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Mirrors > Home > MPE Home > Th. List > isgrpi | Structured version Visualization version GIF version |
Description: Properties that determine a group. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). (Contributed by NM, 3-Sep-2011.) |
Ref | Expression |
---|---|
isgrpi.b | ⊢ 𝐵 = (Base‘𝐺) |
isgrpi.p | ⊢ + = (+g‘𝐺) |
isgrpi.c | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
isgrpi.a | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
isgrpi.z | ⊢ 0 ∈ 𝐵 |
isgrpi.i | ⊢ (𝑥 ∈ 𝐵 → ( 0 + 𝑥) = 𝑥) |
isgrpi.n | ⊢ (𝑥 ∈ 𝐵 → 𝑁 ∈ 𝐵) |
isgrpi.j | ⊢ (𝑥 ∈ 𝐵 → (𝑁 + 𝑥) = 0 ) |
Ref | Expression |
---|---|
isgrpi | ⊢ 𝐺 ∈ Grp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgrpi.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 𝐵 = (Base‘𝐺)) |
3 | isgrpi.p | . . . 4 ⊢ + = (+g‘𝐺) | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → + = (+g‘𝐺)) |
5 | isgrpi.c | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) | |
6 | 5 | 3adant1 1079 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
7 | isgrpi.a | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
8 | 7 | adantl 482 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
9 | isgrpi.z | . . . 4 ⊢ 0 ∈ 𝐵 | |
10 | 9 | a1i 11 | . . 3 ⊢ (⊤ → 0 ∈ 𝐵) |
11 | isgrpi.i | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ( 0 + 𝑥) = 𝑥) | |
12 | 11 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
13 | isgrpi.n | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑁 ∈ 𝐵) | |
14 | 13 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵) |
15 | isgrpi.j | . . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝑁 + 𝑥) = 0 ) | |
16 | 15 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 ) |
17 | 2, 4, 6, 8, 10, 12, 14, 16 | isgrpd 17444 | . 2 ⊢ (⊤ → 𝐺 ∈ Grp) |
18 | 17 | trud 1493 | 1 ⊢ 𝐺 ∈ Grp |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Grpcgrp 17422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 |
This theorem is referenced by: isgrpix 17449 cnaddabl 18272 cncrng 19767 |
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