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| 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 18757 | . 2 ⊢ (𝐺 ∈ Mnd → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)) | 
| 5 | 1, 2, 3, 4 | mgmlrid 18680 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 | 
| This theorem is referenced by: mndlid 18767 mndrid 18768 gsumvallem2 18847 gsumsubm 18848 srgidmlem 20198 ringidmlem 20265 frlmgsum 21792 | 
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