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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 18408 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
5 | eqid 2738 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | eqid 2738 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
7 | grpsubid.m | . . . 4 ⊢ − = (-g‘𝐺) | |
8 | 2, 5, 6, 7 | grpsubval 18426 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 − 0 ) = (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 ))) |
9 | 1, 4, 8 | syl2anr 600 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 ))) |
10 | 3, 6 | grpinvid 18437 | . . . 4 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 ) |
11 | 10 | adantr 484 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝐺)‘ 0 ) = 0 ) |
12 | 11 | oveq2d 7238 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘ 0 )) = (𝑋(+g‘𝐺) 0 )) |
13 | 2, 5, 3 | grprid 18411 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺) 0 ) = 𝑋) |
14 | 9, 12, 13 | 3eqtrd 2782 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ‘cfv 6389 (class class class)co 7222 Basecbs 16773 +gcplusg 16815 0gc0g 16957 Grpcgrp 18378 invgcminusg 18379 -gcsg 18380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-op 4557 df-uni 4829 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-id 5464 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-fv 6397 df-riota 7179 df-ov 7225 df-oprab 7226 df-mpo 7227 df-1st 7770 df-2nd 7771 df-0g 16959 df-mgm 18127 df-sgrp 18176 df-mnd 18187 df-grp 18381 df-minusg 18382 df-sbg 18383 |
This theorem is referenced by: odmod 18951 sylow3lem1 19029 telgsums 19391 dprdfeq0 19422 chp0mat 21756 tsmsxplem1 23063 tngnm 23562 ply1divex 25047 ply1remlem 25073 qqhcn 31666 lcfrlem33 39339 |
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