Theorem grpid 18909

Description: Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)

Hypotheses

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

Assertion

Ref	Expression
grpid	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋))

Proof of Theorem grpid

Step	Hyp	Ref	Expression
1		eqcom 2744	. 2 ⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 )
2		grpinveu.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
3		grpinveu.o	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
4	2, 3	grpidcl 18899	. . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
5		grpinveu.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
6	2, 5	grprcan 18907	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))
7	6	3exp2 1356	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → ( 0 ∈ 𝐵 → (𝑋 ∈ 𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )))))
8	4, 7	mpid 44	. . . . 5 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))))
9	8	pm2.43d 53	. . . 4 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )))
10	9	imp 406	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))
11	2, 5, 3	grplid 18901	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
12	11	eqeq2d 2748	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ (𝑋 + 𝑋) = 𝑋))
13	10, 12	bitr3d 281	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 = 0 ↔ (𝑋 + 𝑋) = 𝑋))
14	1, 13	bitr2id 284	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋))

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:  isgrpid2  18910  grpidd2  18911  subg0  19066  qus0  19122  ghmid  19155  isdrng2  20680  lmod0vid  20849  cnfld0  21351  psr0  21917  psd1  22114  ldual0v  39447  erng0g  41291