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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 17437 | . . 3 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)) |
5 | 1, 2, 3 | cmnpropd 18202 | . . 3 ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd)) |
6 | 4, 5 | anbi12d 747 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd) ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd))) |
7 | isabl 18197 | . 2 ⊢ (𝐾 ∈ Abel ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd)) | |
8 | isabl 18197 | . 2 ⊢ (𝐿 ∈ Abel ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd)) | |
9 | 6, 7, 8 | 3bitr4g 303 | 1 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Grpcgrp 17422 CMndccmn 18193 Abelcabl 18194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: ablprop 18204 |
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