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Theorem mgmcl 17166
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 17164 . . . 4 (𝑀 ∈ Mgm → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
43ibi 256 . . 3 (𝑀 ∈ Mgm → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵)
5 ovrspc2v 6626 . . . 4 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵) → (𝑋 𝑌) ∈ 𝐵)
65expcom 451 . . 3 (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵 → ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵))
74, 6syl 17 . 2 (𝑀 ∈ Mgm → ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵))
873impib 1259 1 ((𝑀 ∈ Mgm ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Mgmcmgm 17161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-mgm 17163
This theorem is referenced by:  isnmgm  17167  mgmplusf  17172  issstrmgm  17173  gsummgmpropd  17196  mndcl  17222  dfgrp2  17368  dfgrp3e  17436  mulgnncl  17477  mulgnndir  17490  mgmhmf1o  41072  idmgmhm  41073  issubmgm2  41075  rabsubmgmd  41076  mgmhmco  41086  mgmhmeql  41088  submgmacs  41089  mgmplusgiopALT  41115  rngcl  41168  c0mgm  41194  c0snmgmhm  41199
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