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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 | joinle.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
2 | joinle.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | joinle.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
4 | joinle.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
5 | joinle.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
6 | joinle.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | joinle.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | joinle.e | . . . . 5 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) | |
9 | 2, 3, 4, 5, 6, 7, 8 | joinlem 17011 | . . . 4 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧))) |
10 | 9 | simprd 479 | . . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)) |
11 | breq2 4657 | . . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍)) | |
12 | breq2 4657 | . . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑌 ≤ 𝑧 ↔ 𝑌 ≤ 𝑍)) | |
13 | 11, 12 | anbi12d 747 | . . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) ↔ (𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍))) |
14 | breq2 4657 | . . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑋 ∨ 𝑌) ≤ 𝑧 ↔ (𝑋 ∨ 𝑌) ≤ 𝑍)) | |
15 | 13, 14 | imbi12d 334 | . . . 4 ⊢ (𝑧 = 𝑍 → (((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) → (𝑋 ∨ 𝑌) ≤ 𝑍))) |
16 | 15 | rspcva 3307 | . . 3 ⊢ ((𝑍 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)) → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) → (𝑋 ∨ 𝑌) ≤ 𝑍)) |
17 | 1, 10, 16 | syl2anc 693 | . 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) → (𝑋 ∨ 𝑌) ≤ 𝑍)) |
18 | 2, 3, 4, 5, 6, 7, 8 | lejoin1 17012 | . . . 4 ⊢ (𝜑 → 𝑋 ≤ (𝑋 ∨ 𝑌)) |
19 | 2, 4, 5, 6, 7, 8 | joincl 17006 | . . . . 5 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵) |
20 | 2, 3 | postr 16953 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
21 | 5, 6, 19, 1, 20 | syl13anc 1328 | . . . 4 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
22 | 18, 21 | mpand 711 | . . 3 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → 𝑋 ≤ 𝑍)) |
23 | 2, 3, 4, 5, 6, 7, 8 | lejoin2 17013 | . . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌)) |
24 | 2, 3 | postr 16953 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑌 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑌 ≤ 𝑍)) |
25 | 5, 7, 19, 1, 24 | syl13anc 1328 | . . . 4 ⊢ (𝜑 → ((𝑌 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑌 ≤ 𝑍)) |
26 | 23, 25 | mpand 711 | . . 3 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → 𝑌 ≤ 𝑍)) |
27 | 22, 26 | jcad 555 | . 2 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → (𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍))) |
28 | 17, 27 | impbid 202 | 1 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) ↔ (𝑋 ∨ 𝑌) ≤ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 〈cop 4183 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 Posetcpo 16940 joincjn 16944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-poset 16946 df-lub 16974 df-join 16976 |
This theorem is referenced by: latjle12 17062 |
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