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| Mirrors > Home > MPE Home > Th. List > xpeq12i | Structured version Visualization version GIF version | ||
| Description: Equality inference for Cartesian 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 5710 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-opab 5206 df-xp 5691 |
| This theorem is referenced by: imainrect 6201 cnvrescnv 6215 cnvssrndm 6291 fpar 8141 ttrclexg 9763 canthwelem 10690 trclublem 15034 pjpm 21728 txbasval 23614 hausdiag 23653 ussval 24268 ex-xp 30455 hh0oi 31922 fcnvgreu 32683 sitgclg 34344 sitmcl 34353 ismgmOLD 37857 isdrngo1 37963 trrelsuperrel2dg 43684 |
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