Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prsss | Structured version Visualization version GIF version |
Description: Relation of a subpreset. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
Ref | Expression |
---|---|
ordtNEW.b | ⊢ 𝐵 = (Base‘𝐾) |
ordtNEW.l | ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) |
Ref | Expression |
---|---|
prsss | ⊢ ((𝐾 ∈ Preset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtNEW.l | . . . . 5 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) | |
2 | 1 | ineq1i 3772 | . . . 4 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) = (((le‘𝐾) ∩ (𝐵 × 𝐵)) ∩ (𝐴 × 𝐴)) |
3 | inass 3785 | . . . 4 ⊢ (((le‘𝐾) ∩ (𝐵 × 𝐵)) ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴))) | |
4 | 2, 3 | eqtri 2632 | . . 3 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴))) |
5 | xpss12 5148 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝐴 × 𝐴) ⊆ (𝐵 × 𝐵)) | |
6 | 5 | anidms 675 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐴) ⊆ (𝐵 × 𝐵)) |
7 | sseqin2 3779 | . . . . 5 ⊢ ((𝐴 × 𝐴) ⊆ (𝐵 × 𝐵) ↔ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) | |
8 | 6, 7 | sylib 207 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) |
9 | 8 | ineq2d 3776 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((le‘𝐾) ∩ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴))) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
10 | 4, 9 | syl5eq 2656 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
11 | 10 | adantl 481 | 1 ⊢ ((𝐾 ∈ Preset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 × cxp 5036 ‘cfv 5804 Basecbs 15695 lecple 15775 Preset cpreset 16749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-opab 4644 df-xp 5044 |
This theorem is referenced by: prsssdm 29291 ordtrestNEW 29295 ordtrest2NEW 29297 |
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