Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xpeq1i | GIF version |
Description: Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
Ref | Expression |
---|---|
xpeq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
xpeq1i | ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | xpeq1 4612 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶) |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 × cxp 4596 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-opab 4038 df-xp 4604 |
This theorem is referenced by: iunxpconst 4658 xpindi 4733 resdmres 5089 mapsnconst 6651 mapsncnv 6652 xp2dju 7162 |
Copyright terms: Public domain | W3C validator |