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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 2734 | . . 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 7417 | . . . 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 18833 | . 2 ⊢ (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)) |
8 | 7 | mptru 1549 | 1 ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 +gcplusg 17193 Grpcgrp 18815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7407 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 |
This theorem is referenced by: grppropstr 18835 grpss 18836 opprring 20150 opprsubg 20155 rmodislmod 20528 rmodislmodOLD 20529 lmod1 47075 |
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