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Mirrors > Home > MPE Home > Th. List > mndid | Structured version Visualization version GIF version |
Description: A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.) |
Ref | Expression |
---|---|
mndcl.b | ⊢ 𝐵 = (Base‘𝐺) |
mndcl.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
mndid | ⊢ (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 | ismnd 18023 | . 2 ⊢ (𝐺 ∈ Mnd ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))) |
4 | 3 | simprbi 500 | 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 1542 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 +gcplusg 16661 Mndcmnd 18020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-iota 6291 df-fv 6341 df-ov 7167 df-mgm 17961 df-sgrp 18010 df-mnd 18021 |
This theorem is referenced by: mndideu 18031 mndidcl 18035 mndlrid 18039 prds0g 18054 |
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