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Mirrors > Home > ILE Home > Th. List > xpeq12d | GIF version |
Description: Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.) |
Ref | Expression |
---|---|
xpeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
xpeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
xpeq12d | ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | xpeq12d.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | xpeq12 4470 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | |
4 | 1, 2, 3 | syl2anc 404 | 1 ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 × cxp 4449 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-11 1443 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-opab 3906 df-xp 4457 |
This theorem is referenced by: opeliunxp 4506 mpt2mptsx 5981 dmmpt2ssx 5983 fmpt2x 5984 disjxp1 6015 erssxp 6329 fsum2dlemstep 10882 fisumcom2 10886 |
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