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Theorem isgrp 18755
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 6843 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 isgrp.b . . . 4 𝐵 = (Base‘𝐺)
31, 2eqtr4di 2795 . . 3 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 fveq2 6843 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
5 isgrp.p . . . . . . 7 + = (+g𝐺)
64, 5eqtr4di 2795 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
76oveqd 7375 . . . . 5 (𝑔 = 𝐺 → (𝑚(+g𝑔)𝑎) = (𝑚 + 𝑎))
8 fveq2 6843 . . . . . 6 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
9 isgrp.z . . . . . 6 0 = (0g𝐺)
108, 9eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 → (0g𝑔) = 0 )
117, 10eqeq12d 2753 . . . 4 (𝑔 = 𝐺 → ((𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ (𝑚 + 𝑎) = 0 ))
123, 11rexeqbidv 3321 . . 3 (𝑔 = 𝐺 → (∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ ∃𝑚𝐵 (𝑚 + 𝑎) = 0 ))
133, 12raleqbidv 3320 . 2 (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))
14 df-grp 18752 . 2 Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)}
1513, 14elrab2 3649 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  cfv 6497  (class class class)co 7358  Basecbs 17084  +gcplusg 17134  0gc0g 17322  Mndcmnd 18557  Grpcgrp 18749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-grp 18752
This theorem is referenced by:  grpmnd  18756  grpinvex  18759  grppropd  18766  isgrpd2e  18770  grp1  18855  ghmgrp  18872  2zrngagrp  46248
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