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Theorem mndprop 12868
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 2190 . . 3 (⊤ → (Base‘𝐾) = (Base‘𝐾))
2 mndprop.b . . . 4 (Base‘𝐾) = (Base‘𝐿)
32a1i 9 . . 3 (⊤ → (Base‘𝐾) = (Base‘𝐿))
4 mndprop.p . . . . 5 (+g𝐾) = (+g𝐿)
54oveqi 5904 . . . 4 (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦)
65a1i 9 . . 3 ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
71, 3, 6mndpropd 12867 . 2 (⊤ → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
87mptru 1373 1 (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1364  wtru 1365  wcel 2160  cfv 5231  (class class class)co 5891  Basecbs 12480  +gcplusg 12555  Mndcmnd 12843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7920  ax-resscn 7921  ax-1re 7923  ax-addrcl 7926
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239  df-ov 5894  df-inn 8938  df-2 8996  df-ndx 12483  df-slot 12484  df-base 12486  df-plusg 12568  df-mgm 12798  df-sgrp 12831  df-mnd 12844
This theorem is referenced by: (None)
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