Theorem iscmnd 19314

Description: Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
iscmnd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
iscmnd.p	⊢ (𝜑 → + = (+g‘𝐺))
iscmnd.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
iscmnd.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))

Assertion

Ref	Expression
iscmnd	⊢ (𝜑 → 𝐺 ∈ CMnd)

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

Allowed substitution hints:   + (𝑥,𝑦)

Proof of Theorem iscmnd

Step	Hyp	Ref	Expression
1		iscmnd.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd)
2		iscmnd.c	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3	2	3expib 1120	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
4	3	ralrimivv 3113	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5		iscmnd.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
6		iscmnd.p	. . . . . . . 8 ⊢ (𝜑 → + = (+g‘𝐺))
7	6	oveqd 7272	. . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦))
8	6	oveqd 7272	. . . . . . 7 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥))
9	7, 8	eqeq12d 2754	. . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
10	5, 9	raleqbidv 3327	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
11	5, 10	raleqbidv 3327	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
12	11	anbi2d 628	. . 3 ⊢ (𝜑 → ((𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)) ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))))
13	1, 4, 12	mpbi2and 708	. 2 ⊢ (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
14		eqid 2738	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
15		eqid 2738	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
16	14, 15	iscmn 19309	. 2 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
17	13, 16	sylibr 233	1 ⊢ (𝜑 → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Mndcmnd 18300  CMndccmn 19301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-cmn 19303

This theorem is referenced by:  isabld  19315  subcmn  19353  cntrcmnd  19358  prdscmnd  19377  iscrngd  19740  xrsmcmn  20533  psrcrng  21092  idlsrgcmnd  31562  2zrngacmnd  45388