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Mirrors > Home > MPE Home > Th. List > grpprop | Structured version Visualization version GIF version |
Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.) |
Ref | Expression |
---|---|
grpprop.b | ⊢ (Base‘𝐾) = (Base‘𝐿) |
grpprop.p | ⊢ (+g‘𝐾) = (+g‘𝐿) |
Ref | Expression |
---|---|
grpprop | ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2736 | . . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐾)) | |
2 | grpprop.b | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐿) | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐿)) |
4 | grpprop.p | . . . . 5 ⊢ (+g‘𝐾) = (+g‘𝐿) | |
5 | 4 | oveqi 7444 | . . . 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 | grppropd 18982 | . 2 ⊢ (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)) |
8 | 7 | mptru 1544 | 1 ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Grpcgrp 18964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 |
This theorem is referenced by: grppropstr 18984 grpss 18985 opprsubg 20369 rmodislmod 20945 rmodislmodOLD 20946 opprgrpb 42499 lmod1 48338 |
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