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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 18207 | . 2 ⊢ (𝐺 ∈ Mnd → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)) |
5 | 1, 2, 3, 4 | mgmlrid 18163 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ‘cfv 6397 (class class class)co 7231 Basecbs 16784 +gcplusg 16826 0gc0g 16968 Mndcmnd 18197 |
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 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
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 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-iota 6355 df-fun 6399 df-fv 6405 df-riota 7188 df-ov 7234 df-0g 16970 df-mgm 18138 df-sgrp 18187 df-mnd 18198 |
This theorem is referenced by: mndlid 18217 mndrid 18218 gsumvallem2 18284 gsumsubm 18285 srgidmlem 19559 ringidmlem 19612 frlmgsum 20758 |
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