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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 5150 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑋 ↔ 𝑍 ≤ 𝑋)) | |
2 | breq1 5150 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑌 ↔ 𝑍 ≤ 𝑌)) | |
3 | 1, 2 | anbi12d 632 | . . . 4 ⊢ (𝑧 = 𝑍 → ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) ↔ (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌))) |
4 | breq1 5150 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ (𝑋 ∧ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌))) | |
5 | 3, 4 | imbi12d 345 | . . 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 18346 | . . . 4 ⊢ (𝜑 → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌)))) |
14 | 13 | simprd 497 | . . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))) |
15 | meetle.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
16 | 5, 14, 15 | rspcdva 3613 | . 2 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) → 𝑍 ≤ (𝑋 ∧ 𝑌))) |
17 | 6, 7, 8, 9, 10, 11, 12 | lemeet1 18347 | . . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋) |
18 | 6, 8, 9, 10, 11, 12 | meetcl 18341 | . . . . 5 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ∈ 𝐵) |
19 | 6, 7 | postr 18269 | . . . . 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 1373 | . . . 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 18348 | . . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑌) |
23 | 6, 7 | postr 18269 | . . . . 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 1373 | . . . 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 514 | . 2 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌))) |
27 | 16, 26 | impbid 211 | 1 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 〈cop 4633 class class class wbr 5147 dom cdm 5675 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 lecple 17200 Posetcpo 18256 meetcmee 18261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-poset 18262 df-glb 18296 df-meet 18298 |
This theorem is referenced by: latlem12 18415 |
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