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Theorem grplmulf1o 12970
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 12906 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑥𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
543expa 1204 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑥𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
6 eqid 2187 . . . 4 (invg𝐺) = (invg𝐺)
72, 6grpinvcl 12944 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((invg𝐺)‘𝑋) ∈ 𝐵)
82, 3grpcl 12906 . . . 4 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑋) ∈ 𝐵𝑦𝐵) → (((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
983expa 1204 . . 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 2189 . . 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 12958 . . . . 5 ((𝐺 ∈ Grp ∧ ((((invg𝐺)‘𝑋) + 𝑦) ∈ 𝐵𝑥𝐵𝑋𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥))
1712, 13, 14, 15, 16syl13anc 1250 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg𝐺)‘𝑋) + 𝑦) = 𝑥))
18 eqid 2187 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
192, 3, 18, 6grprinv 12947 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + ((invg𝐺)‘𝑋)) = (0g𝐺))
2019adantr 276 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑋 + ((invg𝐺)‘𝑋)) = (0g𝐺))
2120oveq1d 5903 . . . . . 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 12907 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ ((invg𝐺)‘𝑋) ∈ 𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)))
2512, 15, 22, 23, 24syl13anc 1250 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑋 + ((invg𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)))
262, 3, 18grplid 12927 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → ((0g𝐺) + 𝑦) = 𝑦)
2726ad2ant2rl 511 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → ((0g𝐺) + 𝑦) = 𝑦)
2821, 25, 273eqtr3d 2228 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑥𝐵𝑦𝐵)) → (𝑋 + (((invg𝐺)‘𝑋) + 𝑦)) = 𝑦)
2928eqeq1d 2196 . . . 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 6089 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → 𝐹:𝐵1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158  cmpt 4076  1-1-ontowf1o 5227  cfv 5228  (class class class)co 5888  Basecbs 12475  +gcplusg 12550  0gc0g 12722  Grpcgrp 12898  invgcminusg 12899
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-minusg 12902
This theorem is referenced by: (None)
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