Theorem iscmnd 18205

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 1268	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
4	3	ralrimivv 2970	. . 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 6667	. . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦))
8	6	oveqd 6667	. . . . . . 7 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥))
9	7, 8	eqeq12d 2637	. . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
10	5, 9	raleqbidv 3152	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
11	5, 10	raleqbidv 3152	. . . 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 740	. . 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 956	. 2 ⊢ (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
14		eqid 2622	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
15		eqid 2622	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
16	14, 15	iscmn 18200	. 2 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
17	13, 16	sylibr 224	1 ⊢ (𝜑 → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  Mndcmnd 17294  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-cmn 18195

This theorem is referenced by:  isabld  18206  subcmn  18242  prdscmnd  18264  iscrngd  18586  psrcrng  19413  xrsmcmn  19769  2zrngacmnd  41942