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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 5061 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍)) | |
2 | breq2 5061 | . . . . 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 5061 | . . . 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 17613 | . . . 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 3623 | . 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) → (𝑋 ∨ 𝑌) ≤ 𝑍)) |
17 | 6, 7, 8, 9, 10, 11, 12 | lejoin1 17614 | . . . 4 ⊢ (𝜑 → 𝑋 ≤ (𝑋 ∨ 𝑌)) |
18 | 6, 8, 9, 10, 11, 12 | joincl 17608 | . . . . 5 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵) |
19 | 6, 7 | postr 17555 | . . . . 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 1367 | . . . 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 17615 | . . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌)) |
23 | 6, 7 | postr 17555 | . . . . 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 1367 | . . . 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 1531 ∈ wcel 2108 ∀wral 3136 〈cop 4565 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 lecple 16564 Posetcpo 17542 joincjn 17546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-poset 17548 df-lub 17576 df-join 17578 |
This theorem is referenced by: latjle12 17664 |
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