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Theorem mndprop 17632
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 (+g𝐾) = (+g𝐿)
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.
StepHypRef Expression
1 eqidd 2800 . . 3 (⊤ → (Base‘𝐾) = (Base‘𝐾))
2 mndprop.b . . . 4 (Base‘𝐾) = (Base‘𝐿)
32a1i 11 . . 3 (⊤ → (Base‘𝐾) = (Base‘𝐿))
4 mndprop.p . . . . 5 (+g𝐾) = (+g𝐿)
54oveqi 6891 . . . 4 (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦)
65a1i 11 . . 3 ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
71, 3, 6mndpropd 17631 . 2 (⊤ → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
87mptru 1661 1 (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wtru 1654  wcel 2157  cfv 6101  (class class class)co 6878  Basecbs 16184  +gcplusg 16267  Mndcmnd 17609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983  ax-pow 5035
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881  df-mgm 17557  df-sgrp 17599  df-mnd 17610
This theorem is referenced by:  ring1  18918
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