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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 18970 | . . 3 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)) | 
| 5 | 1, 2, 3 | cmnpropd 19810 | . . 3 ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd)) | 
| 6 | 4, 5 | anbi12d 632 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd) ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd))) | 
| 7 | isabl 19803 | . 2 ⊢ (𝐾 ∈ Abel ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd)) | |
| 8 | isabl 19803 | . 2 ⊢ (𝐿 ∈ Abel ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd)) | |
| 9 | 6, 7, 8 | 3bitr4g 314 | 1 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 Grpcgrp 18952 CMndccmn 19799 Abelcabl 19800 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-cmn 19801 df-abl 19802 | 
| This theorem is referenced by: ablprop 19812 rngpropd 20172 | 
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