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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 19006 | . . . 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 18996 | . . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
8 | 1, 2, 3 | grplid 18998 | . . . . 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 2827 | . . . 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 2751 | . . . 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 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 Grpcgrp 18964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 |
This theorem is referenced by: drngid2 20769 dchr1 27316 rloc0g 33258 erngdvlem4 40974 erngdvlem4-rN 40982 |
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