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Mirrors > Home > ILE Home > Th. List > isgrpi | 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 9 | . . 3 ⊢ (⊤ → 𝐵 = (Base‘𝐺)) |
3 | isgrpi.p | . . . 4 ⊢ + = (+g‘𝐺) | |
4 | 3 | a1i 9 | . . 3 ⊢ (⊤ → + = (+g‘𝐺)) |
5 | isgrpi.c | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) | |
6 | 5 | 3adant1 1015 | . . 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 277 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
9 | isgrpi.z | . . . 4 ⊢ 0 ∈ 𝐵 | |
10 | 9 | a1i 9 | . . 3 ⊢ (⊤ → 0 ∈ 𝐵) |
11 | isgrpi.i | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ( 0 + 𝑥) = 𝑥) | |
12 | 11 | adantl 277 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
13 | isgrpi.n | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑁 ∈ 𝐵) | |
14 | 13 | adantl 277 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵) |
15 | isgrpi.j | . . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝑁 + 𝑥) = 0 ) | |
16 | 15 | adantl 277 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 ) |
17 | 2, 4, 6, 8, 10, 12, 14, 16 | isgrpd 12831 | . 2 ⊢ (⊤ → 𝐺 ∈ Grp) |
18 | 17 | mptru 1362 | 1 ⊢ 𝐺 ∈ Grp |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ⊤wtru 1354 ∈ wcel 2148 ‘cfv 5215 (class class class)co 5872 Basecbs 12454 +gcplusg 12528 Grpcgrp 12809 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-cnex 7899 ax-resscn 7900 ax-1re 7902 ax-addrcl 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-iota 5177 df-fun 5217 df-fn 5218 df-fv 5223 df-riota 5828 df-ov 5875 df-inn 8916 df-2 8974 df-ndx 12457 df-slot 12458 df-base 12460 df-plusg 12541 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 |
This theorem is referenced by: cncrng 13332 |
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