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Theorem 0subm 12748
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 2177 . . . 4 (Base‘𝐺) = (Base‘𝐺)
2 0subm.z . . . 4 0 = (0g𝐺)
31, 2mndidcl 12710 . . 3 (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺))
43snssd 3736 . 2 (𝐺 ∈ Mnd → { 0 } ⊆ (Base‘𝐺))
5 snidg 3620 . . 3 ( 0 ∈ (Base‘𝐺) → 0 ∈ { 0 })
63, 5syl 14 . 2 (𝐺 ∈ Mnd → 0 ∈ { 0 })
7 velsn 3608 . . . . 5 (𝑎 ∈ { 0 } ↔ 𝑎 = 0 )
8 velsn 3608 . . . . 5 (𝑏 ∈ { 0 } ↔ 𝑏 = 0 )
97, 8anbi12i 460 . . . 4 ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) ↔ (𝑎 = 0𝑏 = 0 ))
10 eqid 2177 . . . . . . . 8 (+g𝐺) = (+g𝐺)
111, 10, 2mndlid 12715 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g𝐺) 0 ) = 0 )
123, 11mpdan 421 . . . . . 6 (𝐺 ∈ Mnd → ( 0 (+g𝐺) 0 ) = 0 )
1312, 3eqeltrd 2254 . . . . . . 7 (𝐺 ∈ Mnd → ( 0 (+g𝐺) 0 ) ∈ (Base‘𝐺))
14 elsng 3606 . . . . . . 7 (( 0 (+g𝐺) 0 ) ∈ (Base‘𝐺) → (( 0 (+g𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g𝐺) 0 ) = 0 ))
1513, 14syl 14 . . . . . 6 (𝐺 ∈ Mnd → (( 0 (+g𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g𝐺) 0 ) = 0 ))
1612, 15mpbird 167 . . . . 5 (𝐺 ∈ Mnd → ( 0 (+g𝐺) 0 ) ∈ { 0 })
17 oveq12 5877 . . . . . 6 ((𝑎 = 0𝑏 = 0 ) → (𝑎(+g𝐺)𝑏) = ( 0 (+g𝐺) 0 ))
1817eleq1d 2246 . . . . 5 ((𝑎 = 0𝑏 = 0 ) → ((𝑎(+g𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g𝐺) 0 ) ∈ { 0 }))
1916, 18syl5ibrcom 157 . . . 4 (𝐺 ∈ Mnd → ((𝑎 = 0𝑏 = 0 ) → (𝑎(+g𝐺)𝑏) ∈ { 0 }))
209, 19biimtrid 152 . . 3 (𝐺 ∈ Mnd → ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) → (𝑎(+g𝐺)𝑏) ∈ { 0 }))
2120ralrimivv 2558 . 2 (𝐺 ∈ Mnd → ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g𝐺)𝑏) ∈ { 0 })
221, 2, 10issubm 12740 . 2 (𝐺 ∈ Mnd → ({ 0 } ∈ (SubMnd‘𝐺) ↔ ({ 0 } ⊆ (Base‘𝐺) ∧ 0 ∈ { 0 } ∧ ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g𝐺)𝑏) ∈ { 0 })))
234, 6, 21, 22mpbir3and 1180 1 (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  wss 3129  {csn 3591  cfv 5211  (class class class)co 5868  Basecbs 12432  +gcplusg 12505  0gc0g 12640  Mndcmnd 12696  SubMndcsubmnd 12727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-submnd 12729
This theorem is referenced by: (None)
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