Theorem 0subm 18929

Description: The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)

Hypothesis

Ref	Expression
0subm.z	⊢ 0 = (0g‘𝐺)

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.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		0subm.z	. . . 4 ⊢ 0 = (0g‘𝐺)
3	1, 2	mndidcl 18855	. . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺))
4	3	snssd 4747	. 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ (Base‘𝐺))
5	2	fvexi 6893	. . . 4 ⊢ 0 ∈ V
6	5	snid 4623	. . 3 ⊢ 0 ∈ { 0 }
7	6	a1i 11	. 2 ⊢ (𝐺 ∈ Mnd → 0 ∈ { 0 })
8		velsn 4600	. . . . 5 ⊢ (𝑎 ∈ { 0 } ↔ 𝑎 = 0 )
9		velsn 4600	. . . . 5 ⊢ (𝑏 ∈ { 0 } ↔ 𝑏 = 0 )
10	8, 9	anbi12i 640	. . . 4 ⊢ ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) ↔ (𝑎 = 0 ∧ 𝑏 = 0 ))
11		eqid 2760	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
12	1, 11, 2	mndlid 18860	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 )
13	3, 12	mpdan 700	. . . . . 6 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) = 0 )
14		ovex 7447	. . . . . . 7 ⊢ ( 0 (+g‘𝐺) 0 ) ∈ V
15	14	elsn 4599	. . . . . 6 ⊢ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) = 0 )
16	13, 15	sylibr 237	. . . . 5 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) ∈ { 0 })
17		oveq12 7423	. . . . . 6 ⊢ ((𝑎 = 0 ∧ 𝑏 = 0 ) → (𝑎(+g‘𝐺)𝑏) = ( 0 (+g‘𝐺) 0 ))
18	17	eleq1d 2845	. . . . 5 ⊢ ((𝑎 = 0 ∧ 𝑏 = 0 ) → ((𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 }))
19	16, 18	syl5ibrcom 250	. . . 4 ⊢ (𝐺 ∈ Mnd → ((𝑎 = 0 ∧ 𝑏 = 0 ) → (𝑎(+g‘𝐺)𝑏) ∈ { 0 }))
20	10, 19	biimtrid 245	. . 3 ⊢ (𝐺 ∈ Mnd → ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) → (𝑎(+g‘𝐺)𝑏) ∈ { 0 }))
21	20	ralrimivv 3203	. 2 ⊢ (𝐺 ∈ Mnd → ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 })
22	1, 2, 11	issubm 18914	. 2 ⊢ (𝐺 ∈ Mnd → ({ 0 } ∈ (SubMnd‘𝐺) ↔ ({ 0 } ⊆ (Base‘𝐺) ∧ 0 ∈ { 0 } ∧ ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 })))
23	4, 7, 21, 22	mpbir3and 1361	1 ⊢ (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  {csn 4584  ‘cfv 6533  (class class class)co 7414  Basecbs 17304  +_gcplusg 17345  0_gc0g 17527  Mndcmnd 18839  SubMndcsubmnd 18893

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-riota 7371  df-ov 7417  df-0g 17529  df-mgm 18733  df-sgrp 18824  df-mnd 18840  df-submnd 18895

This theorem is used by:  idressubmefmnd  19010  0subg  19278  0subgALT  19698