Theorem isgrpid2 17281

Description: Properties showing that an element 𝑍 is the identity element of a group. (Contributed by NM, 7-Aug-2013.)

Hypotheses

Ref	Expression
grpinveu.b	⊢ 𝐵 = (Base‘𝐺)
grpinveu.p	⊢ + = (+g‘𝐺)
grpinveu.o	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
isgrpid2	⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍))

Proof of Theorem isgrpid2

Step	Hyp	Ref	Expression
1		grpinveu.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
2		grpinveu.p	. . . . 5 ⊢ + = (+g‘𝐺)
3		grpinveu.o	. . . . 5 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	grpid 17280	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑍) = 𝑍 ↔ 0 = 𝑍))
5	4	biimpd 218	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑍) = 𝑍 → 0 = 𝑍))
6	5	expimpd 627	. 2 ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) → 0 = 𝑍))
7	1, 3	grpidcl 17273	. . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
8	1, 2, 3	grplid 17275	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 )
9	7, 8	mpdan 699	. . . 4 ⊢ (𝐺 ∈ Grp → ( 0 + 0 ) = 0 )
10	7, 9	jca 553	. . 3 ⊢ (𝐺 ∈ Grp → ( 0 ∈ 𝐵 ∧ ( 0 + 0 ) = 0 ))
11		eleq1 2676	. . . 4 ⊢ ( 0 = 𝑍 → ( 0 ∈ 𝐵 ↔ 𝑍 ∈ 𝐵))
12		id 22	. . . . . 6 ⊢ ( 0 = 𝑍 → 0 = 𝑍)
13	12, 12	oveq12d 6567	. . . . 5 ⊢ ( 0 = 𝑍 → ( 0 + 0 ) = (𝑍 + 𝑍))
14	13, 12	eqeq12d 2625	. . . 4 ⊢ ( 0 = 𝑍 → (( 0 + 0 ) = 0 ↔ (𝑍 + 𝑍) = 𝑍))
15	11, 14	anbi12d 743	. . 3 ⊢ ( 0 = 𝑍 → (( 0 ∈ 𝐵 ∧ ( 0 + 0 ) = 0 ) ↔ (𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
16	10, 15	syl5ibcom 234	. 2 ⊢ (𝐺 ∈ Grp → ( 0 = 𝑍 → (𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
17	6, 16	impbid 201	1 ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923  Grpcgrp 17245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248

This theorem is referenced by:  drngid2  18586  dchr1  24782  erngdvlem4  35297  erngdvlem4-rN  35305