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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 17260 | . . 3 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)) |
5 | 1, 2, 3 | cmnpropd 18025 | . . 3 ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd)) |
6 | 4, 5 | anbi12d 743 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd) ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd))) |
7 | isabl 18020 | . 2 ⊢ (𝐾 ∈ Abel ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd)) | |
8 | isabl 18020 | . 2 ⊢ (𝐿 ∈ Abel ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd)) | |
9 | 6, 7, 8 | 3bitr4g 302 | 1 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Grpcgrp 17245 CMndccmn 18016 Abelcabl 18017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-cmn 18018 df-abl 18019 |
This theorem is referenced by: ablprop 18027 |
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