Theorem iscmn 13096

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 5516	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		iscmn.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	eqtr4di 2228	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		raleq 2673	. . . . 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 2681	. . . 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 5516	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
8		iscmn.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
9	7, 8	eqtr4di 2228	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
10	9	oveqd 5892	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
11	9	oveqd 5892	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
12	10, 11	eqeq12d 2192	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13	12	2ralbidv 2501	. . 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 13090	. 2 ⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g‘𝑔)𝑦) = (𝑦(+g‘𝑔)𝑥)}
16	14, 15	elrab2 2897	1 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ‘cfv 5217  (class class class)co 5875  Basecbs 12462  +_gcplusg 12536  Mndcmnd 12817  CMndccmn 13088

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225  df-ov 5878  df-cmn 13090

This theorem is referenced by:  isabl2  13097  cmnpropd  13098  iscmnd  13101  cmnmnd  13104  cmncom  13105  iscrng2  13198