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Mirrors > Home > MPE Home > Th. List > isgrpd2 | Structured version Visualization version GIF version |
Description: Deduce a group from its properties. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). Note: normally we don't use a 𝜑 antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2735, but we make an exception for theorems such as isgrpd2 18987, ismndd 18782, and islmodd 20881 since theorems using them often rewrite the structure components. (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) |
isgrpd2.n | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵) |
isgrpd2.j | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 ) |
Ref | Expression |
---|---|
isgrpd2 | ⊢ (𝜑 → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgrpd2.b | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
2 | isgrpd2.p | . 2 ⊢ (𝜑 → + = (+g‘𝐺)) | |
3 | isgrpd2.z | . 2 ⊢ (𝜑 → 0 = (0g‘𝐺)) | |
4 | isgrpd2.g | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
5 | isgrpd2.n | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵) | |
6 | isgrpd2.j | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 ) | |
7 | oveq1 7438 | . . . . 5 ⊢ (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥)) | |
8 | 7 | eqeq1d 2737 | . . . 4 ⊢ (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 )) |
9 | 8 | rspcev 3622 | . . 3 ⊢ ((𝑁 ∈ 𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
10 | 5, 6, 9 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
11 | 1, 2, 3, 4, 10 | isgrpd2e 18986 | 1 ⊢ (𝜑 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 Mndcmnd 18760 Grpcgrp 18964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-grp 18967 |
This theorem is referenced by: prdsgrpd 19081 oppggrp 19391 |
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