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Mirrors > Home > MPE Home > Th. List > ablpropd | Structured version Visualization version GIF version |
Description: If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.) |
Ref | Expression |
---|---|
ablpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
ablpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
ablpropd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
Ref | Expression |
---|---|
ablpropd | ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablpropd.1 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | ablpropd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
3 | ablpropd.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
4 | 1, 2, 3 | grppropd 18110 | . . 3 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)) |
5 | 1, 2, 3 | cmnpropd 18908 | . . 3 ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd)) |
6 | 4, 5 | anbi12d 633 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd) ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd))) |
7 | isabl 18902 | . 2 ⊢ (𝐾 ∈ Abel ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd)) | |
8 | isabl 18902 | . 2 ⊢ (𝐿 ∈ Abel ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd)) | |
9 | 6, 7, 8 | 3bitr4g 317 | 1 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Grpcgrp 18095 CMndccmn 18898 Abelcabl 18899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-cmn 18900 df-abl 18901 |
This theorem is referenced by: ablprop 18910 |
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