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Mirrors > Home > MPE Home > Th. List > isgrpd2e | Structured version Visualization version GIF version |
Description: Deduce a group from its properties. In this version of isgrpd2 18341, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 10-Aug-2013.) |
Ref | Expression |
---|---|
isgrpd2.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
isgrpd2.p | ⊢ (𝜑 → + = (+g‘𝐺)) |
isgrpd2.z | ⊢ (𝜑 → 0 = (0g‘𝐺)) |
isgrpd2.g | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
isgrpd2e.n | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
Ref | Expression |
---|---|
isgrpd2e | ⊢ (𝜑 → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgrpd2.g | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
2 | isgrpd2e.n | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) | |
3 | 2 | ralrimiva 3095 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
4 | isgrpd2.b | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
5 | isgrpd2.p | . . . . . . 7 ⊢ (𝜑 → + = (+g‘𝐺)) | |
6 | 5 | oveqd 7208 | . . . . . 6 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥)) |
7 | isgrpd2.z | . . . . . 6 ⊢ (𝜑 → 0 = (0g‘𝐺)) | |
8 | 6, 7 | eqeq12d 2752 | . . . . 5 ⊢ (𝜑 → ((𝑦 + 𝑥) = 0 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
9 | 4, 8 | rexeqbidv 3304 | . . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
10 | 4, 9 | raleqbidv 3303 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ↔ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
11 | 3, 10 | mpbid 235 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) |
12 | eqid 2736 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
13 | eqid 2736 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
14 | eqid 2736 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
15 | 12, 13, 14 | isgrp 18325 | . 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
16 | 1, 11, 15 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 +gcplusg 16749 0gc0g 16898 Mndcmnd 18127 Grpcgrp 18319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 df-grp 18322 |
This theorem is referenced by: isgrpd2 18341 isgrpde 18342 |
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