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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 5707 | . 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 1534 × cxp 5680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-opab 5216 df-xp 5688 |
This theorem is referenced by: imainrect 6192 cnvrescnv 6206 cnvssrndm 6282 fpar 8130 ttrclexg 9766 canthwelem 10693 trclublem 15000 pjpm 21706 txbasval 23601 hausdiag 23640 ussval 24255 ex-xp 30369 hh0oi 31836 fcnvgreu 32590 sitgclg 34176 sitmcl 34185 ismgmOLD 37551 isdrngo1 37657 trrelsuperrel2dg 43338 |
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