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Mirrors > Home > MPE Home > Th. List > meetle | Structured version Visualization version GIF version |
Description: A meet is less than or equal to a third value iff each argument is less than or equal to the third value. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.) |
Ref | Expression |
---|---|
meetle.b | ⊢ 𝐵 = (Base‘𝐾) |
meetle.l | ⊢ ≤ = (le‘𝐾) |
meetle.m | ⊢ ∧ = (meet‘𝐾) |
meetle.k | ⊢ (𝜑 → 𝐾 ∈ Poset) |
meetle.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
meetle.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
meetle.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
meetle.e | ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
Ref | Expression |
---|---|
meetle | ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5033 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑋 ↔ 𝑍 ≤ 𝑋)) | |
2 | breq1 5033 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑌 ↔ 𝑍 ≤ 𝑌)) | |
3 | 1, 2 | anbi12d 633 | . . . 4 ⊢ (𝑧 = 𝑍 → ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) ↔ (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌))) |
4 | breq1 5033 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ (𝑋 ∧ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌))) | |
5 | 3, 4 | imbi12d 348 | . . 3 ⊢ (𝑧 = 𝑍 → (((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌)) ↔ ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) → 𝑍 ≤ (𝑋 ∧ 𝑌)))) |
6 | meetle.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
7 | meetle.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
8 | meetle.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
9 | meetle.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
10 | meetle.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | meetle.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
12 | meetle.e | . . . . 5 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) | |
13 | 6, 7, 8, 9, 10, 11, 12 | meetlem 17627 | . . . 4 ⊢ (𝜑 → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌)))) |
14 | 13 | simprd 499 | . . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))) |
15 | meetle.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
16 | 5, 14, 15 | rspcdva 3573 | . 2 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) → 𝑍 ≤ (𝑋 ∧ 𝑌))) |
17 | 6, 7, 8, 9, 10, 11, 12 | lemeet1 17628 | . . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋) |
18 | 6, 8, 9, 10, 11, 12 | meetcl 17622 | . . . . 5 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ∈ 𝐵) |
19 | 6, 7 | postr 17555 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑍 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑋) → 𝑍 ≤ 𝑋)) |
20 | 9, 15, 18, 10, 19 | syl13anc 1369 | . . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑋) → 𝑍 ≤ 𝑋)) |
21 | 17, 20 | mpan2d 693 | . . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑋)) |
22 | 6, 7, 8, 9, 10, 11, 12 | lemeet2 17629 | . . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑌) |
23 | 6, 7 | postr 17555 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑍 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) → 𝑍 ≤ 𝑌)) |
24 | 9, 15, 18, 11, 23 | syl13anc 1369 | . . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) → 𝑍 ≤ 𝑌)) |
25 | 22, 24 | mpan2d 693 | . . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑌)) |
26 | 21, 25 | jcad 516 | . 2 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌))) |
27 | 16, 26 | impbid 215 | 1 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 〈cop 4531 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 Posetcpo 17542 meetcmee 17547 |
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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
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-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-poset 17548 df-glb 17577 df-meet 17579 |
This theorem is referenced by: latlem12 17680 |
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