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Mirrors > Home > MPE Home > Th. List > xpcoid | Structured version Visualization version GIF version |
Description: Composition of two square Cartesian products. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
Ref | Expression |
---|---|
xpcoid | ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | co01 5650 | . . 3 ⊢ (∅ ∘ ∅) = ∅ | |
2 | id 22 | . . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
3 | 2 | sqxpeqd 5141 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × ∅)) |
4 | 0xp 5199 | . . . . 5 ⊢ (∅ × ∅) = ∅ | |
5 | 3, 4 | syl6eq 2672 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅) |
6 | 5, 5 | coeq12d 5286 | . . 3 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (∅ ∘ ∅)) |
7 | 1, 6, 5 | 3eqtr4a 2682 | . 2 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) |
8 | xpco 5675 | . 2 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) | |
9 | 7, 8 | pm2.61ine 2877 | 1 ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∅c0 3915 × cxp 5112 ∘ ccom 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 |
This theorem is referenced by: utop2nei 22054 |
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