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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 2744 | . 2 ⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 ) | |
| 2 | grpinveu.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
| 3 | grpinveu.o | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
| 4 | 2, 3 | grpidcl 18983 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) | 
| 5 | grpinveu.p | . . . . . . . 8 ⊢ + = (+g‘𝐺) | |
| 6 | 2, 5 | grprcan 18991 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )) | 
| 7 | 6 | 3exp2 1355 | . . . . . 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 406 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )) | 
| 11 | 2, 5, 3 | grplid 18985 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) | 
| 12 | 11 | eqeq2d 2748 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ (𝑋 + 𝑋) = 𝑋)) | 
| 13 | 10, 12 | bitr3d 281 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 = 0 ↔ (𝑋 + 𝑋) = 𝑋)) | 
| 14 | 1, 13 | bitr2id 284 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Grpcgrp 18951 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 | 
| This theorem is referenced by: isgrpid2 18994 grpidd2 18995 subg0 19150 qus0 19207 ghmid 19240 isdrng2 20743 lmod0vid 20892 cnfld0 21405 psr0 21978 psd1 22171 ldual0v 39151 erng0g 40996 | 
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