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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 2738 | . . 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 18969 | . 2 ⊢ (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)) | 
| 8 | 7 | mptru 1547 | 1 ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Grpcgrp 18951 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 | 
| This theorem is referenced by: grppropstr 18971 grpss 18972 opprsubg 20352 rmodislmod 20928 opprgrpb 42522 lmod1 48409 | 
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