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Mirrors > Home > MPE Home > Th. List > grpid | Structured version Visualization version GIF version |
Description: Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.) |
Ref | Expression |
---|---|
grpinveu.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinveu.p | ⊢ + = (+g‘𝐺) |
grpinveu.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
grpid | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2825 | . 2 ⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 ) | |
2 | grpinveu.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpinveu.o | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
4 | 2, 3 | grpidcl 18069 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
5 | grpinveu.p | . . . . . . . 8 ⊢ + = (+g‘𝐺) | |
6 | 2, 5 | grprcan 18075 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )) |
7 | 6 | 3exp2 1346 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → ( 0 ∈ 𝐵 → (𝑋 ∈ 𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))))) |
8 | 4, 7 | mpid 44 | . . . . 5 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )))) |
9 | 8 | pm2.43d 53 | . . . 4 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))) |
10 | 9 | imp 407 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )) |
11 | 2, 5, 3 | grplid 18071 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
12 | 11 | eqeq2d 2829 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ (𝑋 + 𝑋) = 𝑋)) |
13 | 10, 12 | bitr3d 282 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 = 0 ↔ (𝑋 + 𝑋) = 𝑋)) |
14 | 1, 13 | syl5rbb 285 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Grpcgrp 18041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-riota 7103 df-ov 7148 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 |
This theorem is referenced by: isgrpid2 18078 grpidd2 18079 subg0 18223 qus0 18276 ghmid 18302 symgid 18459 isdrng2 19441 lmod0vid 19595 psr0 20107 cnfld0 20497 ldual0v 36166 erng0g 38010 |
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