Theorem iscmn 14096

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 5695	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		iscmn.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	eqtr4di 2289	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		raleq 2749	. . . . 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 2761	. . . 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 14	. . 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 5695	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
8		iscmn.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
9	7, 8	eqtr4di 2289	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
10	9	oveqd 6102	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
11	9	oveqd 6102	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
12	10, 11	eqeq12d 2253	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13	12	2ralbidv 2574	. . 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 188	. 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 14089	. 2 ⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥)}
16	14, 15	elrab2 2985	1 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ‘cfv 5377  (class class class)co 6085  Basecbs 13352  +_gcplusg 13431  Mndcmnd 13729  CMndccmn 14087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088  df-cmn 14089

This theorem is used by:  isabl2  14097  cmnpropd  14098  iscmnd  14101  cmnmnd  14104  cmncom  14105  cmnsubm  14112  ghmcmn  14131  iscrng2  14319