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Theorem grpidd2 17666
Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 17651. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
grpidd2.b (𝜑𝐵 = (Base‘𝐺))
grpidd2.p (𝜑+ = (+g𝐺))
grpidd2.z (𝜑0𝐵)
grpidd2.i ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
grpidd2.j (𝜑𝐺 ∈ Grp)
Assertion
Ref Expression
grpidd2 (𝜑0 = (0g𝐺))
Distinct variable groups:   𝑥,𝐵   𝑥, +   𝜑,𝑥   𝑥, 0
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem grpidd2
StepHypRef Expression
1 grpidd2.p . . . . 5 (𝜑+ = (+g𝐺))
21oveqd 6809 . . . 4 (𝜑 → ( 0 + 0 ) = ( 0 (+g𝐺) 0 ))
3 oveq2 6800 . . . . . 6 (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
4 id 22 . . . . . 6 (𝑥 = 0𝑥 = 0 )
53, 4eqeq12d 2785 . . . . 5 (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
6 grpidd2.i . . . . . 6 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
76ralrimiva 3114 . . . . 5 (𝜑 → ∀𝑥𝐵 ( 0 + 𝑥) = 𝑥)
8 grpidd2.z . . . . 5 (𝜑0𝐵)
95, 7, 8rspcdva 3464 . . . 4 (𝜑 → ( 0 + 0 ) = 0 )
102, 9eqtr3d 2806 . . 3 (𝜑 → ( 0 (+g𝐺) 0 ) = 0 )
11 grpidd2.j . . . 4 (𝜑𝐺 ∈ Grp)
12 grpidd2.b . . . . 5 (𝜑𝐵 = (Base‘𝐺))
138, 12eleqtrd 2851 . . . 4 (𝜑0 ∈ (Base‘𝐺))
14 eqid 2770 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2770 . . . . 5 (+g𝐺) = (+g𝐺)
16 eqid 2770 . . . . 5 (0g𝐺) = (0g𝐺)
1714, 15, 16grpid 17664 . . . 4 ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
1811, 13, 17syl2anc 565 . . 3 (𝜑 → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
1910, 18mpbid 222 . 2 (𝜑 → (0g𝐺) = 0 )
2019eqcomd 2776 1 (𝜑0 = (0g𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  0gc0g 16307  Grpcgrp 17629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-riota 6753  df-ov 6795  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632
This theorem is referenced by:  imasgrp2  17737
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