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Mirrors > Home > MPE Home > Th. List > ablgrpd | Structured version Visualization version GIF version |
Description: An Abelian group is a group, deduction form of ablgrp 19574. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
ablgrpd.1 | ⊢ (𝜑 → 𝐺 ∈ Abel) |
Ref | Expression |
---|---|
ablgrpd | ⊢ (𝜑 → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablgrpd.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
2 | ablgrp 19574 | . 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Grpcgrp 18755 Abelcabl 19570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-in 3922 df-abl 19572 |
This theorem is referenced by: ablsimpgd 19902 |
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