Theorem 0subm 18843

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 2735	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		0subm.z	. . . 4 ⊢ 0 = (0g‘𝐺)
3	1, 2	mndidcl 18775	. . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺))
4	3	snssd 4814	. 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ (Base‘𝐺))
5	2	fvexi 6921	. . . 4 ⊢ 0 ∈ V
6	5	snid 4667	. . 3 ⊢ 0 ∈ { 0 }
7	6	a1i 11	. 2 ⊢ (𝐺 ∈ Mnd → 0 ∈ { 0 })
8		velsn 4647	. . . . 5 ⊢ (𝑎 ∈ { 0 } ↔ 𝑎 = 0 )
9		velsn 4647	. . . . 5 ⊢ (𝑏 ∈ { 0 } ↔ 𝑏 = 0 )
10	8, 9	anbi12i 628	. . . 4 ⊢ ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) ↔ (𝑎 = 0 ∧ 𝑏 = 0 ))
11		eqid 2735	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
12	1, 11, 2	mndlid 18780	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 )
13	3, 12	mpdan 687	. . . . . 6 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) = 0 )
14		ovex 7464	. . . . . . 7 ⊢ ( 0 (+g‘𝐺) 0 ) ∈ V
15	14	elsn 4646	. . . . . 6 ⊢ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) = 0 )
16	13, 15	sylibr 234	. . . . 5 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) ∈ { 0 })
17		oveq12 7440	. . . . . 6 ⊢ ((𝑎 = 0 ∧ 𝑏 = 0 ) → (𝑎(+g‘𝐺)𝑏) = ( 0 (+g‘𝐺) 0 ))
18	17	eleq1d 2824	. . . . 5 ⊢ ((𝑎 = 0 ∧ 𝑏 = 0 ) → ((𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 }))
19	16, 18	syl5ibrcom 247	. . . 4 ⊢ (𝐺 ∈ Mnd → ((𝑎 = 0 ∧ 𝑏 = 0 ) → (𝑎(+g‘𝐺)𝑏) ∈ { 0 }))
20	10, 19	biimtrid 242	. . 3 ⊢ (𝐺 ∈ Mnd → ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) → (𝑎(+g‘𝐺)𝑏) ∈ { 0 }))
21	20	ralrimivv 3198	. 2 ⊢ (𝐺 ∈ Mnd → ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 })
22	1, 2, 11	issubm 18829	. 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 1341	1 ⊢ (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ⊆ wss 3963  {csn 4631  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  0_gc0g 17486  Mndcmnd 18760  SubMndcsubmnd 18808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810

This theorem is referenced by:  idressubmefmnd  18924  0subg  19182  0subgALT  19601