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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 18885 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
5 | eqid 2724 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | eqid 2724 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
7 | grpsubid.m | . . . 4 ⊢ − = (-g‘𝐺) | |
8 | 2, 5, 6, 7 | grpsubval 18905 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 − 0 ) = (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 ))) |
9 | 1, 4, 8 | syl2anr 596 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 ))) |
10 | 3, 6 | grpinvid 18919 | . . . 4 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 ) |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝐺)‘ 0 ) = 0 ) |
12 | 11 | oveq2d 7417 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 )) = (𝑋(+g‘𝐺) 0 )) |
13 | 2, 5, 3 | grprid 18888 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺) 0 ) = 𝑋) |
14 | 9, 12, 13 | 3eqtrd 2768 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 +gcplusg 17196 0gc0g 17384 Grpcgrp 18853 invgcminusg 18854 -gcsg 18855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-sbg 18858 |
This theorem is referenced by: odmod 19456 sylow3lem1 19537 telgsums 19903 dprdfeq0 19934 rngqiprngimf1lem 21137 chp0mat 22670 tsmsxplem1 23979 tngnm 24490 ply1divex 25994 ply1remlem 26020 r1pid2 33145 qqhcn 33460 lcfrlem33 40936 |
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