Theorem isgrpid2 18910

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 18909	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑍) = 𝑍 ↔ 0 = 𝑍))
5	4	biimpd 229	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑍) = 𝑍 → 0 = 𝑍))
6	5	expimpd 453	. 2 ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) → 0 = 𝑍))
7	1, 3	grpidcl 18899	. . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
8	1, 2, 3	grplid 18901	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 )
9	7, 8	mpdan 688	. . . 4 ⊢ (𝐺 ∈ Grp → ( 0 + 0 ) = 0 )
10	7, 9	jca 511	. . 3 ⊢ (𝐺 ∈ Grp → ( 0 ∈ 𝐵 ∧ ( 0 + 0 ) = 0 ))
11		eleq1 2825	. . . 4 ⊢ ( 0 = 𝑍 → ( 0 ∈ 𝐵 ↔ 𝑍 ∈ 𝐵))
12		id 22	. . . . . 6 ⊢ ( 0 = 𝑍 → 0 = 𝑍)
13	12, 12	oveq12d 7378	. . . . 5 ⊢ ( 0 = 𝑍 → ( 0 + 0 ) = (𝑍 + 𝑍))
14	13, 12	eqeq12d 2753	. . . 4 ⊢ ( 0 = 𝑍 → (( 0 + 0 ) = 0 ↔ (𝑍 + 𝑍) = 𝑍))
15	11, 14	anbi12d 633	. . 3 ⊢ ( 0 = 𝑍 → (( 0 ∈ 𝐵 ∧ ( 0 + 0 ) = 0 ) ↔ (𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
16	10, 15	syl5ibcom 245	. 2 ⊢ (𝐺 ∈ Grp → ( 0 = 𝑍 → (𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍)))
17	6, 16	impbid 212	1 ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  +_gcplusg 17181  0_gc0g 17363  Grpcgrp 18867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-riota 7317  df-ov 7363  df-0g 17365  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-grp 18870

This theorem is referenced by:  drngid2  20689  dchr1  27228  rloc0g  33334  erngdvlem4  41288  erngdvlem4-rN  41296