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Theorem 0subm 18738
Description: The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
Hypothesis
Ref Expression
0subm.z 0 = (0gβ€˜πΊ)
Assertion
Ref Expression
0subm (𝐺 ∈ Mnd β†’ { 0 } ∈ (SubMndβ€˜πΊ))

Proof of Theorem 0subm
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2724 . . . 4 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
2 0subm.z . . . 4 0 = (0gβ€˜πΊ)
31, 2mndidcl 18678 . . 3 (𝐺 ∈ Mnd β†’ 0 ∈ (Baseβ€˜πΊ))
43snssd 4805 . 2 (𝐺 ∈ Mnd β†’ { 0 } βŠ† (Baseβ€˜πΊ))
52fvexi 6896 . . . 4 0 ∈ V
65snid 4657 . . 3 0 ∈ { 0 }
76a1i 11 . 2 (𝐺 ∈ Mnd β†’ 0 ∈ { 0 })
8 velsn 4637 . . . . 5 (π‘Ž ∈ { 0 } ↔ π‘Ž = 0 )
9 velsn 4637 . . . . 5 (𝑏 ∈ { 0 } ↔ 𝑏 = 0 )
108, 9anbi12i 626 . . . 4 ((π‘Ž ∈ { 0 } ∧ 𝑏 ∈ { 0 }) ↔ (π‘Ž = 0 ∧ 𝑏 = 0 ))
11 eqid 2724 . . . . . . . 8 (+gβ€˜πΊ) = (+gβ€˜πΊ)
121, 11, 2mndlid 18683 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 0 ∈ (Baseβ€˜πΊ)) β†’ ( 0 (+gβ€˜πΊ) 0 ) = 0 )
133, 12mpdan 684 . . . . . 6 (𝐺 ∈ Mnd β†’ ( 0 (+gβ€˜πΊ) 0 ) = 0 )
14 ovex 7435 . . . . . . 7 ( 0 (+gβ€˜πΊ) 0 ) ∈ V
1514elsn 4636 . . . . . 6 (( 0 (+gβ€˜πΊ) 0 ) ∈ { 0 } ↔ ( 0 (+gβ€˜πΊ) 0 ) = 0 )
1613, 15sylibr 233 . . . . 5 (𝐺 ∈ Mnd β†’ ( 0 (+gβ€˜πΊ) 0 ) ∈ { 0 })
17 oveq12 7411 . . . . . 6 ((π‘Ž = 0 ∧ 𝑏 = 0 ) β†’ (π‘Ž(+gβ€˜πΊ)𝑏) = ( 0 (+gβ€˜πΊ) 0 ))
1817eleq1d 2810 . . . . 5 ((π‘Ž = 0 ∧ 𝑏 = 0 ) β†’ ((π‘Ž(+gβ€˜πΊ)𝑏) ∈ { 0 } ↔ ( 0 (+gβ€˜πΊ) 0 ) ∈ { 0 }))
1916, 18syl5ibrcom 246 . . . 4 (𝐺 ∈ Mnd β†’ ((π‘Ž = 0 ∧ 𝑏 = 0 ) β†’ (π‘Ž(+gβ€˜πΊ)𝑏) ∈ { 0 }))
2010, 19biimtrid 241 . . 3 (𝐺 ∈ Mnd β†’ ((π‘Ž ∈ { 0 } ∧ 𝑏 ∈ { 0 }) β†’ (π‘Ž(+gβ€˜πΊ)𝑏) ∈ { 0 }))
2120ralrimivv 3190 . 2 (𝐺 ∈ Mnd β†’ βˆ€π‘Ž ∈ { 0 }βˆ€π‘ ∈ { 0 } (π‘Ž(+gβ€˜πΊ)𝑏) ∈ { 0 })
221, 2, 11issubm 18724 . 2 (𝐺 ∈ Mnd β†’ ({ 0 } ∈ (SubMndβ€˜πΊ) ↔ ({ 0 } βŠ† (Baseβ€˜πΊ) ∧ 0 ∈ { 0 } ∧ βˆ€π‘Ž ∈ { 0 }βˆ€π‘ ∈ { 0 } (π‘Ž(+gβ€˜πΊ)𝑏) ∈ { 0 })))
234, 7, 21, 22mpbir3and 1339 1 (𝐺 ∈ Mnd β†’ { 0 } ∈ (SubMndβ€˜πΊ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βŠ† wss 3941  {csn 4621  β€˜cfv 6534  (class class class)co 7402  Basecbs 17149  +gcplusg 17202  0gc0g 17390  Mndcmnd 18663  SubMndcsubmnd 18708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-riota 7358  df-ov 7405  df-0g 17392  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-submnd 18710
This theorem is referenced by:  idressubmefmnd  18819  0subg  19074  0subgALT  19484
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