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Mirrors > Home > MPE Home > Th. List > isgrp | Structured version Visualization version GIF version |
Description: The predicate "is a group". (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
isgrp.b | ⊢ 𝐵 = (Base‘𝐺) |
isgrp.p | ⊢ + = (+g‘𝐺) |
isgrp.z | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
isgrp | ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎 ∈ 𝐵 ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6891 | . . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
2 | isgrp.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 1, 2 | eqtr4di 2785 | . . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
4 | fveq2 6891 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
5 | isgrp.p | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
6 | 4, 5 | eqtr4di 2785 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
7 | 6 | oveqd 7431 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑚(+g‘𝑔)𝑎) = (𝑚 + 𝑎)) |
8 | fveq2 6891 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺)) | |
9 | isgrp.z | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
10 | 8, 9 | eqtr4di 2785 | . . . . 5 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 ) |
11 | 7, 10 | eqeq12d 2743 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ (𝑚 + 𝑎) = 0 )) |
12 | 3, 11 | rexeqbidv 3338 | . . 3 ⊢ (𝑔 = 𝐺 → (∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 )) |
13 | 3, 12 | raleqbidv 3337 | . 2 ⊢ (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ ∀𝑎 ∈ 𝐵 ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 )) |
14 | df-grp 18884 | . 2 ⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)} | |
15 | 13, 14 | elrab2 3683 | 1 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎 ∈ 𝐵 ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 ‘cfv 6542 (class class class)co 7414 Basecbs 17171 +gcplusg 17224 0gc0g 17412 Mndcmnd 18685 Grpcgrp 18881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-ov 7417 df-grp 18884 |
This theorem is referenced by: grpmnd 18888 grpinvex 18891 grppropd 18899 isgrpd2e 18903 grp1 18994 ghmgrp 19013 primrootsunit1 41504 2zrngagrp 47234 |
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