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Theorem grpsubf 18822
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 2736 . . . . . . 7 (invg𝐺) = (invg𝐺)
31, 2grpinvcl 18795 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → ((invg𝐺)‘𝑦) ∈ 𝐵)
433adant2 1131 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((invg𝐺)‘𝑦) ∈ 𝐵)
5 eqid 2736 . . . . . 6 (+g𝐺) = (+g𝐺)
61, 5grpcl 18753 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵 ∧ ((invg𝐺)‘𝑦) ∈ 𝐵) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
74, 6syld3an3 1409 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
873expb 1120 . . 3 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
98ralrimivva 3196 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
10 grpsubcl.m . . . 4 = (-g𝐺)
111, 5, 2, 10grpsubfval 18791 . . 3 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦)))
1211fmpo 7997 . 2 (∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵 :(𝐵 × 𝐵)⟶𝐵)
139, 12sylib 217 1 (𝐺 ∈ Grp → :(𝐵 × 𝐵)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wral 3063   × cxp 5630  wf 6490  cfv 6494  (class class class)co 7354  Basecbs 17080  +gcplusg 17130  Grpcgrp 18745  invgcminusg 18746  -gcsg 18747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-riota 7310  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7918  df-2nd 7919  df-0g 17320  df-mgm 18494  df-sgrp 18543  df-mnd 18554  df-grp 18748  df-minusg 18749  df-sbg 18750
This theorem is referenced by:  grpsubcl  18823  cnfldsub  20821  distgp  23446  indistgp  23447  clssubg  23456  tgphaus  23464  qustgplem  23468  nrmmetd  23926  isngp2  23949  isngp3  23950  ngpds  23956  ngptgp  23988  tngnm  24011  tngngp2  24012  rrxds  24753
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