Theorem isgrp 18913

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 6834	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		isgrp.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	eqtr4di 2793	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		fveq2 6834	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
5		isgrp.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
6	4, 5	eqtr4di 2793	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
7	6	oveqd 7380	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑚(+g‘𝑔)𝑎) = (𝑚 + 𝑎))
8		fveq2 6834	. . . . . 6 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
9		isgrp.z	. . . . . 6 ⊢ 0 = (0g‘𝐺)
10	8, 9	eqtr4di 2793	. . . . 5 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
11	7, 10	eqeq12d 2756	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ (𝑚 + 𝑎) = 0 ))
12	3, 11	rexeqbidv 3315	. . 3 ⊢ (𝑔 = 𝐺 → (∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 ))
13	3, 12	raleqbidv 3314	. 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 18910	. 2 ⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)}
15	13, 14	elrab2 3639	1 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎 ∈ 𝐵 ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 ))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  0_gc0g 17400  Mndcmnd 18700  Grpcgrp 18907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-grp 18910

This theorem is referenced by:  grpmnd  18914  grpinvex  18917  grppropd  18925  isgrpd2e  18929  grp1  19021  ghmgrp  19040  primrootsunit1  42589  2zrngagrp  48747