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 subproset. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
Ref | Expression |
---|---|
ordtNEW.b | ⊢ 𝐵 = (Base‘𝐾) |
ordtNEW.l | ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) |
Ref | Expression |
---|---|
prsss | ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtNEW.l | . . . . 5 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) | |
2 | 1 | ineq1i 4182 | . . . 4 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) = (((le‘𝐾) ∩ (𝐵 × 𝐵)) ∩ (𝐴 × 𝐴)) |
3 | inass 4193 | . . . 4 ⊢ (((le‘𝐾) ∩ (𝐵 × 𝐵)) ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴))) | |
4 | 2, 3 | eqtri 2841 | . . 3 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴))) |
5 | xpss12 5563 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝐴 × 𝐴) ⊆ (𝐵 × 𝐵)) | |
6 | 5 | anidms 567 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐴) ⊆ (𝐵 × 𝐵)) |
7 | sseqin2 4189 | . . . . 5 ⊢ ((𝐴 × 𝐴) ⊆ (𝐵 × 𝐵) ↔ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) | |
8 | 6, 7 | sylib 219 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) |
9 | 8 | ineq2d 4186 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((le‘𝐾) ∩ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴))) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
10 | 4, 9 | syl5eq 2865 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
11 | 10 | adantl 482 | 1 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 × cxp 5546 ‘cfv 6348 Basecbs 16471 lecple 16560 Proset cproset 17524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 df-opab 5120 df-xp 5554 |
This theorem is referenced by: prsssdm 31059 ordtrestNEW 31063 ordtrest2NEW 31065 |
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