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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 17909 | . 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 499 | 1 ⊢ (𝐺 ∈ Mnd → ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∃wrex 3138 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 +gcplusg 16560 Mndcmnd 17906 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-nul 5203 ax-pow 5259 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-iota 6307 df-fv 6356 df-ov 7152 df-mgm 17847 df-sgrp 17896 df-mnd 17907 |
This theorem is referenced by: mndideu 17917 mndidcl 17921 mndlrid 17925 prds0g 17940 |
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