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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 1131 | . . 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 483 | . . 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 483 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
13 | isgrpi.n | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑁 ∈ 𝐵) | |
14 | 13 | adantl 483 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵) |
15 | isgrpi.j | . . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝑁 + 𝑥) = 0 ) | |
16 | 15 | adantl 483 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 ) |
17 | 2, 4, 6, 8, 10, 12, 14, 16 | isgrpd 18777 | . 2 ⊢ (⊤ → 𝐺 ∈ Grp) |
18 | 17 | mptru 1549 | 1 ⊢ 𝐺 ∈ Grp |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Grpcgrp 18753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-riota 7314 df-ov 7361 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 |
This theorem is referenced by: isgrpix 18782 cnaddabl 19652 cncrng 20834 |
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