Theorem isgrp 13078

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	⊢ + = (+g‘𝐺)
isgrp.z	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
isgrp	⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎 ∈ 𝐵 ∃𝑚 ∈ 𝐵 (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.

Step	Hyp	Ref	Expression
1		fveq2 5554	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		isgrp.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	eqtr4di 2244	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		fveq2 5554	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
5		isgrp.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
6	4, 5	eqtr4di 2244	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
7	6	oveqd 5935	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑚(+g‘𝑔)𝑎) = (𝑚 + 𝑎))
8		fveq2 5554	. . . . . 6 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
9		isgrp.z	. . . . . 6 ⊢ 0 = (0g‘𝐺)
10	8, 9	eqtr4di 2244	. . . . 5 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
11	7, 10	eqeq12d 2208	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ (𝑚 + 𝑎) = 0 ))
12	3, 11	rexeqbidv 2707	. . 3 ⊢ (𝑔 = 𝐺 → (∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 ))
13	3, 12	raleqbidv 2706	. 2 ⊢ (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ ∀𝑎 ∈ 𝐵 ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 ))
14		df-grp 13075	. 2 ⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)}
15	13, 14	elrab2 2919	1 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎 ∈ 𝐵 ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 ))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ‘cfv 5254  (class class class)co 5918  Basecbs 12618  +_gcplusg 12695  0_gc0g 12867  Mndcmnd 12997  Grpcgrp 13072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921  df-grp 13075

This theorem is referenced by:  grpmnd  13079  grpinvex  13082  grppropd  13089  isgrpd2e  13092  grp1  13178  ghmgrp  13188