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Mirrors > Home > MPE Home > Th. List > xpeq2i | Structured version Visualization version GIF version |
Description: Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.) |
Ref | Expression |
---|---|
xpeq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
xpeq2i | ⊢ (𝐶 × 𝐴) = (𝐶 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | xpeq2 5053 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐶 × 𝐴) = (𝐶 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-opab 4644 df-xp 5044 |
This theorem is referenced by: xpindir 5178 xpssres 5354 difxp1 5478 xpima 5495 xpexgALT 7052 curry1 7156 fparlem3 7166 fparlem4 7167 xp1en 7931 xp2cda 8885 xpcdaen 8888 pwcda1 8899 pwcdandom 9368 yonedalem3b 16742 yonedalem3 16743 pws1 18439 pwsmgp 18441 xkoinjcn 21300 imasdsf1olem 21988 df0op2 27995 ho01i 28071 nmop0h 28234 mbfmcst 29648 0rrv 29840 cvmlift2lem12 30550 zrdivrng 32922 |
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