![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0subm | Structured version Visualization version GIF version |
Description: The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.) |
Ref | Expression |
---|---|
0subm.z | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
0subm | ⊢ (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 0subm.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | mndidcl 18674 | . . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺)) |
4 | 3 | snssd 4805 | . 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ (Base‘𝐺)) |
5 | 2 | fvexi 6896 | . . . 4 ⊢ 0 ∈ V |
6 | 5 | snid 4657 | . . 3 ⊢ 0 ∈ { 0 } |
7 | 6 | a1i 11 | . 2 ⊢ (𝐺 ∈ Mnd → 0 ∈ { 0 }) |
8 | velsn 4637 | . . . . 5 ⊢ (𝑎 ∈ { 0 } ↔ 𝑎 = 0 ) | |
9 | velsn 4637 | . . . . 5 ⊢ (𝑏 ∈ { 0 } ↔ 𝑏 = 0 ) | |
10 | 8, 9 | anbi12i 626 | . . . 4 ⊢ ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) ↔ (𝑎 = 0 ∧ 𝑏 = 0 )) |
11 | eqid 2724 | . . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
12 | 1, 11, 2 | mndlid 18679 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
13 | 3, 12 | mpdan 684 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) = 0 ) |
14 | ovex 7435 | . . . . . . 7 ⊢ ( 0 (+g‘𝐺) 0 ) ∈ V | |
15 | 14 | elsn 4636 | . . . . . 6 ⊢ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) = 0 ) |
16 | 13, 15 | sylibr 233 | . . . . 5 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) ∈ { 0 }) |
17 | oveq12 7411 | . . . . . 6 ⊢ ((𝑎 = 0 ∧ 𝑏 = 0 ) → (𝑎(+g‘𝐺)𝑏) = ( 0 (+g‘𝐺) 0 )) | |
18 | 17 | eleq1d 2810 | . . . . 5 ⊢ ((𝑎 = 0 ∧ 𝑏 = 0 ) → ((𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 })) |
19 | 16, 18 | syl5ibrcom 246 | . . . 4 ⊢ (𝐺 ∈ Mnd → ((𝑎 = 0 ∧ 𝑏 = 0 ) → (𝑎(+g‘𝐺)𝑏) ∈ { 0 })) |
20 | 10, 19 | biimtrid 241 | . . 3 ⊢ (𝐺 ∈ Mnd → ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) → (𝑎(+g‘𝐺)𝑏) ∈ { 0 })) |
21 | 20 | ralrimivv 3190 | . 2 ⊢ (𝐺 ∈ Mnd → ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 }) |
22 | 1, 2, 11 | issubm 18720 | . 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 1339 | 1 ⊢ (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ⊆ wss 3941 {csn 4621 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 +gcplusg 17198 0gc0g 17386 Mndcmnd 18659 SubMndcsubmnd 18704 |
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 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 |
This theorem is referenced by: idressubmefmnd 18815 0subg 19070 0subgALT 19480 |
Copyright terms: Public domain | W3C validator |