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

Theorem xpeq1i 4308
 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 4302 . 2 (A = B → (A × 𝐶) = (B × 𝐶))
31, 2ax-mp 7 1 (A × 𝐶) = (B × 𝐶)
 Colors of variables: wff set class Syntax hints:   = wceq 1242   × cxp 4286 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-opab 3810  df-xp 4294 This theorem is referenced by:  iunxpconst  4343  xpindi  4414  resdmres  4755
 Copyright terms: Public domain W3C validator