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Mirrors > Home > MPE Home > Th. List > mndlrid | Structured version Visualization version GIF version |
Description: A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.) |
Ref | Expression |
---|---|
mndlrid.b | ⊢ 𝐵 = (Base‘𝐺) |
mndlrid.p | ⊢ + = (+g‘𝐺) |
mndlrid.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
mndlrid | ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndlrid.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mndlrid.o | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | mndlrid.p | . 2 ⊢ + = (+g‘𝐺) | |
4 | 1, 3 | mndid 17913 | . 2 ⊢ (𝐺 ∈ Mnd → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)) |
5 | 1, 2, 3, 4 | mgmlrid 17869 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Mndcmnd 17903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 |
This theorem is referenced by: mndlid 17923 mndrid 17924 gsumvallem2 17990 gsumsubm 17991 srgidmlem 19263 ringidmlem 19316 frlmgsum 20461 |
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