Theorem cmnpropd 19732

Description: If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
ablpropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
ablpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
ablpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))

Assertion

Ref	Expression
cmnpropd	⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem cmnpropd

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablpropd.1	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		ablpropd.2	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		ablpropd.3	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
4	1, 2, 3	mndpropd 18696	. . 3 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
5	3	oveqrspc2v 7395	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
6	3	oveqrspc2v 7395	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑢) = (𝑣(+g‘𝐿)𝑢))
7	6	ancom2s 651	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑢) = (𝑣(+g‘𝐿)𝑢))
8	5, 7	eqeq12d 2753	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢) ↔ (𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
9	8	2ralbidva 3200	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
10	1	raleqdv 3298	. . . . 5 ⊢ (𝜑 → (∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢)))
11	1, 10	raleqbidv 3318	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢)))
12	2	raleqdv 3298	. . . . 5 ⊢ (𝜑 → (∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐿)(𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
13	2, 12	raleqbidv 3318	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
14	9, 11, 13	3bitr3d 309	. . 3 ⊢ (𝜑 → (∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
15	4, 14	anbi12d 633	. 2 ⊢ (𝜑 → ((𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢)) ↔ (𝐿 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢))))
16		eqid 2737	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
17		eqid 2737	. . 3 ⊢ (+g‘𝐾) = (+g‘𝐾)
18	16, 17	iscmn 19730	. 2 ⊢ (𝐾 ∈ CMnd ↔ (𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢)))
19		eqid 2737	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
20		eqid 2737	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
21	19, 20	iscmn 19730	. 2 ⊢ (𝐿 ∈ CMnd ↔ (𝐿 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
22	15, 18, 21	3bitr4g 314	1 ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  Mndcmnd 18671  CMndccmn 19721

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  ax-nul 5253  ax-pr 5379

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-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-cmn 19723

This theorem is referenced by:  ablpropd  19733  crngpropd  20236  prdscrngd  20269  resvcmn  33432