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Theorem grplmulf1o 12990
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 12925 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑥𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
543expa 1205 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑥𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
6 eqid 2189 . . . 4 (invg𝐺) = (invg𝐺)
72, 6grpinvcl 12964 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((invg𝐺)‘𝑋) ∈ 𝐵)
82, 3grpcl 12925 . . . 4 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑋) ∈ 𝐵𝑦𝐵) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
983expa 1205 . . 3 (((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑋) ∈ 𝐵) ∧ 𝑦𝐵) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
107, 9syldanl 449 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
11 eqcom 2191 . . 3 (𝑥 = (((invg𝐺)‘𝑋) + 𝑦) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥)
12 simpll 527 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝐺 ∈ Grp)
1310adantrl 478 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
14 simprl 529 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
15 simplr 528 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝑋𝐵)
162, 3grplcan 12978 . . . . 5 ((𝐺 ∈ Grp ∧ ((((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵𝑥𝐵𝑋𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥))
1712, 13, 14, 15, 16syl13anc 1251 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥))
18 eqid 2189 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
192, 3, 18, 6grprinv 12967 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + ((invg𝐺)‘𝑋)) = (0g𝐺))
2019adantr 276 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑋 + ((invg𝐺)‘𝑋)) = (0g𝐺))
2120oveq1d 5906 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = ((0g𝐺) + 𝑦))
227adantr 276 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((invg𝐺)‘𝑋) ∈ 𝐵)
23 simprr 531 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
242, 3grpass 12926 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ ((invg𝐺)‘𝑋) ∈ 𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)))
2512, 15, 22, 23, 24syl13anc 1251 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)))
262, 3, 18grplid 12947 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → ((0g𝐺) + 𝑦) = 𝑦)
2726ad2ant2rl 511 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((0g𝐺) + 𝑦) = 𝑦)
2821, 25, 273eqtr3d 2230 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = 𝑦)
2928eqeq1d 2198 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ 𝑦 = (𝑋 + 𝑥)))
3017, 29bitr3d 190 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((((invg𝐺)‘𝑋) + 𝑦) = 𝑥𝑦 = (𝑋 + 𝑥)))
3111, 30bitrid 192 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 = (((invg𝐺)‘𝑋) + 𝑦) ↔ 𝑦 = (𝑋 + 𝑥)))
321, 5, 10, 31f1o2d 6094 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → 𝐹:𝐵1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  cmpt 4079  1-1-ontowf1o 5230  cfv 5231  (class class class)co 5891  Basecbs 12486  +gcplusg 12561  0gc0g 12733  Grpcgrp 12917  invgcminusg 12918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-inn 8939  df-2 8997  df-ndx 12489  df-slot 12490  df-base 12492  df-plusg 12574  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-grp 12920  df-minusg 12921
This theorem is referenced by: (None)
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