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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 5103 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑋 ↔ 𝑍 ≤ 𝑋)) | |
| 2 | breq1 5103 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑌 ↔ 𝑍 ≤ 𝑌)) | |
| 3 | 1, 2 | anbi12d 641 | . . . 4 ⊢ (𝑧 = 𝑍 → ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) ↔ (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌))) |
| 4 | breq1 5103 | . . . 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 18427 | . . . 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 3582 | . 2 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) → 𝑍 ≤ (𝑋 ∧ 𝑌))) |
| 17 | 6, 7, 8, 9, 10, 11, 12 | lemeet1 18428 | . . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋) |
| 18 | 6, 8, 9, 10, 11, 12 | meetcl 18422 | . . . . 5 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ∈ 𝐵) |
| 19 | 6, 7 | postr 18352 | . . . . 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 1391 | . . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑋) → 𝑍 ≤ 𝑋)) |
| 21 | 17, 20 | mpan2d 704 | . . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑋)) |
| 22 | 6, 7, 8, 9, 10, 11, 12 | lemeet2 18429 | . . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑌) |
| 23 | 6, 7 | postr 18352 | . . . . 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 1391 | . . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) → 𝑍 ≤ 𝑌)) |
| 25 | 22, 24 | mpan2d 704 | . . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑌)) |
| 26 | 21, 25 | jcad 520 | . 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 399 = wceq 1560 ∈ wcel 2142 ∀wral 3076 〈cop 4588 class class class wbr 5100 dom cdm 5647 ‘cfv 6521 (class class class)co 7396 Basecbs 17245 lecple 17293 Posetcpo 18339 meetcmee 18344 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-poset 18345 df-glb 18377 df-meet 18379 |
| This theorem is referenced by: latlem12 18498 |
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