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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 5574 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 × cxp 5547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-opab 5121 df-xp 5555 |
This theorem is referenced by: imainrect 6032 cnvrescnv 6046 cnvssrndm 6116 fpar 7805 canthwelem 10066 trclublem 14349 pjpm 20846 txbasval 22208 hausdiag 22247 ussval 22862 ex-xp 28209 hh0oi 29674 fcnvgreu 30412 sitgclg 31595 sitmcl 31604 ismgmOLD 35122 isdrngo1 35228 trrelsuperrel2dg 40009 |
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