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Theorem grpsubf 18116
Description: Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
Hypotheses
Ref Expression
grpsubcl.b 𝐵 = (Base‘𝐺)
grpsubcl.m = (-g𝐺)
Assertion
Ref Expression
grpsubf (𝐺 ∈ Grp → :(𝐵 × 𝐵)⟶𝐵)

Proof of Theorem grpsubf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpsubcl.b . . . . . . 7 𝐵 = (Base‘𝐺)
2 eqid 2818 . . . . . . 7 (invg𝐺) = (invg𝐺)
31, 2grpinvcl 18089 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → ((invg𝐺)‘𝑦) ∈ 𝐵)
433adant2 1123 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((invg𝐺)‘𝑦) ∈ 𝐵)
5 eqid 2818 . . . . . 6 (+g𝐺) = (+g𝐺)
61, 5grpcl 18049 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵 ∧ ((invg𝐺)‘𝑦) ∈ 𝐵) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
74, 6syld3an3 1401 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
873expb 1112 . . 3 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
98ralrimivva 3188 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
10 grpsubcl.m . . . 4 = (-g𝐺)
111, 5, 2, 10grpsubfval 18085 . . 3 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦)))
1211fmpo 7755 . 2 (∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵 :(𝐵 × 𝐵)⟶𝐵)
139, 12sylib 219 1 (𝐺 ∈ Grp → :(𝐵 × 𝐵)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135   × cxp 5546  wf 6344  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Grpcgrp 18041  invgcminusg 18042  -gcsg 18043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-sbg 18046
This theorem is referenced by:  grpsubcl  18117  cnfldsub  20501  distgp  22635  indistgp  22636  clssubg  22644  tgphaus  22652  qustgplem  22656  nrmmetd  23111  isngp2  23133  isngp3  23134  ngpds  23140  ngptgp  23172  tngnm  23187  tngngp2  23188  rrxds  23923
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