Theorem iscmn 19865

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 6881	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		iscmn.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	eqtr4di 2815	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		raleq 3319	. . . . 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 3332	. . . 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 6881	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
8		iscmn.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
9	7, 8	eqtr4di 2815	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
10	9	oveqd 7429	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
11	9	oveqd 7429	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
12	10, 11	eqeq12d 2778	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13	12	2ralbidv 3228	. . 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 19858	. 2 ⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥)}
16	14, 15	elrab2 3653	1 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  +_gcplusg 17316  Mndcmnd 18798  CMndccmn 19856

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415  df-cmn 19858

This theorem is used by:  isabl2  19866  cmnpropd  19867  iscmnd  19870  cmnmnd  19873  cmncom  19874  ghmcmn  19907  submcmn2  19915  cycsubmcmn  19965  iscrng2  20340  xrs1cmn  21603  abliso  33364  gicabl  43854  pgrpgt2nabl  49174