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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlsupr3 | Structured version Visualization version GIF version |
Description: Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄, which can replace the superposition part of ishlat1 36503, (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) ), with the simpler ∃𝑧 ∈ 𝐴(𝑥 ∨ 𝑧) = (𝑦 ∨ 𝑧) as shown in ishlat3N 36505. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvlsupr2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cvlsupr2.l | ⊢ ≤ = (le‘𝐾) |
cvlsupr2.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
cvlsupr3 | ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ≠ 𝑄 → (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 3017 | . . . 4 ⊢ (𝑃 ≠ 𝑄 ↔ ¬ 𝑃 = 𝑄) | |
2 | 1 | imbi1i 352 | . . 3 ⊢ ((𝑃 ≠ 𝑄 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ↔ (¬ 𝑃 = 𝑄 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) |
3 | oveq1 7163 | . . . 4 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) | |
4 | 3 | biantrur 533 | . . 3 ⊢ ((¬ 𝑃 = 𝑄 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ↔ ((𝑃 = 𝑄 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (¬ 𝑃 = 𝑄 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))) |
5 | pm4.83 1021 | . . 3 ⊢ (((𝑃 = 𝑄 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (¬ 𝑃 = 𝑄 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) | |
6 | 2, 4, 5 | 3bitrri 300 | . 2 ⊢ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ≠ 𝑄 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) |
7 | cvlsupr2.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | cvlsupr2.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
9 | cvlsupr2.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
10 | 7, 8, 9 | cvlsupr2 36494 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))) |
11 | 10 | 3expia 1117 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃 ≠ 𝑄 → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))) |
12 | 11 | pm5.74d 275 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ↔ (𝑃 ≠ 𝑄 → (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))) |
13 | 6, 12 | syl5bb 285 | 1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ≠ 𝑄 → (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 lecple 16572 joincjn 17554 Atomscatm 36414 CvLatclc 36416 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-lat 17656 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 |
This theorem is referenced by: ishlat3N 36505 hlsupr2 36538 |
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