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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 6108 | . . 3 ⊢ (∅ ∘ ∅) = ∅ | |
2 | id 22 | . . . . . 6 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
3 | 2 | sqxpeqd 5581 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × ∅)) |
4 | 0xp 5643 | . . . . 5 ⊢ (∅ × ∅) = ∅ | |
5 | 3, 4 | syl6eq 2872 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅) |
6 | 5, 5 | coeq12d 5729 | . . 3 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (∅ ∘ ∅)) |
7 | 1, 6, 5 | 3eqtr4a 2882 | . 2 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) |
8 | xpco 6134 | . 2 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) | |
9 | 7, 8 | pm2.61ine 3100 | 1 ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∅c0 4290 × cxp 5547 ∘ ccom 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 |
This theorem is referenced by: utop2nei 22853 |
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