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Mirrors > Home > ILE Home > Th. List > xpeq12 | GIF version |
Description: Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) |
Ref | Expression |
---|---|
xpeq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1 4636 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | |
2 | xpeq2 4637 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 × 𝐶) = (𝐵 × 𝐷)) | |
3 | 1, 2 | sylan9eq 2230 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 × cxp 4620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-opab 4062 df-xp 4628 |
This theorem is referenced by: xpeq12i 4644 xpeq12d 4647 xpid11 4845 xp11m 5062 txtopon 13395 txbasval 13400 ismet 13477 isxmet 13478 |
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