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Theorem isgrp 17636
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 6332 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 isgrp.b . . . 4 𝐵 = (Base‘𝐺)
31, 2syl6eqr 2823 . . 3 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 fveq2 6332 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
5 isgrp.p . . . . . . 7 + = (+g𝐺)
64, 5syl6eqr 2823 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
76oveqd 6810 . . . . 5 (𝑔 = 𝐺 → (𝑚(+g𝑔)𝑎) = (𝑚 + 𝑎))
8 fveq2 6332 . . . . . 6 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
9 isgrp.z . . . . . 6 0 = (0g𝐺)
108, 9syl6eqr 2823 . . . . 5 (𝑔 = 𝐺 → (0g𝑔) = 0 )
117, 10eqeq12d 2786 . . . 4 (𝑔 = 𝐺 → ((𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ (𝑚 + 𝑎) = 0 ))
123, 11rexeqbidv 3302 . . 3 (𝑔 = 𝐺 → (∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ ∃𝑚𝐵 (𝑚 + 𝑎) = 0 ))
133, 12raleqbidv 3301 . 2 (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))
14 df-grp 17633 . 2 Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)}
1513, 14elrab2 3518 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Mndcmnd 17502  Grpcgrp 17630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-grp 17633
This theorem is referenced by:  grpmnd  17637  grpinvex  17640  grppropd  17645  isgrpd2e  17649  grp1  17730  ghmgrp  17747  2zrngagrp  42471
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