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Mirrors > Home > ILE Home > Th. List > isgrpde | GIF version |
Description: Deduce a group from its properties. In this version of isgrpd 12904, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 6-Jan-2015.) |
Ref | Expression |
---|---|
isgrpd.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
isgrpd.p | ⊢ (𝜑 → + = (+g‘𝐺)) |
isgrpd.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
isgrpd.a | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
isgrpd.z | ⊢ (𝜑 → 0 ∈ 𝐵) |
isgrpd.i | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
isgrpde.n | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
Ref | Expression |
---|---|
isgrpde | ⊢ (𝜑 → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgrpd.b | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
2 | isgrpd.p | . 2 ⊢ (𝜑 → + = (+g‘𝐺)) | |
3 | isgrpd.z | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) | |
4 | isgrpd.i | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | |
5 | isgrpd.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) | |
6 | isgrpd.a | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
7 | isgrpde.n | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) | |
8 | 5, 3, 4, 6, 7 | grpridd 12811 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥) |
9 | 1, 2, 3, 4, 8 | grpidd 12807 | . 2 ⊢ (𝜑 → 0 = (0g‘𝐺)) |
10 | 1, 2, 5, 6, 3, 4, 8 | ismndd 12843 | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
11 | 1, 2, 9, 10, 7 | isgrpd2e 12901 | 1 ⊢ (𝜑 → 𝐺 ∈ Grp) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 ‘cfv 5218 (class class class)co 5877 Basecbs 12464 +gcplusg 12538 Grpcgrp 12882 |
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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-cnex 7904 ax-resscn 7905 ax-1re 7907 ax-addrcl 7910 |
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 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-iota 5180 df-fun 5220 df-fn 5221 df-fv 5226 df-riota 5833 df-ov 5880 df-inn 8922 df-2 8980 df-ndx 12467 df-slot 12468 df-base 12470 df-plusg 12551 df-0g 12712 df-mgm 12780 df-sgrp 12813 df-mnd 12823 df-grp 12885 |
This theorem is referenced by: isgrpd 12904 dfgrp2 12907 unitgrp 13290 |
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