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Theorem isgrp 17356
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 6153 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 isgrp.b . . . 4 𝐵 = (Base‘𝐺)
31, 2syl6eqr 2673 . . 3 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 fveq2 6153 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
5 isgrp.p . . . . . . 7 + = (+g𝐺)
64, 5syl6eqr 2673 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
76oveqd 6627 . . . . 5 (𝑔 = 𝐺 → (𝑚(+g𝑔)𝑎) = (𝑚 + 𝑎))
8 fveq2 6153 . . . . . 6 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
9 isgrp.z . . . . . 6 0 = (0g𝐺)
108, 9syl6eqr 2673 . . . . 5 (𝑔 = 𝐺 → (0g𝑔) = 0 )
117, 10eqeq12d 2636 . . . 4 (𝑔 = 𝐺 → ((𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ (𝑚 + 𝑎) = 0 ))
123, 11rexeqbidv 3145 . . 3 (𝑔 = 𝐺 → (∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ ∃𝑚𝐵 (𝑚 + 𝑎) = 0 ))
133, 12raleqbidv 3144 . 2 (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔) ↔ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))
14 df-grp 17353 . 2 Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)}
1513, 14elrab2 3352 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cfv 5852  (class class class)co 6610  Basecbs 15788  +gcplusg 15869  0gc0g 16028  Mndcmnd 17222  Grpcgrp 17350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613  df-grp 17353
This theorem is referenced by:  grpmnd  17357  grpinvex  17360  grppropd  17365  isgrpd2e  17369  grp1  17450  ghmgrp  17467  2zrngagrp  41252
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