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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 2732, but we make an exception for theorems such as isgrpd2 18841, ismndd 18646, and islmodd 20476 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 7415 | . . . . 5 ⊢ (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥)) | |
8 | 7 | eqeq1d 2734 | . . . 4 ⊢ (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 )) |
9 | 8 | rspcev 3612 | . . 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 18840 | 1 ⊢ (𝜑 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ‘cfv 6543 (class class class)co 7408 Basecbs 17143 +gcplusg 17196 0gc0g 17384 Mndcmnd 18624 Grpcgrp 18818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7411 df-grp 18821 |
This theorem is referenced by: prdsgrpd 18932 oppggrp 19223 |
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