Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpeq1i GIF version

Theorem xpeq1i 4568
 Description: Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
Hypothesis
Ref Expression
xpeq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
xpeq1i (𝐴 × 𝐶) = (𝐵 × 𝐶)

Proof of Theorem xpeq1i
StepHypRef Expression
1 xpeq1i.1 . 2 𝐴 = 𝐵
2 xpeq1 4562 . 2 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
31, 2ax-mp 5 1 (𝐴 × 𝐶) = (𝐵 × 𝐶)
 Colors of variables: wff set class Syntax hints:   = wceq 1332   × cxp 4546 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-opab 3999  df-xp 4554 This theorem is referenced by:  iunxpconst  4608  xpindi  4683  resdmres  5039  mapsnconst  6597  mapsncnv  6598  xp2dju  7091
 Copyright terms: Public domain W3C validator