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Mirrors > Home > ILE Home > Th. List > 0subm | 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 2177 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 0subm.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | mndidcl 12710 | . . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺)) |
4 | 3 | snssd 3736 | . 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ (Base‘𝐺)) |
5 | snidg 3620 | . . 3 ⊢ ( 0 ∈ (Base‘𝐺) → 0 ∈ { 0 }) | |
6 | 3, 5 | syl 14 | . 2 ⊢ (𝐺 ∈ Mnd → 0 ∈ { 0 }) |
7 | velsn 3608 | . . . . 5 ⊢ (𝑎 ∈ { 0 } ↔ 𝑎 = 0 ) | |
8 | velsn 3608 | . . . . 5 ⊢ (𝑏 ∈ { 0 } ↔ 𝑏 = 0 ) | |
9 | 7, 8 | anbi12i 460 | . . . 4 ⊢ ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) ↔ (𝑎 = 0 ∧ 𝑏 = 0 )) |
10 | eqid 2177 | . . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
11 | 1, 10, 2 | mndlid 12715 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
12 | 3, 11 | mpdan 421 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) = 0 ) |
13 | 12, 3 | eqeltrd 2254 | . . . . . . 7 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) ∈ (Base‘𝐺)) |
14 | elsng 3606 | . . . . . . 7 ⊢ (( 0 (+g‘𝐺) 0 ) ∈ (Base‘𝐺) → (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) = 0 )) | |
15 | 13, 14 | syl 14 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) = 0 )) |
16 | 12, 15 | mpbird 167 | . . . . 5 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) ∈ { 0 }) |
17 | oveq12 5877 | . . . . . 6 ⊢ ((𝑎 = 0 ∧ 𝑏 = 0 ) → (𝑎(+g‘𝐺)𝑏) = ( 0 (+g‘𝐺) 0 )) | |
18 | 17 | eleq1d 2246 | . . . . 5 ⊢ ((𝑎 = 0 ∧ 𝑏 = 0 ) → ((𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 })) |
19 | 16, 18 | syl5ibrcom 157 | . . . 4 ⊢ (𝐺 ∈ Mnd → ((𝑎 = 0 ∧ 𝑏 = 0 ) → (𝑎(+g‘𝐺)𝑏) ∈ { 0 })) |
20 | 9, 19 | biimtrid 152 | . . 3 ⊢ (𝐺 ∈ Mnd → ((𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 }) → (𝑎(+g‘𝐺)𝑏) ∈ { 0 })) |
21 | 20 | ralrimivv 2558 | . 2 ⊢ (𝐺 ∈ Mnd → ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 }) |
22 | 1, 2, 10 | issubm 12740 | . 2 ⊢ (𝐺 ∈ Mnd → ({ 0 } ∈ (SubMnd‘𝐺) ↔ ({ 0 } ⊆ (Base‘𝐺) ∧ 0 ∈ { 0 } ∧ ∀𝑎 ∈ { 0 }∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 }))) |
23 | 4, 6, 21, 22 | mpbir3and 1180 | 1 ⊢ (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 {csn 3591 ‘cfv 5211 (class class class)co 5868 Basecbs 12432 +gcplusg 12505 0gc0g 12640 Mndcmnd 12696 SubMndcsubmnd 12727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-riota 5824 df-ov 5871 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-submnd 12729 |
This theorem is referenced by: (None) |
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