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Theorem grpidd2 12791
Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 12776. (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 5885 . . . 4 (𝜑 → ( 0 + 0 ) = ( 0 (+g𝐺) 0 ))
3 oveq2 5876 . . . . . 6 (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 ))
4 id 19 . . . . . 6 (𝑥 = 0𝑥 = 0 )
53, 4eqeq12d 2192 . . . . 5 (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 ))
6 grpidd2.i . . . . . 6 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
76ralrimiva 2550 . . . . 5 (𝜑 → ∀𝑥𝐵 ( 0 + 𝑥) = 𝑥)
8 grpidd2.z . . . . 5 (𝜑0𝐵)
95, 7, 8rspcdva 2846 . . . 4 (𝜑 → ( 0 + 0 ) = 0 )
102, 9eqtr3d 2212 . . 3 (𝜑 → ( 0 (+g𝐺) 0 ) = 0 )
11 grpidd2.j . . . 4 (𝜑𝐺 ∈ Grp)
12 grpidd2.b . . . . 5 (𝜑𝐵 = (Base‘𝐺))
138, 12eleqtrd 2256 . . . 4 (𝜑0 ∈ (Base‘𝐺))
14 eqid 2177 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2177 . . . . 5 (+g𝐺) = (+g𝐺)
16 eqid 2177 . . . . 5 (0g𝐺) = (0g𝐺)
1714, 15, 16grpid 12789 . . . 4 ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
1811, 13, 17syl2anc 411 . . 3 (𝜑 → (( 0 (+g𝐺) 0 ) = 0 ↔ (0g𝐺) = 0 ))
1910, 18mpbid 147 . 2 (𝜑 → (0g𝐺) = 0 )
2019eqcomd 2183 1 (𝜑0 = (0g𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  cfv 5211  (class class class)co 5868  Basecbs 12432  +gcplusg 12505  0gc0g 12640  Grpcgrp 12754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757
This theorem is referenced by: (None)
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