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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 19702. (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 19702 | . 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 2098 Grpcgrp 18860 Abelcabl 19698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-in 3950 df-abl 19700 |
This theorem is referenced by: imasabl 19793 ablsimpgd 20035 rnggrp 20060 primrootscoprmpow 41478 |
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