Theorem mndprop 18326

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 2739	. . 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 7268	. . . 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 18325	. 2 ⊢ (⊤ → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
8	7	mptru 1546	1 ⊢ (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539  ⊤wtru 1540   ∈ wcel 2108  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Mndcmnd 18300

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

This theorem is referenced by:  ring1  19756