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Mirrors > Home > MPE Home > Th. List > isabl | Structured version Visualization version GIF version |
Description: The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.) |
Ref | Expression |
---|---|
isabl | ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-abl 18196 | . 2 ⊢ Abel = (Grp ∩ CMnd) | |
2 | 1 | elin2 3801 | 1 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∈ wcel 1990 Grpcgrp 17422 CMndccmn 18193 Abelcabl 18194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-abl 18196 |
This theorem is referenced by: ablgrp 18198 ablcmn 18199 isabl2 18201 ablpropd 18203 isabld 18206 ghmabl 18238 prdsabld 18265 unitabl 18668 tsmsinv 21951 tgptsmscls 21953 tsmsxplem1 21956 tsmsxplem2 21957 abliso 29696 gicabl 37669 2zrngaabl 41944 pgrpgt2nabl 42147 |
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