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Theorem grplmulf1o 12833
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 12775 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑥𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
543expa 1203 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑥𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
6 eqid 2177 . . . 4 (invg𝐺) = (invg𝐺)
72, 6grpinvcl 12811 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((invg𝐺)‘𝑋) ∈ 𝐵)
82, 3grpcl 12775 . . . 4 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑋) ∈ 𝐵𝑦𝐵) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
983expa 1203 . . 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 2179 . . 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 12821 . . . . 5 ((𝐺 ∈ Grp ∧ ((((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵𝑥𝐵𝑋𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥))
1712, 13, 14, 15, 16syl13anc 1240 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥))
18 eqid 2177 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
192, 3, 18, 6grprinv 12813 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + ((invg𝐺)‘𝑋)) = (0g𝐺))
2019adantr 276 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑋 + ((invg𝐺)‘𝑋)) = (0g𝐺))
2120oveq1d 5884 . . . . . 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 12776 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ ((invg𝐺)‘𝑋) ∈ 𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)))
2512, 15, 22, 23, 24syl13anc 1240 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)))
262, 3, 18grplid 12796 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → ((0g𝐺) + 𝑦) = 𝑦)
2726ad2ant2rl 511 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((0g𝐺) + 𝑦) = 𝑦)
2821, 25, 273eqtr3d 2218 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = 𝑦)
2928eqeq1d 2186 . . . 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 6070 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → 𝐹:𝐵1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  cmpt 4061  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  Basecbs 12445  +gcplusg 12518  0gc0g 12653  Grpcgrp 12767  invgcminusg 12768
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-0g 12655  df-mgm 12667  df-sgrp 12700  df-mnd 12710  df-grp 12770  df-minusg 12771
This theorem is referenced by: (None)
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