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Theorem grpsubf 18831
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 2733 . . . . . . 7 (invg𝐺) = (invg𝐺)
31, 2grpinvcl 18803 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → ((invg𝐺)‘𝑦) ∈ 𝐵)
433adant2 1132 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((invg𝐺)‘𝑦) ∈ 𝐵)
5 eqid 2733 . . . . . 6 (+g𝐺) = (+g𝐺)
61, 5grpcl 18761 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵 ∧ ((invg𝐺)‘𝑦) ∈ 𝐵) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
74, 6syld3an3 1410 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
873expb 1121 . . 3 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
98ralrimivva 3194 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
10 grpsubcl.m . . . 4 = (-g𝐺)
111, 5, 2, 10grpsubfval 18799 . . 3 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦)))
1211fmpo 8001 . 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 1542  wcel 2107  wral 3061   × cxp 5632  wf 6493  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Grpcgrp 18753  invgcminusg 18754  -gcsg 18755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757  df-sbg 18758
This theorem is referenced by:  grpsubcl  18832  cnfldsub  20841  distgp  23466  indistgp  23467  clssubg  23476  tgphaus  23484  qustgplem  23488  nrmmetd  23946  isngp2  23969  isngp3  23970  ngpds  23976  ngptgp  24008  tngnm  24031  tngngp2  24032  rrxds  24773
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