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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 2732 | . . 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 7425 | . . . 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 19708 | . 2 ⊢ (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)) |
8 | 7 | mptru 1547 | 1 ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 Abelcabl 19697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-cmn 19698 df-abl 19699 |
This theorem is referenced by: opprrng 20243 zlmlmod 21386 dvaabl 40361 cznabel 47099 |
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