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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 2820 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 0subm.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | mndidcl 17919 | . . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺)) |
4 | 3 | snssd 4735 | . 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ (Base‘𝐺)) |
5 | 2 | fvexi 6677 | . . . 4 ⊢ 0 ∈ V |
6 | 5 | snid 4594 | . . 3 ⊢ 0 ∈ { 0 } |
7 | 6 | a1i 11 | . 2 ⊢ (𝐺 ∈ Mnd → 0 ∈ { 0 }) |
8 | velsn 4576 | . . . . 5 ⊢ (𝑎 ∈ { 0 } ↔ 𝑎 = 0 ) | |
9 | velsn 4576 | . . . . 5 ⊢ (𝑏 ∈ { 0 } ↔ 𝑏 = 0 ) | |
10 | 8, 9 | anbi12i 628 | . . . 4 ⊢ ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) ↔ (𝑎 = 0 ∧ 𝑏 = 0 )) |
11 | eqid 2820 | . . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
12 | 1, 11, 2 | mndlid 17924 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
13 | 3, 12 | mpdan 685 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) = 0 ) |
14 | ovex 7182 | . . . . . . 7 ⊢ ( 0 (+g‘𝐺) 0 ) ∈ V | |
15 | 14 | elsn 4575 | . . . . . 6 ⊢ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) = 0 ) |
16 | 13, 15 | sylibr 236 | . . . . 5 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) ∈ { 0 }) |
17 | oveq12 7158 | . . . . . 6 ⊢ ((𝑎 = 0 ∧ 𝑏 = 0 ) → (𝑎(+g‘𝐺)𝑏) = ( 0 (+g‘𝐺) 0 )) | |
18 | 17 | eleq1d 2896 | . . . . 5 ⊢ ((𝑎 = 0 ∧ 𝑏 = 0 ) → ((𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 })) |
19 | 16, 18 | syl5ibrcom 249 | . . . 4 ⊢ (𝐺 ∈ Mnd → ((𝑎 = 0 ∧ 𝑏 = 0 ) → (𝑎(+g‘𝐺)𝑏) ∈ { 0 })) |
20 | 10, 19 | syl5bi 244 | . . 3 ⊢ (𝐺 ∈ Mnd → ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) → (𝑎(+g‘𝐺)𝑏) ∈ { 0 })) |
21 | 20 | ralrimivv 3189 | . 2 ⊢ (𝐺 ∈ Mnd → ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 }) |
22 | 1, 2, 11 | issubm 17961 | . 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 1337 | 1 ⊢ (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ⊆ wss 3929 {csn 4560 ‘cfv 6348 (class class class)co 7149 Basecbs 16476 +gcplusg 16558 0gc0g 16706 Mndcmnd 17904 SubMndcsubmnd 17948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-riota 7107 df-ov 7152 df-0g 16708 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-submnd 17950 |
This theorem is referenced by: idressubmefmnd 18056 |
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