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Mirrors > Home > MPE Home > Th. List > isgrpid2 | Structured version Visualization version GIF version |
Description: Properties showing that an element 𝑍 is the identity element of a group. (Contributed by NM, 7-Aug-2013.) |
Ref | Expression |
---|---|
grpinveu.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinveu.p | ⊢ + = (+g‘𝐺) |
grpinveu.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
isgrpid2 | ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinveu.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpinveu.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
3 | grpinveu.o | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
4 | 1, 2, 3 | grpid 18131 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑍) = 𝑍 ↔ 0 = 𝑍)) |
5 | 4 | biimpd 232 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑍) = 𝑍 → 0 = 𝑍)) |
6 | 5 | expimpd 457 | . 2 ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) → 0 = 𝑍)) |
7 | 1, 3 | grpidcl 18123 | . . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
8 | 1, 2, 3 | grplid 18125 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 ) |
9 | 7, 8 | mpdan 686 | . . . 4 ⊢ (𝐺 ∈ Grp → ( 0 + 0 ) = 0 ) |
10 | 7, 9 | jca 515 | . . 3 ⊢ (𝐺 ∈ Grp → ( 0 ∈ 𝐵 ∧ ( 0 + 0 ) = 0 )) |
11 | eleq1 2877 | . . . 4 ⊢ ( 0 = 𝑍 → ( 0 ∈ 𝐵 ↔ 𝑍 ∈ 𝐵)) | |
12 | id 22 | . . . . . 6 ⊢ ( 0 = 𝑍 → 0 = 𝑍) | |
13 | 12, 12 | oveq12d 7153 | . . . . 5 ⊢ ( 0 = 𝑍 → ( 0 + 0 ) = (𝑍 + 𝑍)) |
14 | 13, 12 | eqeq12d 2814 | . . . 4 ⊢ ( 0 = 𝑍 → (( 0 + 0 ) = 0 ↔ (𝑍 + 𝑍) = 𝑍)) |
15 | 11, 14 | anbi12d 633 | . . 3 ⊢ ( 0 = 𝑍 → (( 0 ∈ 𝐵 ∧ ( 0 + 0 ) = 0 ) ↔ (𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍))) |
16 | 10, 15 | syl5ibcom 248 | . 2 ⊢ (𝐺 ∈ Grp → ( 0 = 𝑍 → (𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍))) |
17 | 6, 16 | impbid 215 | 1 ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Grpcgrp 18095 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 |
This theorem is referenced by: drngid2 19511 dchr1 25841 erngdvlem4 38287 erngdvlem4-rN 38295 |
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