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Theorem isgrp 17695
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 6375 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 isgrp.b . . . 4 𝐵 = (Base‘𝐺)
31, 2syl6eqr 2817 . . 3 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 fveq2 6375 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
5 isgrp.p . . . . . . 7 + = (+g𝐺)
64, 5syl6eqr 2817 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
76oveqd 6859 . . . . 5 (𝑔 = 𝐺 → (𝑚(+g𝑔)𝑎) = (𝑚 + 𝑎))
8 fveq2 6375 . . . . . 6 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
9 isgrp.z . . . . . 6 0 = (0g𝐺)
108, 9syl6eqr 2817 . . . . 5 (𝑔 = 𝐺 → (0g𝑔) = 0 )
117, 10eqeq12d 2780 . . . 4 (𝑔 = 𝐺 → ((𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ (𝑚 + 𝑎) = 0 ))
123, 11rexeqbidv 3301 . . 3 (𝑔 = 𝐺 → (∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ ∃𝑚𝐵 (𝑚 + 𝑎) = 0 ))
133, 12raleqbidv 3300 . 2 (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))
14 df-grp 17692 . 2 Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)}
1513, 14elrab2 3523 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  cfv 6068  (class class class)co 6842  Basecbs 16130  +gcplusg 16214  0gc0g 16366  Mndcmnd 17560  Grpcgrp 17689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-iota 6031  df-fv 6076  df-ov 6845  df-grp 17692
This theorem is referenced by:  grpmnd  17696  grpinvex  17699  grppropd  17704  isgrpd2e  17708  grp1  17789  ghmgrp  17806  2zrngagrp  42612
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