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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-grpssmnd | Structured version Visualization version GIF version |
Description: Groups are monoids. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-grpssmnd | ⊢ Grp ⊆ Mnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-grp 18101 | . 2 ⊢ Grp = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∃𝑧 ∈ (Base‘𝑥)(𝑧(+g‘𝑥)𝑦) = (0g‘𝑥)} | |
2 | 1 | ssrab3 4050 | 1 ⊢ Grp ⊆ Mnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∀wral 3137 ∃wrex 3138 ⊆ wss 3929 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 +gcplusg 16560 0gc0g 16708 Mndcmnd 17906 Grpcgrp 18098 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-in 3936 df-ss 3945 df-grp 18101 |
This theorem is referenced by: bj-grpssmndel 34581 |
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