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Theorem grpidd2 18119
Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 18103. (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 7147 . . . 4 (𝜑 → ( 0 + 0 ) = ( 0 (+g𝐺) 0 ))
3 oveq2 7138 . . . . . 6 (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
4 id 22 . . . . . 6 (𝑥 = 0𝑥 = 0 )
53, 4eqeq12d 2837 . . . . 5 (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
6 grpidd2.i . . . . . 6 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
76ralrimiva 3170 . . . . 5 (𝜑 → ∀𝑥𝐵 ( 0 + 𝑥) = 𝑥)
8 grpidd2.z . . . . 5 (𝜑0𝐵)
95, 7, 8rspcdva 3602 . . . 4 (𝜑 → ( 0 + 0 ) = 0 )
102, 9eqtr3d 2858 . . 3 (𝜑 → ( 0 (+g𝐺) 0 ) = 0 )
11 grpidd2.j . . . 4 (𝜑𝐺 ∈ Grp)
12 grpidd2.b . . . . 5 (𝜑𝐵 = (Base‘𝐺))
138, 12eleqtrd 2914 . . . 4 (𝜑0 ∈ (Base‘𝐺))
14 eqid 2821 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2821 . . . . 5 (+g𝐺) = (+g𝐺)
16 eqid 2821 . . . . 5 (0g𝐺) = (0g𝐺)
1714, 15, 16grpid 18117 . . . 4 ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
1811, 13, 17syl2anc 587 . . 3 (𝜑 → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
1910, 18mpbid 235 . 2 (𝜑 → (0g𝐺) = 0 )
2019eqcomd 2827 1 (𝜑0 = (0g𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cfv 6328  (class class class)co 7130  Basecbs 16461  +gcplusg 16543  0gc0g 16691  Grpcgrp 18081
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-riota 7088  df-ov 7133  df-0g 16693  df-mgm 17830  df-sgrp 17879  df-mnd 17890  df-grp 18084
This theorem is referenced by:  imasgrp2  18192
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