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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 6670 | . . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
2 | isgrp.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 1, 2 | syl6eqr 2874 | . . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
4 | fveq2 6670 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
5 | isgrp.p | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
6 | 4, 5 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
7 | 6 | oveqd 7173 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑚(+g‘𝑔)𝑎) = (𝑚 + 𝑎)) |
8 | fveq2 6670 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺)) | |
9 | isgrp.z | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
10 | 8, 9 | syl6eqr 2874 | . . . . 5 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 ) |
11 | 7, 10 | eqeq12d 2837 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ (𝑚 + 𝑎) = 0 )) |
12 | 3, 11 | rexeqbidv 3402 | . . 3 ⊢ (𝑔 = 𝐺 → (∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 )) |
13 | 3, 12 | raleqbidv 3401 | . 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 18106 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Mndcmnd 17911 Grpcgrp 18103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-grp 18106 |
This theorem is referenced by: grpmnd 18110 grpinvex 18113 grppropd 18118 isgrpd2e 18122 grp1 18206 ghmgrp 18223 2zrngagrp 44234 |
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