![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvrat42 | Structured version Visualization version GIF version |
Description: Commuted version of cvrat4 37906. (Contributed by NM, 28-Jan-2012.) |
Ref | Expression |
---|---|
cvrat4.b | ⊢ 𝐵 = (Base‘𝐾) |
cvrat4.l | ⊢ ≤ = (le‘𝐾) |
cvrat4.j | ⊢ ∨ = (join‘𝐾) |
cvrat4.z | ⊢ 0 = (0.‘𝐾) |
cvrat4.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
cvrat42 | ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑟 ∨ 𝑄)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvrat4.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cvrat4.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | cvrat4.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | cvrat4.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
5 | cvrat4.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 1, 2, 3, 4, 5 | cvrat4 37906 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))) |
7 | hllat 37825 | . . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
8 | 7 | ad2antrr 724 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝐾 ∈ Lat) |
9 | simplr3 1217 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑄 ∈ 𝐴) | |
10 | 1, 5 | atbase 37751 | . . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑄 ∈ 𝐵) |
12 | 1, 5 | atbase 37751 | . . . . . . 7 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ 𝐵) |
13 | 12 | adantl 482 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐵) |
14 | 1, 3 | latjcom 18336 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵) → (𝑄 ∨ 𝑟) = (𝑟 ∨ 𝑄)) |
15 | 8, 11, 13, 14 | syl3anc 1371 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → (𝑄 ∨ 𝑟) = (𝑟 ∨ 𝑄)) |
16 | 15 | breq2d 5117 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑟 ∨ 𝑄))) |
17 | 16 | anbi2d 629 | . . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → ((𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)) ↔ (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑟 ∨ 𝑄)))) |
18 | 17 | rexbidva 3173 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)) ↔ ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑟 ∨ 𝑄)))) |
19 | 6, 18 | sylibd 238 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑟 ∨ 𝑄)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 lecple 17140 joincjn 18200 0.cp0 18312 Latclat 18320 Atomscatm 37725 HLchlt 37812 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-lat 18321 df-clat 18388 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 |
This theorem is referenced by: pmapjat1 38316 djhcvat42 39878 |
Copyright terms: Public domain | W3C validator |