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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 5367 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | |
4 | 1, 2, 3 | mp2an 683 | 1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 × cxp 5340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-opab 4936 df-xp 5348 |
This theorem is referenced by: imainrect 5816 cnvssrndm 5898 idssxpOLD 6242 fpar 7545 canthwelem 9787 trclublem 14113 pjpm 20415 txbasval 21780 hausdiag 21819 ussval 22433 ex-xp 27840 hh0oi 29306 fcnvgreu 30009 sitgclg 30938 sitmcl 30947 ismgmOLD 34184 isdrngo1 34290 rtrclex 38758 rtrclexi 38762 trrelsuperrel2dg 38797 |
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