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Theorem grpinvid 18551

Description: The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.)

**Hypotheses**
Ref	Expression
grpinvid.u	⊢ 0 = (0_g‘𝐺)
grpinvid.n	⊢ 𝑁 = (inv_g‘𝐺)

**Assertion**
Ref	Expression
grpinvid	⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 )

**Proof of Theorem grpinvid**
Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		grpinvid.u	. . . 4 ⊢ 0 = (0_g‘𝐺)
3	1, 2	grpidcl 18522	. . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺))
4		eqid 2738	. . . 4 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
5	1, 4, 2	grplid 18524	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+_g‘𝐺) 0 ) = 0 )
6	3, 5	mpdan 683	. 2 ⊢ (𝐺 ∈ Grp → ( 0 (+_g‘𝐺) 0 ) = 0 )
7		grpinvid.n	. . . 4 ⊢ 𝑁 = (inv_g‘𝐺)
8	1, 4, 2, 7	grpinvid1 18545	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺) ∧ 0 ∈ (Base‘𝐺)) → ((𝑁‘ 0 ) = 0 ↔ ( 0 (+_g‘𝐺) 0 ) = 0 ))
9	3, 3, 8	mpd3an23 1461	. 2 ⊢ (𝐺 ∈ Grp → ((𝑁‘ 0 ) = 0 ↔ ( 0 (+_g‘𝐺) 0 ) = 0 ))
10	6, 9	mpbird 256	1 ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 +_gcplusg 16888 0_gc0g 17067 Grpcgrp 18492 inv_gcminusg 18493

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-riota 7212 df-ov 7258 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496

This theorem is referenced by: grpinvnz 18561 grpsubid1 18575 mulgneg 18637 mulginvcom 18643 mulgz 18646 0subg 18695 eqgid 18723 odnncl 19068 gexdvds 19104 gsumzinv 19461 gsumsub 19464 dprdfinv 19537 dsmmsubg 20860 mplsubglem 21115 mhpinvcl 21252 dchrisum0re 26566 qusker 31451 baerlem3lem1 39648

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