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Mirrors > Home > MPE Home > Th. List > ablprop | 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, 11-Oct-2013.) |
Ref | Expression |
---|---|
ablprop.b | ⊢ (Base‘𝐾) = (Base‘𝐿) |
ablprop.p | ⊢ (+g‘𝐾) = (+g‘𝐿) |
Ref | Expression |
---|---|
ablprop | ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2819 | . . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐾)) | |
2 | ablprop.b | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐿) | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐿)) |
4 | ablprop.p | . . . . 5 ⊢ (+g‘𝐾) = (+g‘𝐿) | |
5 | 4 | oveqi 7158 | . . . 4 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦) |
6 | 5 | a1i 11 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
7 | 1, 3, 6 | ablpropd 18846 | . 2 ⊢ (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)) |
8 | 7 | mptru 1535 | 1 ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Abelcabl 18836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-cmn 18837 df-abl 18838 |
This theorem is referenced by: zlmlmod 20598 dvaabl 38040 cznabel 44153 |
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