![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isgrpde | Structured version Visualization version GIF version |
Description: Deduce a group from its properties. In this version of isgrpd 18840, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 6-Jan-2015.) |
Ref | Expression |
---|---|
isgrpd.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
isgrpd.p | ⊢ (𝜑 → + = (+g‘𝐺)) |
isgrpd.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
isgrpd.a | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
isgrpd.z | ⊢ (𝜑 → 0 ∈ 𝐵) |
isgrpd.i | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
isgrpde.n | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
Ref | Expression |
---|---|
isgrpde | ⊢ (𝜑 → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgrpd.b | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
2 | isgrpd.p | . 2 ⊢ (𝜑 → + = (+g‘𝐺)) | |
3 | isgrpd.z | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) | |
4 | isgrpd.i | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | |
5 | isgrpd.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) | |
6 | isgrpd.a | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
7 | isgrpde.n | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) | |
8 | 5, 3, 4, 6, 7 | grprida 18590 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥) |
9 | 1, 2, 3, 4, 8 | grpidd 18586 | . 2 ⊢ (𝜑 → 0 = (0g‘𝐺)) |
10 | 1, 2, 5, 6, 3, 4, 8 | ismndd 18643 | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
11 | 1, 2, 9, 10, 7 | isgrpd2e 18837 | 1 ⊢ (𝜑 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ‘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-rmo 3377 df-reu 3378 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-riota 7360 df-ov 7407 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 |
This theorem is referenced by: isgrpd 18840 dfgrp2 18843 imasgrp2 18934 unitgrp 20186 |
Copyright terms: Public domain | W3C validator |