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Theorem grpsubf 19050
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 2735 . . . . . . 7 (invg𝐺) = (invg𝐺)
31, 2grpinvcl 19018 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → ((invg𝐺)‘𝑦) ∈ 𝐵)
433adant2 1130 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((invg𝐺)‘𝑦) ∈ 𝐵)
5 eqid 2735 . . . . . 6 (+g𝐺) = (+g𝐺)
61, 5grpcl 18972 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵 ∧ ((invg𝐺)‘𝑦) ∈ 𝐵) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
74, 6syld3an3 1408 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
873expb 1119 . . 3 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
98ralrimivva 3200 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)((invg𝐺)‘𝑦)) ∈ 𝐵)
10 grpsubcl.m . . . 4 = (-g𝐺)
111, 5, 2, 10grpsubfval 19014 . . 3 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥(+g𝐺)((invg𝐺)‘𝑦)))
1211fmpo 8092 . 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 1537  wcel 2106  wral 3059   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  invgcminusg 18965  -gcsg 18966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969
This theorem is referenced by:  grpsubcl  19051  cnfldsub  21428  distgp  24123  indistgp  24124  clssubg  24133  tgphaus  24141  qustgplem  24145  nrmmetd  24603  isngp2  24626  isngp3  24627  ngpds  24633  ngptgp  24665  tngnm  24688  tngngp2  24689  rrxds  25441
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