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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 2823, but we make an exception for theorems such as isgrpd2 18125, ismndd 17935, and islmodd 19642 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 7165 | . . . . 5 ⊢ (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥)) | |
8 | 7 | eqeq1d 2825 | . . . 4 ⊢ (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 )) |
9 | 8 | rspcev 3625 | . . 3 ⊢ ((𝑁 ∈ 𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
10 | 5, 6, 9 | syl2anc 586 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
11 | 1, 2, 3, 4, 10 | isgrpd2e 18124 | 1 ⊢ (𝜑 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 0gc0g 16715 Mndcmnd 17913 Grpcgrp 18105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-grp 18108 |
This theorem is referenced by: prdsgrpd 18211 oppggrp 18487 |
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