Theorem cmnpropd 19311

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 18325	. . 3 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
5	3	oveqrspc2v 7282	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
6	3	oveqrspc2v 7282	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑢) = (𝑣(+g‘𝐿)𝑢))
7	6	ancom2s 646	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑢) = (𝑣(+g‘𝐿)𝑢))
8	5, 7	eqeq12d 2754	. . . . 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 3121	. . . 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 3339	. . . . 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 3327	. . . 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 3339	. . . . 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 3327	. . . 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 308	. . 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 630	. 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 2738	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
17		eqid 2738	. . 3 ⊢ (+g‘𝐾) = (+g‘𝐾)
18	16, 17	iscmn 19309	. 2 ⊢ (𝐾 ∈ CMnd ↔ (𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢)))
19		eqid 2738	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
20		eqid 2738	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
21	19, 20	iscmn 19309	. 2 ⊢ (𝐿 ∈ CMnd ↔ (𝐿 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
22	15, 18, 21	3bitr4g 313	1 ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = 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-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

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-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  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-mgm 18241  df-sgrp 18290  df-mnd 18301  df-cmn 19303

This theorem is referenced by:  ablpropd  19312  crngpropd  19737  prdscrngd  19767  resvcmn  31444