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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 5706 | . 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 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: xpindir 5845 xpssres 6036 difxp1 6185 xpima 6202 xpexgALT 8006 curry1 8129 fparlem3 8139 fparlem4 8140 xp1en 9097 djuunxp 9961 dju1dif 10213 djuassen 10219 xpdjuen 10220 infdju1 10230 yonedalem3b 18324 yonedalem3 18325 pws1 20322 pwsmgp 20324 xkoinjcn 23695 imasdsf1olem 24383 df0op2 31771 ho01i 31847 nmop0h 32010 mbfmcst 34261 0rrv 34453 cvmlift2lem12 35319 zrdivrng 37960 |
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