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Theorem mndprop 18691
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 2727 . . 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 7417 . . . 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 18690 . 2 (⊤ → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
87mptru 1540 1 (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wtru 1534  wcel 2098  cfv 6536  (class class class)co 7404  Basecbs 17151  +gcplusg 17204  Mndcmnd 18665
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 2163  ax-ext 2697  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-mgm 18571  df-sgrp 18650  df-mnd 18666
This theorem is referenced by:  ring1  20207
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