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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 17909 | . 2 ⊢ (𝐺 ∈ Mnd → ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)) |
4 | mgmidmo 17858 | . 2 ⊢ ∃*𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) | |
5 | reu5 3428 | . 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 590 | 1 ⊢ (𝐺 ∈ Mnd → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ∃!wreu 3137 ∃*wrmo 3138 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Mndcmnd 17899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 ax-pow 5257 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-mgm 17840 df-sgrp 17889 df-mnd 17900 |
This theorem is referenced by: grpideu 18052 srgideu 19193 ringideu 19244 |
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