![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xpcoid | Structured version Visualization version GIF version |
Description: Composition of two Cartesian squares. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
Ref | Expression |
---|---|
xpcoid | ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | co01 6282 | . . 3 ⊢ (∅ ∘ ∅) = ∅ | |
2 | id 22 | . . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
3 | 2 | sqxpeqd 5720 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × ∅)) |
4 | 0xp 5786 | . . . . 5 ⊢ (∅ × ∅) = ∅ | |
5 | 3, 4 | eqtrdi 2790 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅) |
6 | 5, 5 | coeq12d 5877 | . . 3 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (∅ ∘ ∅)) |
7 | 1, 6, 5 | 3eqtr4a 2800 | . 2 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) |
8 | xpco 6310 | . 2 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) | |
9 | 7, 8 | pm2.61ine 3022 | 1 ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∅c0 4338 × cxp 5686 ∘ ccom 5692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 |
This theorem is referenced by: utop2nei 24274 |
Copyright terms: Public domain | W3C validator |