Theorem 0subm 18871

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 2763	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		0subm.z	. . . 4 ⊢ 0 = (0g‘𝐺)
3	1, 2	mndidcl 18802	. . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺))
4	3	snssd 4752	. 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ (Base‘𝐺))
5	2	fvexi 6895	. . . 4 ⊢ 0 ∈ V
6	5	snid 4628	. . 3 ⊢ 0 ∈ { 0 }
7	6	a1i 11	. 2 ⊢ (𝐺 ∈ Mnd → 0 ∈ { 0 })
8		velsn 4605	. . . . 5 ⊢ (𝑎 ∈ { 0 } ↔ 𝑎 = 0 )
9		velsn 4605	. . . . 5 ⊢ (𝑏 ∈ { 0 } ↔ 𝑏 = 0 )
10	8, 9	anbi12i 639	. . . 4 ⊢ ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) ↔ (𝑎 = 0 ∧ 𝑏 = 0 ))
11		eqid 2763	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
12	1, 11, 2	mndlid 18807	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 )
13	3, 12	mpdan 699	. . . . . 6 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) = 0 )
14		ovex 7443	. . . . . . 7 ⊢ ( 0 (+g‘𝐺) 0 ) ∈ V
15	14	elsn 4604	. . . . . 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 7419	. . . . . 6 ⊢ ((𝑎 = 0 ∧ 𝑏 = 0 ) → (𝑎(+g‘𝐺)𝑏) = ( 0 (+g‘𝐺) 0 ))
18	17	eleq1d 2848	. . . . 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 3206	. 2 ⊢ (𝐺 ∈ Mnd → ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 })
22	1, 2, 11	issubm 18856	. 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))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3905  {csn 4589  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  +_gcplusg 17305  0_gc0g 17487  Mndcmnd 18787  SubMndcsubmnd 18835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-riota 7367  df-ov 7413  df-0g 17489  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-submnd 18837

This theorem is referenced by:  idressubmefmnd  18952  0subg  19213  0subgALT  19633