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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 18993 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑍) = 𝑍 ↔ 0 = 𝑍)) |
| 5 | 4 | biimpd 229 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑍) = 𝑍 → 0 = 𝑍)) |
| 6 | 5 | expimpd 453 | . 2 ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) → 0 = 𝑍)) |
| 7 | 1, 3 | grpidcl 18983 | . . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
| 8 | 1, 2, 3 | grplid 18985 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 ) |
| 9 | 7, 8 | mpdan 687 | . . . 4 ⊢ (𝐺 ∈ Grp → ( 0 + 0 ) = 0 ) |
| 10 | 7, 9 | jca 511 | . . 3 ⊢ (𝐺 ∈ Grp → ( 0 ∈ 𝐵 ∧ ( 0 + 0 ) = 0 )) |
| 11 | eleq1 2829 | . . . 4 ⊢ ( 0 = 𝑍 → ( 0 ∈ 𝐵 ↔ 𝑍 ∈ 𝐵)) | |
| 12 | id 22 | . . . . . 6 ⊢ ( 0 = 𝑍 → 0 = 𝑍) | |
| 13 | 12, 12 | oveq12d 7449 | . . . . 5 ⊢ ( 0 = 𝑍 → ( 0 + 0 ) = (𝑍 + 𝑍)) |
| 14 | 13, 12 | eqeq12d 2753 | . . . 4 ⊢ ( 0 = 𝑍 → (( 0 + 0 ) = 0 ↔ (𝑍 + 𝑍) = 𝑍)) |
| 15 | 11, 14 | anbi12d 632 | . . 3 ⊢ ( 0 = 𝑍 → (( 0 ∈ 𝐵 ∧ ( 0 + 0 ) = 0 ) ↔ (𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍))) |
| 16 | 10, 15 | syl5ibcom 245 | . 2 ⊢ (𝐺 ∈ Grp → ( 0 = 𝑍 → (𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍))) |
| 17 | 6, 16 | impbid 212 | 1 ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍)) |
| 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: drngid2 20752 dchr1 27301 rloc0g 33275 erngdvlem4 40993 erngdvlem4-rN 41001 |
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