Theorem isgrp 18101

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 6663	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		isgrp.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	syl6eqr 2872	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		fveq2 6663	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
5		isgrp.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
6	4, 5	syl6eqr 2872	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
7	6	oveqd 7165	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑚(+g‘𝑔)𝑎) = (𝑚 + 𝑎))
8		fveq2 6663	. . . . . 6 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
9		isgrp.z	. . . . . 6 ⊢ 0 = (0g‘𝐺)
10	8, 9	syl6eqr 2872	. . . . 5 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
11	7, 10	eqeq12d 2835	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ (𝑚 + 𝑎) = 0 ))
12	3, 11	rexeqbidv 3401	. . 3 ⊢ (𝑔 = 𝐺 → (∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔) ↔ ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 ))
13	3, 12	raleqbidv 3400	. 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 18098	. 2 ⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)}
15	13, 14	elrab2 3681	1 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎 ∈ 𝐵 ∃𝑚 ∈ 𝐵 (𝑚 + 𝑎) = 0 ))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∀wral 3136  ∃wrex 3137  ‘cfv 6348  (class class class)co 7148  Basecbs 16475  +_gcplusg 16557  0_gc0g 16705  Mndcmnd 17903  Grpcgrp 18095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7151  df-grp 18098

This theorem is referenced by:  grpmnd  18102  grpinvex  18105  grppropd  18110  isgrpd2e  18114  grp1  18198  ghmgrp  18215  2zrngagrp  44205