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Mirrors > Home > MPE Home > Th. List > mndideu | Structured version Visualization version GIF version |
Description: The two-sided identity element of a monoid is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by Mario Carneiro, 8-Dec-2014.) |
Ref | Expression |
---|---|
mndcl.b | ⊢ 𝐵 = (Base‘𝐺) |
mndcl.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
mndideu | ⊢ (𝐺 ∈ Mnd → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mndcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
3 | 1, 2 | mndid 18137 | . 2 ⊢ (𝐺 ∈ Mnd → ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)) |
4 | mgmidmo 18086 | . 2 ⊢ ∃*𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) | |
5 | reu5 3327 | . 2 ⊢ (∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ (∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∃*𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))) | |
6 | 3, 4, 5 | sylanblrc 593 | 1 ⊢ (𝐺 ∈ Mnd → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ∃!wreu 3053 ∃*wrmo 3054 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 +gcplusg 16749 Mndcmnd 18127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 df-mgm 18068 df-sgrp 18117 df-mnd 18128 |
This theorem is referenced by: grpideu 18330 srgideu 19483 ringideu 19537 |
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