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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 6081 | . . 3 ⊢ (∅ ∘ ∅) = ∅ | |
2 | id 22 | . . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
3 | 2 | sqxpeqd 5551 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × ∅)) |
4 | 0xp 5613 | . . . . 5 ⊢ (∅ × ∅) = ∅ | |
5 | 3, 4 | eqtrdi 2849 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅) |
6 | 5, 5 | coeq12d 5699 | . . 3 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (∅ ∘ ∅)) |
7 | 1, 6, 5 | 3eqtr4a 2859 | . 2 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) |
8 | xpco 6108 | . 2 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) | |
9 | 7, 8 | pm2.61ine 3070 | 1 ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∅c0 4243 × cxp 5517 ∘ ccom 5523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 |
This theorem is referenced by: utop2nei 22856 |
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