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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemsly | Structured version Visualization version GIF version |
Description: Lemma for dath 36876. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
Ref | Expression |
---|---|
dalema.ph | ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈))))) |
dalemc.l | ⊢ ≤ = (le‘𝐾) |
dalemc.j | ⊢ ∨ = (join‘𝐾) |
dalemc.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dalemsly.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
Ref | Expression |
---|---|
dalemsly | ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalema.ph | . . . . . . 7 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈))))) | |
2 | 1 | dalemkelat 36764 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat) |
3 | dalemc.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 1, 3 | dalemseb 36782 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾)) |
5 | dalemc.j | . . . . . . 7 ⊢ ∨ = (join‘𝐾) | |
6 | 1, 5, 3 | dalemtjueb 36787 | . . . . . 6 ⊢ (𝜑 → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾)) |
7 | eqid 2824 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | dalemc.l | . . . . . . 7 ⊢ ≤ = (le‘𝐾) | |
9 | 7, 8, 5 | latlej1 17673 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾)) → 𝑆 ≤ (𝑆 ∨ (𝑇 ∨ 𝑈))) |
10 | 2, 4, 6, 9 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → 𝑆 ≤ (𝑆 ∨ (𝑇 ∨ 𝑈))) |
11 | 1 | dalemkehl 36763 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) |
12 | 1 | dalemsea 36769 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
13 | 1 | dalemtea 36770 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴) |
14 | 1 | dalemuea 36771 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
15 | 5, 3 | hlatjass 36510 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) = (𝑆 ∨ (𝑇 ∨ 𝑈))) |
16 | 11, 12, 13, 14, 15 | syl13anc 1368 | . . . . 5 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) = (𝑆 ∨ (𝑇 ∨ 𝑈))) |
17 | 10, 16 | breqtrrd 5097 | . . . 4 ⊢ (𝜑 → 𝑆 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) |
18 | dalemsly.z | . . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
19 | 17, 18 | breqtrrdi 5111 | . . 3 ⊢ (𝜑 → 𝑆 ≤ 𝑍) |
20 | 19 | adantr 483 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑍) |
21 | simpr 487 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑌 = 𝑍) | |
22 | 20, 21 | breqtrrd 5097 | 1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 lecple 16575 joincjn 17557 Latclat 17658 Atomscatm 36403 HLchlt 36490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-proset 17541 df-poset 17559 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-lat 17659 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 |
This theorem is referenced by: dalem21 36834 dalem23 36836 dalem24 36837 dalem25 36838 |
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