Theorem iscmnd 19763

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 1123	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
4	3	ralrimivv 3179	. . 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 7378	. . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦))
8	6	oveqd 7378	. . . . . . 7 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥))
9	7, 8	eqeq12d 2753	. . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
10	5, 9	raleqbidv 3312	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
11	5, 10	raleqbidv 3312	. . . 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 631	. . 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 713	. 2 ⊢ (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
14		eqid 2737	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
15		eqid 2737	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
16	14, 15	iscmn 19758	. 2 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
17	13, 16	sylibr 234	1 ⊢ (𝜑 → 𝐺 ∈ CMnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  +_gcplusg 17214  Mndcmnd 18696  CMndccmn 19749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6449  df-fv 6501  df-ov 7364  df-cmn 19751

This theorem is referenced by:  isabld  19764  subcmn  19806  cntrcmnd  19811  prdscmnd  19830  iscrngd  20267  xrsmcmn  21384  psrcrng  21963  0ringcring  33331  idlsrgcmnd  33593  primrootsunit1  42553  2zrngacmnd  48739