0g𝐺) + 𝑦) = 𝑦)
2821, 25, 273eqtr3d 2781 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = 𝑦)
2928eqeq1d 2735 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ 𝑦 = (𝑋 + 𝑥)))
3017, 29bitr3d 281 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((((invg𝐺)‘𝑋) + 𝑦) = 𝑥𝑦 = (𝑋 + 𝑥)))
3111, 30bitrid 283 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 = (((invg𝐺)‘𝑋) + 𝑦) ↔ 𝑦 = (𝑋 + 𝑥)))
321, 5, 10, 31f1o2d 7608 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → 𝐹:𝐵1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cmpt 5189  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Grpcgrp 18753  invgcminusg 18754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757
This theorem is referenced by:  sylow1lem2  19386  sylow2blem1  19407
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