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Mirrors > Home > MPE Home > Th. List > joinle | Structured version Visualization version GIF version |
Description: A join 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.) |
Ref | Expression |
---|---|
joinle.b | ⊢ 𝐵 = (Base‘𝐾) |
joinle.l | ⊢ ≤ = (le‘𝐾) |
joinle.j | ⊢ ∨ = (join‘𝐾) |
joinle.k | ⊢ (𝜑 → 𝐾 ∈ Poset) |
joinle.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
joinle.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
joinle.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
joinle.e | ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
Ref | Expression |
---|---|
joinle | ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) ↔ (𝑋 ∨ 𝑌) ≤ 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5148 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍)) | |
2 | breq2 5148 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑌 ≤ 𝑧 ↔ 𝑌 ≤ 𝑍)) | |
3 | 1, 2 | anbi12d 632 | . . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) ↔ (𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍))) |
4 | breq2 5148 | . . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋 ∨ 𝑌) ≤ 𝑧 ↔ (𝑋 ∨ 𝑌) ≤ 𝑍)) | |
5 | 3, 4 | imbi12d 345 | . . 3 ⊢ (𝑧 = 𝑍 → (((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) → (𝑋 ∨ 𝑌) ≤ 𝑍))) |
6 | joinle.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
7 | joinle.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
8 | joinle.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
9 | joinle.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
10 | joinle.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | joinle.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
12 | joinle.e | . . . . 5 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) | |
13 | 6, 7, 8, 9, 10, 11, 12 | joinlem 18323 | . . . 4 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧))) |
14 | 13 | simprd 497 | . . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)) |
15 | joinle.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
16 | 5, 14, 15 | rspcdva 3612 | . 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) → (𝑋 ∨ 𝑌) ≤ 𝑍)) |
17 | 6, 7, 8, 9, 10, 11, 12 | lejoin1 18324 | . . . 4 ⊢ (𝜑 → 𝑋 ≤ (𝑋 ∨ 𝑌)) |
18 | 6, 8, 9, 10, 11, 12 | joincl 18318 | . . . . 5 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵) |
19 | 6, 7 | postr 18260 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
20 | 9, 10, 18, 15, 19 | syl13anc 1373 | . . . 4 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
21 | 17, 20 | mpand 694 | . . 3 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → 𝑋 ≤ 𝑍)) |
22 | 6, 7, 8, 9, 10, 11, 12 | lejoin2 18325 | . . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌)) |
23 | 6, 7 | postr 18260 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑌 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑌 ≤ 𝑍)) |
24 | 9, 11, 18, 15, 23 | syl13anc 1373 | . . . 4 ⊢ (𝜑 → ((𝑌 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑌 ≤ 𝑍)) |
25 | 22, 24 | mpand 694 | . . 3 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → 𝑌 ≤ 𝑍)) |
26 | 21, 25 | jcad 514 | . 2 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → (𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍))) |
27 | 16, 26 | impbid 211 | 1 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) ↔ (𝑋 ∨ 𝑌) ≤ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 〈cop 4630 class class class wbr 5144 dom cdm 5672 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 lecple 17191 Posetcpo 18247 joincjn 18251 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-poset 18253 df-lub 18286 df-join 18288 |
This theorem is referenced by: latjle12 18390 |
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