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Theorem mgmcl 17446
Description: Closure of the operation of a magma. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 13-Jan-2020.)
Hypotheses
Ref Expression
mgmcl.b 𝐵 = (Base‘𝑀)
mgmcl.o = (+g𝑀)
Assertion
Ref Expression
mgmcl ((𝑀 ∈ Mgm ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)

Proof of Theorem mgmcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmcl.b . . . . 5 𝐵 = (Base‘𝑀)
2 mgmcl.o . . . . 5 = (+g𝑀)
31, 2ismgm 17444 . . . 4 (𝑀 ∈ Mgm → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
43ibi 256 . . 3 (𝑀 ∈ Mgm → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵)
5 ovrspc2v 6835 . . . 4 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵) → (𝑋 𝑌) ∈ 𝐵)
65expcom 450 . . 3 (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵 → ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵))
74, 6syl 17 . 2 (𝑀 ∈ Mgm → ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵))
873impib 1109 1 ((𝑀 ∈ Mgm ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  Mgmcmgm 17441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816  df-mgm 17443
This theorem is referenced by:  isnmgm  17447  mgmplusf  17452  issstrmgm  17453  gsummgmpropd  17476  mndcl  17502  dfgrp2  17648  dfgrp3e  17716  mulgnncl  17757  mulgnndir  17770  mgmhmf1o  42297  idmgmhm  42298  issubmgm2  42300  rabsubmgmd  42301  mgmhmco  42311  mgmhmeql  42313  submgmacs  42314  mgmplusgiopALT  42340  rngcl  42393  c0mgm  42419  c0snmgmhm  42424
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