Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  grpsubcld Structured version   Visualization version   GIF version

Theorem grpsubcld 33004
Description: Closure of group subtraction. (Contributed by Thierry Arnoux, 3-Aug-2025.)
Hypotheses
Ref Expression
grpsubcld.b 𝐵 = (Base‘𝐺)
grpsubcld.m = (-g𝐺)
grpsubcld.g (𝜑𝐺 ∈ Grp)
grpsubcld.x (𝜑𝑋𝐵)
grpsubcld.y (𝜑𝑌𝐵)
Assertion
Ref Expression
grpsubcld (𝜑 → (𝑋 𝑌) ∈ 𝐵)

Proof of Theorem grpsubcld
StepHypRef Expression
1 grpsubcld.g . 2 (𝜑𝐺 ∈ Grp)
2 grpsubcld.x . 2 (𝜑𝑋𝐵)
3 grpsubcld.y . 2 (𝜑𝑌𝐵)
4 grpsubcld.b . . 3 𝐵 = (Base‘𝐺)
5 grpsubcld.m . . 3 = (-g𝐺)
64, 5grpsubcl 19037 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
71, 2, 3, 6syl3anc 1369 1 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1535  wcel 2104  cfv 6559  (class class class)co 7426  Basecbs 17235  Grpcgrp 18950  -gcsg 18952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7748
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3479  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 4916  df-iun 5001  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6511  df-fun 6561  df-fn 6562  df-f 6563  df-fv 6567  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8008  df-2nd 8009  df-0g 17478  df-mgm 18655  df-sgrp 18734  df-mnd 18750  df-grp 18953  df-minusg 18954  df-sbg 18955
This theorem is referenced by:  assalactf1o  33626
  Copyright terms: Public domain W3C validator