Theorem mndprop 18728

Description: If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)

Hypotheses

Ref	Expression
mndprop.b	⊢ (Base‘𝐾) = (Base‘𝐿)
mndprop.p	⊢ (+g‘𝐾) = (+g‘𝐿)

Assertion

Ref	Expression
mndprop	⊢ (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)

Proof of Theorem mndprop

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2726	. . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐾))
2		mndprop.b	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐿)
3	2	a1i 11	. . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐿))
4		mndprop.p	. . . . 5 ⊢ (+g‘𝐾) = (+g‘𝐿)
5	4	oveqi 7432	. . . 4 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)
6	5	a1i 11	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
7	1, 3, 6	mndpropd 18727	. 2 ⊢ (⊤ → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
8	7	mptru 1540	1 ⊢ (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533  ⊤wtru 1534   ∈ wcel 2098  ‘cfv 6549  (class class class)co 7419  Basecbs 17188  +_gcplusg 17241  Mndcmnd 18702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-mgm 18608  df-sgrp 18687  df-mnd 18703

This theorem is referenced by:  ring1  20263