Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5072 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍)) | |
2 | breq2 5072 | . . . . 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 5072 | . . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋 ∨ 𝑌) ≤ 𝑧 ↔ (𝑋 ∨ 𝑌) ≤ 𝑍)) | |
5 | 3, 4 | imbi12d 347 | . . 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 17623 | . . . 4 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧))) |
14 | 13 | simprd 498 | . . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)) |
15 | joinle.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
16 | 5, 14, 15 | rspcdva 3627 | . 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) → (𝑋 ∨ 𝑌) ≤ 𝑍)) |
17 | 6, 7, 8, 9, 10, 11, 12 | lejoin1 17624 | . . . 4 ⊢ (𝜑 → 𝑋 ≤ (𝑋 ∨ 𝑌)) |
18 | 6, 8, 9, 10, 11, 12 | joincl 17618 | . . . . 5 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵) |
19 | 6, 7 | postr 17565 | . . . . 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 1368 | . . . 4 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
21 | 17, 20 | mpand 693 | . . 3 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → 𝑋 ≤ 𝑍)) |
22 | 6, 7, 8, 9, 10, 11, 12 | lejoin2 17625 | . . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌)) |
23 | 6, 7 | postr 17565 | . . . . 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 1368 | . . . 4 ⊢ (𝜑 → ((𝑌 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑌 ≤ 𝑍)) |
25 | 22, 24 | mpand 693 | . . 3 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → 𝑌 ≤ 𝑍)) |
26 | 21, 25 | jcad 515 | . 2 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → (𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍))) |
27 | 16, 26 | impbid 214 | 1 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) ↔ (𝑋 ∨ 𝑌) ≤ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 〈cop 4575 class class class wbr 5068 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 lecple 16574 Posetcpo 17552 joincjn 17556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-poset 17558 df-lub 17586 df-join 17588 |
This theorem is referenced by: latjle12 17674 |
Copyright terms: Public domain | W3C validator |