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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 18843 | . 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 6880 | . . . . . . . 8 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
6 | 1 | eleq2i 2825 | . . . . . . . . 9 ⊢ (𝑛 ∈ 𝐵 ↔ 𝑛 ∈ (Base‘𝐺)) |
7 | eleq2 2822 | . . . . . . . . . 10 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) ↔ 𝑛 ∈ ∅)) | |
8 | noel 4329 | . . . . . . . . . . 11 ⊢ ¬ 𝑛 ∈ ∅ | |
9 | 8 | pm2.21i 119 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ∅ → 𝐺 ∈ V) |
10 | 7, 9 | syl6bi 252 | . . . . . . . . 9 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) → 𝐺 ∈ V)) |
11 | 6, 10 | biimtrid 241 | . . . . . . . 8 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ 𝐵 → 𝐺 ∈ V)) |
12 | 5, 11 | syl 17 | . . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (𝑛 ∈ 𝐵 → 𝐺 ∈ V)) |
13 | 4, 12 | pm2.61i 182 | . . . . . 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 3148 | . . . 4 ⊢ (∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → 𝐺 ∈ V) |
16 | 1, 2 | issgrpv 18608 | . . . 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 274 | 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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∅c0 4321 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 +gcplusg 17193 Smgrpcsgrp 18605 Grpcgrp 18815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-riota 7361 df-ov 7408 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 |
This theorem is referenced by: (None) |
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