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Theorem cntri 17809
Description: Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
cntri.b 𝐵 = (Base‘𝑀)
cntri.p + = (+g𝑀)
cntri.z 𝑍 = (Cntr‘𝑀)
Assertion
Ref Expression
cntri ((𝑋𝑍𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Proof of Theorem cntri
StepHypRef Expression
1 cntri.z . . . 4 𝑍 = (Cntr‘𝑀)
2 cntri.b . . . . 5 𝐵 = (Base‘𝑀)
3 eqid 2651 . . . . 5 (Cntz‘𝑀) = (Cntz‘𝑀)
42, 3cntrval 17798 . . . 4 ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀)
51, 4eqtr4i 2676 . . 3 𝑍 = ((Cntz‘𝑀)‘𝐵)
65eleq2i 2722 . 2 (𝑋𝑍𝑋 ∈ ((Cntz‘𝑀)‘𝐵))
7 cntri.p . . 3 + = (+g𝑀)
87, 3cntzi 17808 . 2 ((𝑋 ∈ ((Cntz‘𝑀)‘𝐵) ∧ 𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
96, 8sylanb 488 1 ((𝑋𝑍𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Cntzccntz 17794  Cntrccntr 17795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-cntz 17796  df-cntr 17797
This theorem is referenced by: (None)
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