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Mirrors > Home > MPE Home > Th. List > dfgrp2e | Structured version Visualization version GIF version |
Description: Alternate definition of a group as a set with a closed, associative operation, a left identity and a left inverse for each element. Alternate definition in [Lang] p. 7. (Contributed by NM, 10-Oct-2006.) (Revised by AV, 28-Aug-2021.) |
Ref | Expression |
---|---|
dfgrp2.b | ⊢ 𝐵 = (Base‘𝐺) |
dfgrp2.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
dfgrp2e | ⊢ (𝐺 ∈ Grp ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfgrp2.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | dfgrp2.p | . . 3 ⊢ + = (+g‘𝐺) | |
3 | 1, 2 | dfgrp2 18066 | . 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛))) |
4 | ax-1 6 | . . . . . . 7 ⊢ (𝐺 ∈ V → (𝑛 ∈ 𝐵 → 𝐺 ∈ V)) | |
5 | fvprc 6656 | . . . . . . . 8 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
6 | 1 | eleq2i 2901 | . . . . . . . . 9 ⊢ (𝑛 ∈ 𝐵 ↔ 𝑛 ∈ (Base‘𝐺)) |
7 | eleq2 2898 | . . . . . . . . . 10 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) ↔ 𝑛 ∈ ∅)) | |
8 | noel 4293 | . . . . . . . . . . 11 ⊢ ¬ 𝑛 ∈ ∅ | |
9 | 8 | pm2.21i 119 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ∅ → 𝐺 ∈ V) |
10 | 7, 9 | syl6bi 254 | . . . . . . . . 9 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) → 𝐺 ∈ V)) |
11 | 6, 10 | syl5bi 243 | . . . . . . . 8 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ 𝐵 → 𝐺 ∈ V)) |
12 | 5, 11 | syl 17 | . . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (𝑛 ∈ 𝐵 → 𝐺 ∈ V)) |
13 | 4, 12 | pm2.61i 183 | . . . . . 6 ⊢ (𝑛 ∈ 𝐵 → 𝐺 ∈ V) |
14 | 13 | a1d 25 | . . . . 5 ⊢ (𝑛 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → 𝐺 ∈ V)) |
15 | 14 | rexlimiv 3277 | . . . 4 ⊢ (∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → 𝐺 ∈ V) |
16 | 1, 2 | issgrpv 17891 | . . . 4 ⊢ (𝐺 ∈ V → (𝐺 ∈ Smgrp ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))) |
17 | 15, 16 | syl 17 | . . 3 ⊢ (∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → (𝐺 ∈ Smgrp ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))) |
18 | 17 | pm5.32ri 576 | . 2 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛))) |
19 | 3, 18 | bitri 276 | 1 ⊢ (𝐺 ∈ Grp ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 Vcvv 3492 ∅c0 4288 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Smgrpcsgrp 17888 Grpcgrp 18041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-riota 7103 df-ov 7148 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 |
This theorem is referenced by: (None) |
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