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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 29239 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 36510 | . . . . . . 7 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
2 | 1 | adantr 484 | . . . . . 6 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) |
3 | eqid 2798 | . . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
4 | olm1.u | . . . . . . 7 ⊢ 1 = (1.‘𝐾) | |
5 | eqid 2798 | . . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
6 | 3, 4, 5 | opoc1 36498 | . . . . . 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 7151 | . . . 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 36490 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵) |
11 | 1, 10 | sylan 583 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵) |
12 | eqid 2798 | . . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾) | |
13 | 9, 12, 3 | olj01 36521 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)(0.‘𝐾)) = ((oc‘𝐾)‘𝑋)) |
14 | 11, 13 | syldan 594 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)(0.‘𝐾)) = ((oc‘𝐾)‘𝑋)) |
15 | 8, 14 | eqtrd 2833 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘ 1 )) = ((oc‘𝐾)‘𝑋)) |
16 | 15 | fveq2d 6649 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘ 1 ))) = ((oc‘𝐾)‘((oc‘𝐾)‘𝑋))) |
17 | 9, 4 | op1cl 36481 | . . . 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 36520 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 1 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘ 1 ))) = (𝑋 ∧ 1 )) |
21 | 18, 20 | mpd3an3 1459 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘ 1 ))) = (𝑋 ∧ 1 )) |
22 | 9, 5 | opococ 36491 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑋)) = 𝑋) |
23 | 1, 22 | sylan 583 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑋)) = 𝑋) |
24 | 16, 21, 23 | 3eqtr3d 2841 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 1 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 occoc 16565 joincjn 17546 meetcmee 17547 0.cp0 17639 1.cp1 17640 OPcops 36468 OLcol 36470 |
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-p0 17641 df-p1 17642 df-lat 17648 df-oposet 36472 df-ol 36474 |
This theorem is referenced by: olm12 36524 lhpmcvr3 37321 trljat1 37462 trljat2 37463 cdlemc1 37487 cdlemc6 37492 cdleme0cp 37510 cdleme0cq 37511 cdleme1 37523 cdleme4 37534 cdleme5 37536 cdleme8 37546 cdleme9 37549 cdleme10 37550 cdleme20c 37607 cdleme20j 37614 cdleme22e 37640 cdleme22eALTN 37641 cdleme30a 37674 cdleme35b 37746 cdleme35e 37749 cdleme42a 37767 trlcoabs2N 38018 trlcolem 38022 cdlemi1 38114 cdlemk4 38130 dia2dimlem1 38360 cdlemn10 38502 dihglbcpreN 38596 |
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