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Mirrors > Home > MPE Home > Th. List > ismnd | Structured version Visualization version GIF version |
Description: The predicate "is a monoid". This is the definig theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 17124), whose operation is associative (so, a semigroup, see also mndass 17125) and has a two-sided neutral element (see mndid 17126). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.) |
Ref | Expression |
---|---|
ismnd.b | ⊢ 𝐵 = (Base‘𝐺) |
ismnd.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
ismnd | ⊢ (𝐺 ∈ Mnd ↔ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismnd.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | ismnd.p | . . 3 ⊢ + = (+g‘𝐺) | |
3 | 1, 2 | ismnddef 17119 | . 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ SGrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))) |
4 | rexn0 4026 | . . . 4 ⊢ (∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → 𝐵 ≠ ∅) | |
5 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
6 | 1, 5 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
7 | 6 | necon1ai 2809 | . . . 4 ⊢ (𝐵 ≠ ∅ → 𝐺 ∈ V) |
8 | 1, 2 | issgrpv 17109 | . . . 4 ⊢ (𝐺 ∈ V → (𝐺 ∈ SGrp ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))))) |
9 | 4, 7, 8 | 3syl 18 | . . 3 ⊢ (∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → (𝐺 ∈ SGrp ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))))) |
10 | 9 | pm5.32ri 668 | . 2 ⊢ ((𝐺 ∈ SGrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)) ↔ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))) |
11 | 3, 10 | bitri 263 | 1 ⊢ (𝐺 ∈ Mnd ↔ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 SGrpcsgrp 17106 Mndcmnd 17117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-mgm 17065 df-sgrp 17107 df-mnd 17118 |
This theorem is referenced by: mndid 17126 ismndd 17136 mndpropd 17139 mhmmnd 17360 signswmnd 29960 |
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