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Mirrors > Home > MPE Home > Th. List > odumeet | Structured version Visualization version GIF version |
Description: Meets in a dual order are joins in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
oduglb.d | ⊢ 𝐷 = (ODual‘𝑂) |
odumeet.j | ⊢ ∨ = (join‘𝑂) |
Ref | Expression |
---|---|
odumeet | ⊢ ∨ = (meet‘𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odumeet.j | . 2 ⊢ ∨ = (join‘𝑂) | |
2 | oduglb.d | . . . . . . 7 ⊢ 𝐷 = (ODual‘𝑂) | |
3 | eqid 2818 | . . . . . . 7 ⊢ (lub‘𝑂) = (lub‘𝑂) | |
4 | 2, 3 | oduglb 17737 | . . . . . 6 ⊢ (𝑂 ∈ V → (lub‘𝑂) = (glb‘𝐷)) |
5 | 4 | breqd 5068 | . . . . 5 ⊢ (𝑂 ∈ V → ({𝑎, 𝑏} (lub‘𝑂)𝑐 ↔ {𝑎, 𝑏} (glb‘𝐷)𝑐)) |
6 | 5 | oprabbidv 7209 | . . . 4 ⊢ (𝑂 ∈ V → {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (lub‘𝑂)𝑐} = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (glb‘𝐷)𝑐}) |
7 | eqid 2818 | . . . . 5 ⊢ (join‘𝑂) = (join‘𝑂) | |
8 | 3, 7 | joinfval 17599 | . . . 4 ⊢ (𝑂 ∈ V → (join‘𝑂) = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (lub‘𝑂)𝑐}) |
9 | 2 | fvexi 6677 | . . . . 5 ⊢ 𝐷 ∈ V |
10 | eqid 2818 | . . . . . 6 ⊢ (glb‘𝐷) = (glb‘𝐷) | |
11 | eqid 2818 | . . . . . 6 ⊢ (meet‘𝐷) = (meet‘𝐷) | |
12 | 10, 11 | meetfval 17613 | . . . . 5 ⊢ (𝐷 ∈ V → (meet‘𝐷) = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (glb‘𝐷)𝑐}) |
13 | 9, 12 | mp1i 13 | . . . 4 ⊢ (𝑂 ∈ V → (meet‘𝐷) = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (glb‘𝐷)𝑐}) |
14 | 6, 8, 13 | 3eqtr4d 2863 | . . 3 ⊢ (𝑂 ∈ V → (join‘𝑂) = (meet‘𝐷)) |
15 | fvprc 6656 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (join‘𝑂) = ∅) | |
16 | fvprc 6656 | . . . . . . 7 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
17 | 2, 16 | syl5eq 2865 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → 𝐷 = ∅) |
18 | 17 | fveq2d 6667 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → (meet‘𝐷) = (meet‘∅)) |
19 | meet0 17735 | . . . . 5 ⊢ (meet‘∅) = ∅ | |
20 | 18, 19 | syl6eq 2869 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (meet‘𝐷) = ∅) |
21 | 15, 20 | eqtr4d 2856 | . . 3 ⊢ (¬ 𝑂 ∈ V → (join‘𝑂) = (meet‘𝐷)) |
22 | 14, 21 | pm2.61i 183 | . 2 ⊢ (join‘𝑂) = (meet‘𝐷) |
23 | 1, 22 | eqtri 2841 | 1 ⊢ ∨ = (meet‘𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 {cpr 4559 class class class wbr 5057 ‘cfv 6348 {coprab 7146 lubclub 17540 glbcglb 17541 joincjn 17542 meetcmee 17543 ODualcodu 17726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-dec 12087 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ple 16573 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-odu 17727 |
This theorem is referenced by: odulatb 17741 latdisd 17788 odudlatb 17794 dlatjmdi 17795 |
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