Theorem iscmn 20003

Description: The predicate "is a commutative monoid". (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
iscmn.b	⊢ 𝐵 = (Base‘𝐺)
iscmn.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
iscmn	⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦

Allowed substitution hints:   + (𝑥, 𝑦)

Proof of Theorem iscmn

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		iscmn.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	eqtr4di 2813	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		raleq 3316	. . . . 5 ⊢ ((Base‘𝑔) = 𝐵 → (∀𝑦 ∈ (Base‘𝑔)(𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥) ↔ ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥)))
5	4	raleqbi1dv 3329	. . . 4 ⊢ ((Base‘𝑔) = 𝐵 → (∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥)))
6	3, 5	syl 18	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥)))
7		fveq2 6885	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
8		iscmn.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
9	7, 8	eqtr4di 2813	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
10	9	oveqd 7437	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
11	9	oveqd 7437	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
12	10, 11	eqeq12d 2776	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13	12	2ralbidv 3226	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
14	6, 13	bitrd 282	. 2 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
15		df-cmn 19996	. 2 ⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥)}
16	14, 15	elrab2 3648	1 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  Mndcmnd 18923  CMndccmn 19994

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-cmn 19996

This theorem is used by:  isabl2  20004  cmnpropd  20005  iscmnd  20008  cmnmnd  20011  cmncom  20012  ghmcmn  20045  submcmn2  20053  cycsubmcmn  20103  iscrng2  20479  xrs1cmn  21748  abliso  33596  gicabl  44101  pgrpgt2nabl  49478