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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 2737, but we make an exception for theorems such as isgrpd2 18974, ismndd 18769, and islmodd 20864 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 2739 | . . . 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 18973 | 1 ⊢ (𝜑 → 𝐺 ∈ Grp) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 Grpcgrp 18951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-grp 18954 |
| This theorem is referenced by: prdsgrpd 19068 oppggrp 19376 |
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