Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6191 | . . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
2 | isgrp.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 1, 2 | syl6eqr 2674 | . . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
4 | fveq2 6191 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
5 | isgrp.p | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
6 | 4, 5 | syl6eqr 2674 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
7 | 6 | oveqd 6667 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑚(+g‘𝑔)𝑎) = (𝑚 + 𝑎)) |
8 | fveq2 6191 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺)) | |
9 | isgrp.z | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
10 | 8, 9 | syl6eqr 2674 | . . . . 5 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 ) |
11 | 7, 10 | eqeq12d 2637 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ (𝑚 + 𝑎) = 0 )) |
12 | 3, 11 | rexeqbidv 3153 | . . 3 ⊢ (𝑔 = 𝐺 → (∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 )) |
13 | 3, 12 | raleqbidv 3152 | . 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 17425 | . 2 ⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)} | |
15 | 13, 14 | elrab2 3366 | 1 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎 ∈ 𝐵 ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 0gc0g 16100 Mndcmnd 17294 Grpcgrp 17422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-grp 17425 |
This theorem is referenced by: grpmnd 17429 grpinvex 17432 grppropd 17437 isgrpd2e 17441 grp1 17522 ghmgrp 17539 2zrngagrp 41943 |
Copyright terms: Public domain | W3C validator |