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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 5078 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑋 ↔ 𝑍 ≤ 𝑋)) | |
| 2 | breq1 5078 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑌 ↔ 𝑍 ≤ 𝑌)) | |
| 3 | 1, 2 | anbi12d 639 | . . . 4 ⊢ (𝑧 = 𝑍 → ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) ↔ (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌))) |
| 4 | breq1 5078 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ (𝑋 ∧ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌))) | |
| 5 | 3, 4 | imbi12d 346 | . . 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 18356 | . . . 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 3563 | . 2 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) → 𝑍 ≤ (𝑋 ∧ 𝑌))) |
| 17 | 6, 7, 8, 9, 10, 11, 12 | lemeet1 18357 | . . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋) |
| 18 | 6, 8, 9, 10, 11, 12 | meetcl 18351 | . . . . 5 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ∈ 𝐵) |
| 19 | 6, 7 | postr 18281 | . . . . 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 1381 | . . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑋) → 𝑍 ≤ 𝑋)) |
| 21 | 17, 20 | mpan2d 701 | . . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑋)) |
| 22 | 6, 7, 8, 9, 10, 11, 12 | lemeet2 18358 | . . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑌) |
| 23 | 6, 7 | postr 18281 | . . . . 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 1381 | . . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) → 𝑍 ≤ 𝑌)) |
| 25 | 22, 24 | mpan2d 701 | . . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑌)) |
| 26 | 21, 25 | jcad 518 | . 2 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌))) |
| 27 | 16, 26 | impbid 214 | 1 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ∀wral 3055 〈cop 4564 class class class wbr 5075 dom cdm 5621 ‘cfv 6489 (class class class)co 7360 Basecbs 17174 lecple 17222 Posetcpo 18268 meetcmee 18273 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 |
| This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-poset 18274 df-glb 18306 df-meet 18308 |
| This theorem is referenced by: latlem12 18427 |
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