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Theorem mndprop 12861
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 2188 . . 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 5901 . . . 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 12860 . 2 (⊤ → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
87mptru 1372 1 (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1363  wtru 1364  wcel 2158  cfv 5228  (class class class)co 5888  Basecbs 12475  +gcplusg 12550  Mndcmnd 12836
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-ov 5891  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-mgm 12793  df-sgrp 12826  df-mnd 12837
This theorem is referenced by: (None)
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