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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > olm11 | Structured version Visualization version GIF version |
Description: The meet of an ortholattice element with one equals itself. (chm1i 31338 analog.) (Contributed by NM, 22-May-2012.) |
Ref | Expression |
---|---|
olm1.b | ⊢ 𝐵 = (Base‘𝐾) |
olm1.m | ⊢ ∧ = (meet‘𝐾) |
olm1.u | ⊢ 1 = (1.‘𝐾) |
Ref | Expression |
---|---|
olm11 | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 1 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olop 38816 | . . . . . . 7 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
2 | 1 | adantr 479 | . . . . . 6 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) |
3 | eqid 2725 | . . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
4 | olm1.u | . . . . . . 7 ⊢ 1 = (1.‘𝐾) | |
5 | eqid 2725 | . . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
6 | 3, 4, 5 | opoc1 38804 | . . . . . 6 ⊢ (𝐾 ∈ OP → ((oc‘𝐾)‘ 1 ) = (0.‘𝐾)) |
7 | 2, 6 | syl 17 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘ 1 ) = (0.‘𝐾)) |
8 | 7 | oveq2d 7435 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘ 1 )) = (((oc‘𝐾)‘𝑋)(join‘𝐾)(0.‘𝐾))) |
9 | olm1.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
10 | 9, 5 | opoccl 38796 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵) |
11 | 1, 10 | sylan 578 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵) |
12 | eqid 2725 | . . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾) | |
13 | 9, 12, 3 | olj01 38827 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)(0.‘𝐾)) = ((oc‘𝐾)‘𝑋)) |
14 | 11, 13 | syldan 589 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)(0.‘𝐾)) = ((oc‘𝐾)‘𝑋)) |
15 | 8, 14 | eqtrd 2765 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘ 1 )) = ((oc‘𝐾)‘𝑋)) |
16 | 15 | fveq2d 6900 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘ 1 ))) = ((oc‘𝐾)‘((oc‘𝐾)‘𝑋))) |
17 | 9, 4 | op1cl 38787 | . . . 4 ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵) |
18 | 2, 17 | syl 17 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 1 ∈ 𝐵) |
19 | olm1.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
20 | 9, 12, 19, 5 | oldmj4 38826 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 1 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘ 1 ))) = (𝑋 ∧ 1 )) |
21 | 18, 20 | mpd3an3 1458 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘ 1 ))) = (𝑋 ∧ 1 )) |
22 | 9, 5 | opococ 38797 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑋)) = 𝑋) |
23 | 1, 22 | sylan 578 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑋)) = 𝑋) |
24 | 16, 21, 23 | 3eqtr3d 2773 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 1 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 occoc 17244 joincjn 18306 meetcmee 18307 0.cp0 18418 1.cp1 18419 OPcops 38774 OLcol 38776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-proset 18290 df-poset 18308 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-oposet 38778 df-ol 38780 |
This theorem is referenced by: olm12 38830 lhpmcvr3 39628 trljat1 39769 trljat2 39770 cdlemc1 39794 cdlemc6 39799 cdleme0cp 39817 cdleme0cq 39818 cdleme1 39830 cdleme4 39841 cdleme5 39843 cdleme8 39853 cdleme9 39856 cdleme10 39857 cdleme20c 39914 cdleme20j 39921 cdleme22e 39947 cdleme22eALTN 39948 cdleme30a 39981 cdleme35b 40053 cdleme35e 40056 cdleme42a 40074 trlcoabs2N 40325 trlcolem 40329 cdlemi1 40421 cdlemk4 40437 dia2dimlem1 40667 cdlemn10 40809 dihglbcpreN 40903 |
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