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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemsly | Structured version Visualization version GIF version |
Description: Lemma for dath 37032. 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 36920 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat) |
3 | dalemc.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 1, 3 | dalemseb 36938 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾)) |
5 | dalemc.j | . . . . . . 7 ⊢ ∨ = (join‘𝐾) | |
6 | 1, 5, 3 | dalemtjueb 36943 | . . . . . 6 ⊢ (𝜑 → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾)) |
7 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | dalemc.l | . . . . . . 7 ⊢ ≤ = (le‘𝐾) | |
9 | 7, 8, 5 | latlej1 17662 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾)) → 𝑆 ≤ (𝑆 ∨ (𝑇 ∨ 𝑈))) |
10 | 2, 4, 6, 9 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → 𝑆 ≤ (𝑆 ∨ (𝑇 ∨ 𝑈))) |
11 | 1 | dalemkehl 36919 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) |
12 | 1 | dalemsea 36925 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
13 | 1 | dalemtea 36926 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴) |
14 | 1 | dalemuea 36927 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
15 | 5, 3 | hlatjass 36666 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) = (𝑆 ∨ (𝑇 ∨ 𝑈))) |
16 | 11, 12, 13, 14, 15 | syl13anc 1369 | . . . . 5 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) = (𝑆 ∨ (𝑇 ∨ 𝑈))) |
17 | 10, 16 | breqtrrd 5058 | . . . 4 ⊢ (𝜑 → 𝑆 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) |
18 | dalemsly.z | . . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
19 | 17, 18 | breqtrrdi 5072 | . . 3 ⊢ (𝜑 → 𝑆 ≤ 𝑍) |
20 | 19 | adantr 484 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑍) |
21 | simpr 488 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑌 = 𝑍) | |
22 | 20, 21 | breqtrrd 5058 | 1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 joincjn 17546 Latclat 17647 Atomscatm 36559 HLchlt 36646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-proset 17530 df-poset 17548 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-lat 17648 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 |
This theorem is referenced by: dalem21 36990 dalem23 36992 dalem24 36993 dalem25 36994 |
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