Theorem iscmn 13879

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 5639	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		iscmn.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	eqtr4di 2282	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		raleq 2730	. . . . 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 2742	. . . 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 5639	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
8		iscmn.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
9	7, 8	eqtr4di 2282	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
10	9	oveqd 6034	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
11	9	oveqd 6034	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
12	10, 11	eqeq12d 2246	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13	12	2ralbidv 2556	. . 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 13872	. 2 ⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥)}
16	14, 15	elrab2 2965	1 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ‘cfv 5326  (class class class)co 6017  Basecbs 13081  +_gcplusg 13159  Mndcmnd 13498  CMndccmn 13870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020  df-cmn 13872

This theorem is referenced by:  isabl2  13880  cmnpropd  13881  iscmnd  13884  cmnmnd  13887  cmncom  13888  ghmcmn  13913  iscrng2  14027