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Theorem grpprop 18110
 Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)
Hypotheses
Ref Expression
grpprop.b (Base‘𝐾) = (Base‘𝐿)
grpprop.p (+g𝐾) = (+g𝐿)
Assertion
Ref Expression
grpprop (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Proof of Theorem grpprop
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2823 . . 3 (⊤ → (Base‘𝐾) = (Base‘𝐾))
2 grpprop.b . . . 4 (Base‘𝐾) = (Base‘𝐿)
32a1i 11 . . 3 (⊤ → (Base‘𝐾) = (Base‘𝐿))
4 grpprop.p . . . . 5 (+g𝐾) = (+g𝐿)
54oveqi 7153 . . . 4 (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦)
65a1i 11 . . 3 ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
71, 3, 6grppropd 18109 . 2 (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
87mptru 1545 1 (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ⊤wtru 1539   ∈ wcel 2114  ‘cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  Grpcgrp 18094 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097 This theorem is referenced by:  grppropstr  18111  grpss  18112  opprring  19375  opprsubg  19380  rmodislmod  19693  lmod1  44844
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