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Theorem grpsubf 19037
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 2737 . . . . . . 7 (invg𝐺) = (invg𝐺)
31, 2grpinvcl 19005 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → ((invg𝐺)‘𝑦) ∈ 𝐵)
433adant2 1132 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((invg𝐺)‘𝑦) ∈ 𝐵)
5 eqid 2737 . . . . . 6 (+g𝐺) = (+g𝐺)
61, 5grpcl 18959 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵 ∧ ((invg𝐺)‘𝑦) ∈ 𝐵) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
74, 6syld3an3 1411 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
873expb 1121 . . 3 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
98ralrimivva 3202 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
10 grpsubcl.m . . . 4 = (-g𝐺)
111, 5, 2, 10grpsubfval 19001 . . 3 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦)))
1211fmpo 8093 . 2 (∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵 :(𝐵 × 𝐵)⟶𝐵)
139, 12sylib 218 1 (𝐺 ∈ Grp → :(𝐵 × 𝐵)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951  invgcminusg 18952  -gcsg 18953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956
This theorem is referenced by:  grpsubcl  19038  cnfldsub  21410  distgp  24107  indistgp  24108  clssubg  24117  tgphaus  24125  qustgplem  24129  nrmmetd  24587  isngp2  24610  isngp3  24611  ngpds  24617  ngptgp  24649  tngnm  24672  tngngp2  24673  rrxds  25427
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