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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 17923, 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 3125 | . . 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 6991 | . . . . . 6 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥)) |
7 | isgrpd2.z | . . . . . 6 ⊢ (𝜑 → 0 = (0g‘𝐺)) | |
8 | 6, 7 | eqeq12d 2786 | . . . . 5 ⊢ (𝜑 → ((𝑦 + 𝑥) = 0 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
9 | 4, 8 | rexeqbidv 3335 | . . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
10 | 4, 9 | raleqbidv 3334 | . . 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 224 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) |
12 | eqid 2771 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
13 | eqid 2771 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
14 | eqid 2771 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
15 | 12, 13, 14 | isgrp 17909 | . 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
16 | 1, 11, 15 | sylanbrc 575 | 1 ⊢ (𝜑 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 ∃wrex 3082 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 +gcplusg 16419 0gc0g 16567 Mndcmnd 17774 Grpcgrp 17903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-iota 6149 df-fv 6193 df-ov 6977 df-grp 17906 |
This theorem is referenced by: isgrpd2 17923 isgrpde 17924 |
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