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Mirrors > Home > MPE Home > Th. List > grpsubid1 | Structured version Visualization version GIF version |
Description: Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.) |
Ref | Expression |
---|---|
grpsubid.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubid.o | ⊢ 0 = (0g‘𝐺) |
grpsubid.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpsubid1 | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵) | |
2 | grpsubid.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpsubid.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
4 | 2, 3 | grpidcl 18995 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
5 | eqid 2734 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | eqid 2734 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
7 | grpsubid.m | . . . 4 ⊢ − = (-g‘𝐺) | |
8 | 2, 5, 6, 7 | grpsubval 19015 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 − 0 ) = (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 ))) |
9 | 1, 4, 8 | syl2anr 597 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 ))) |
10 | 3, 6 | grpinvid 19029 | . . . 4 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 ) |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝐺)‘ 0 ) = 0 ) |
12 | 11 | oveq2d 7446 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 )) = (𝑋(+g‘𝐺) 0 )) |
13 | 2, 5, 3 | grprid 18998 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺) 0 ) = 𝑋) |
14 | 9, 12, 13 | 3eqtrd 2778 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 0gc0g 17485 Grpcgrp 18963 invgcminusg 18964 -gcsg 18965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-sbg 18968 |
This theorem is referenced by: odmod 19578 sylow3lem1 19659 telgsums 20025 dprdfeq0 20056 rngqiprngimf1lem 21321 chp0mat 22867 tsmsxplem1 24176 tngnm 24687 ply1divex 26190 r1pid2 26215 ply1remlem 26218 fracfld 33289 r1pid2OLD 33608 irredminply 33721 qqhcn 33953 lcfrlem33 41557 |
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