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Mirrors > Home > ILE Home > Th. List > xpeq12i | GIF version |
Description: Equality inference for cross product. (Contributed by FL, 31-Aug-2009.) |
Ref | Expression |
---|---|
xpeq12i.1 | ⊢ 𝐴 = 𝐵 |
xpeq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
xpeq12i | ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq12i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | xpeq12i.2 | . 2 ⊢ 𝐶 = 𝐷 | |
3 | xpeq12 4553 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | |
4 | 1, 2, 3 | mp2an 422 | 1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 × cxp 4532 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-opab 3985 df-xp 4540 |
This theorem is referenced by: xpssres 4849 imainrect 4979 cnvssrndm 5055 txbasval 12425 |
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