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Theorem 0subm 18053
 Description: The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
Hypothesis
Ref Expression
0subm.z 0 = (0g𝐺)
Assertion
Ref Expression
0subm (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))

Proof of Theorem 0subm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2758 . . . 4 (Base‘𝐺) = (Base‘𝐺)
2 0subm.z . . . 4 0 = (0g𝐺)
31, 2mndidcl 17997 . . 3 (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺))
43snssd 4702 . 2 (𝐺 ∈ Mnd → { 0 } ⊆ (Base‘𝐺))
52fvexi 6676 . . . 4 0 ∈ V
65snid 4561 . . 3 0 ∈ { 0 }
76a1i 11 . 2 (𝐺 ∈ Mnd → 0 ∈ { 0 })
8 velsn 4541 . . . . 5 (𝑎 ∈ { 0 } ↔ 𝑎 = 0 )
9 velsn 4541 . . . . 5 (𝑏 ∈ { 0 } ↔ 𝑏 = 0 )
108, 9anbi12i 629 . . . 4 ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) ↔ (𝑎 = 0𝑏 = 0 ))
11 eqid 2758 . . . . . . . 8 (+g𝐺) = (+g𝐺)
121, 11, 2mndlid 18002 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g𝐺) 0 ) = 0 )
133, 12mpdan 686 . . . . . 6 (𝐺 ∈ Mnd → ( 0 (+g𝐺) 0 ) = 0 )
14 ovex 7188 . . . . . . 7 ( 0 (+g𝐺) 0 ) ∈ V
1514elsn 4540 . . . . . 6 (( 0 (+g𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g𝐺) 0 ) = 0 )
1613, 15sylibr 237 . . . . 5 (𝐺 ∈ Mnd → ( 0 (+g𝐺) 0 ) ∈ { 0 })
17 oveq12 7164 . . . . . 6 ((𝑎 = 0𝑏 = 0 ) → (𝑎(+g𝐺)𝑏) = ( 0 (+g𝐺) 0 ))
1817eleq1d 2836 . . . . 5 ((𝑎 = 0𝑏 = 0 ) → ((𝑎(+g𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g𝐺) 0 ) ∈ { 0 }))
1916, 18syl5ibrcom 250 . . . 4 (𝐺 ∈ Mnd → ((𝑎 = 0𝑏 = 0 ) → (𝑎(+g𝐺)𝑏) ∈ { 0 }))
2010, 19syl5bi 245 . . 3 (𝐺 ∈ Mnd → ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) → (𝑎(+g𝐺)𝑏) ∈ { 0 }))
2120ralrimivv 3119 . 2 (𝐺 ∈ Mnd → ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g𝐺)𝑏) ∈ { 0 })
221, 2, 11issubm 18039 . 2 (𝐺 ∈ Mnd → ({ 0 } ∈ (SubMnd‘𝐺) ↔ ({ 0 } ⊆ (Base‘𝐺) ∧ 0 ∈ { 0 } ∧ ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g𝐺)𝑏) ∈ { 0 })))
234, 7, 21, 22mpbir3and 1339 1 (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ⊆ wss 3860  {csn 4525  ‘cfv 6339  (class class class)co 7155  Basecbs 16546  +gcplusg 16628  0gc0g 16776  Mndcmnd 17982  SubMndcsubmnd 18026 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6298  df-fun 6341  df-fv 6347  df-riota 7113  df-ov 7158  df-0g 16778  df-mgm 17923  df-sgrp 17972  df-mnd 17983  df-submnd 18028 This theorem is referenced by:  idressubmefmnd  18134
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