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Theorem isgrp 18109
 Description: The predicate "is a group." (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
isgrp.b 𝐵 = (Base‘𝐺)
isgrp.p + = (+g𝐺)
isgrp.z 0 = (0g𝐺)
Assertion
Ref Expression
isgrp (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))
Distinct variable groups:   𝑚,𝑎,𝐵   𝐺,𝑎,𝑚
Allowed substitution hints:   + (𝑚,𝑎)   0 (𝑚,𝑎)

Proof of Theorem isgrp
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6661 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 isgrp.b . . . 4 𝐵 = (Base‘𝐺)
31, 2eqtr4di 2877 . . 3 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 fveq2 6661 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
5 isgrp.p . . . . . . 7 + = (+g𝐺)
64, 5eqtr4di 2877 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
76oveqd 7166 . . . . 5 (𝑔 = 𝐺 → (𝑚(+g𝑔)𝑎) = (𝑚 + 𝑎))
8 fveq2 6661 . . . . . 6 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
9 isgrp.z . . . . . 6 0 = (0g𝐺)
108, 9eqtr4di 2877 . . . . 5 (𝑔 = 𝐺 → (0g𝑔) = 0 )
117, 10eqeq12d 2840 . . . 4 (𝑔 = 𝐺 → ((𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ (𝑚 + 𝑎) = 0 ))
123, 11rexeqbidv 3393 . . 3 (𝑔 = 𝐺 → (∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ ∃𝑚𝐵 (𝑚 + 𝑎) = 0 ))
133, 12raleqbidv 3392 . 2 (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))
14 df-grp 18106 . 2 Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)}
1513, 14elrab2 3669 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ∃wrex 3134  ‘cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  0gc0g 16713  Mndcmnd 17911  Grpcgrp 18103 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 2179  ax-ext 2796 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-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-grp 18106 This theorem is referenced by:  grpmnd  18110  grpinvex  18113  grppropd  18118  isgrpd2e  18122  grp1  18206  ghmgrp  18223  2zrngagrp  44498
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