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Mirrors > Home > MPE Home > Th. List > Mathboxes > olm01 | Structured version Visualization version GIF version |
Description: Meet with lattice zero is zero. (chm0 29268 analog.) (Contributed by NM, 8-Nov-2011.) |
Ref | Expression |
---|---|
olm0.b | ⊢ 𝐵 = (Base‘𝐾) |
olm0.m | ⊢ ∧ = (meet‘𝐾) |
olm0.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
olm01 | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olm0.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2821 | . 2 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | ollat 36364 | . . 3 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
4 | 3 | adantr 483 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) |
5 | simpr 487 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
6 | olop 36365 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) |
8 | olm0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
9 | 1, 8 | op0cl 36335 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
10 | 7, 9 | syl 17 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
11 | olm0.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
12 | 1, 11 | latmcl 17662 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∧ 0 ) ∈ 𝐵) |
13 | 4, 5, 10, 12 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 ) ∈ 𝐵) |
14 | 1, 2, 11 | latmle2 17687 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∧ 0 )(le‘𝐾) 0 ) |
15 | 4, 5, 10, 14 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 )(le‘𝐾) 0 ) |
16 | 1, 2, 8 | op0le 36337 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
17 | 6, 16 | sylan 582 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
18 | 1, 2 | latref 17663 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 0 ∈ 𝐵) → 0 (le‘𝐾) 0 ) |
19 | 4, 10, 18 | syl2anc 586 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾) 0 ) |
20 | 1, 2, 11 | latlem12 17688 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵)) → (( 0 (le‘𝐾)𝑋 ∧ 0 (le‘𝐾) 0 ) ↔ 0 (le‘𝐾)(𝑋 ∧ 0 ))) |
21 | 4, 10, 5, 10, 20 | syl13anc 1368 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (( 0 (le‘𝐾)𝑋 ∧ 0 (le‘𝐾) 0 ) ↔ 0 (le‘𝐾)(𝑋 ∧ 0 ))) |
22 | 17, 19, 21 | mpbi2and 710 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)(𝑋 ∧ 0 )) |
23 | 1, 2, 4, 13, 10, 15, 22 | latasymd 17667 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 lecple 16572 meetcmee 17555 0.cp0 17647 Latclat 17655 OPcops 36323 OLcol 36325 |
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-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-lat 17656 df-oposet 36327 df-ol 36329 |
This theorem is referenced by: olm02 36388 omlfh1N 36409 |
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