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Theorem grpsubid1 18981

Description: Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.)

**Hypotheses**
Ref	Expression
grpsubid.b	⊢ 𝐵 = (Base‘𝐺)
grpsubid.o	⊢ 0 = (0_g‘𝐺)
grpsubid.m	⊢ − = (-_g‘𝐺)

**Assertion**
Ref	Expression
grpsubid1	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = 𝑋)

**Proof of Theorem grpsubid1**
Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵)
2		grpsubid.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3		grpsubid.o	. . . 4 ⊢ 0 = (0_g‘𝐺)
4	2, 3	grpidcl 18922	. . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
5		eqid 2728	. . . 4 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
6		eqid 2728	. . . 4 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
7		grpsubid.m	. . . 4 ⊢ − = (-_g‘𝐺)
8	2, 5, 6, 7	grpsubval 18942	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 − 0 ) = (𝑋(+_g‘𝐺)((inv_g‘𝐺)‘ 0 )))
9	1, 4, 8	syl2anr 596	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = (𝑋(+_g‘𝐺)((inv_g‘𝐺)‘ 0 )))
10	3, 6	grpinvid 18956	. . . 4 ⊢ (𝐺 ∈ Grp → ((inv_g‘𝐺)‘ 0 ) = 0 )
11	10	adantr 480	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((inv_g‘𝐺)‘ 0 ) = 0 )
12	11	oveq2d 7436	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+_g‘𝐺)((inv_g‘𝐺)‘ 0 )) = (𝑋(+_g‘𝐺) 0 ))
13	2, 5, 3	grprid 18925	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+_g‘𝐺) 0 ) = 𝑋)
14	9, 12, 13	3eqtrd 2772	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 +_gcplusg 17233 0_gc0g 17421 Grpcgrp 18890 inv_gcminusg 18891 -_gcsg 18892

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-sbg 18895

This theorem is referenced by: odmod 19501 sylow3lem1 19582 telgsums 19948 dprdfeq0 19979 rngqiprngimf1lem 21184 chp0mat 22761 tsmsxplem1 24070 tngnm 24581 ply1divex 26085 ply1remlem 26112 fracfld 33007 r1pid2 33279 irredminply 33384 qqhcn 33592 lcfrlem33 41048

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