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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 2823 | . . . . . . 7 ⊢ (lub‘𝑂) = (lub‘𝑂) | |
4 | 2, 3 | oduglb 17751 | . . . . . 6 ⊢ (𝑂 ∈ V → (lub‘𝑂) = (glb‘𝐷)) |
5 | 4 | breqd 5079 | . . . . 5 ⊢ (𝑂 ∈ V → ({𝑎, 𝑏} (lub‘𝑂)𝑐 ↔ {𝑎, 𝑏} (glb‘𝐷)𝑐)) |
6 | 5 | oprabbidv 7222 | . . . 4 ⊢ (𝑂 ∈ V → {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (lub‘𝑂)𝑐} = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (glb‘𝐷)𝑐}) |
7 | eqid 2823 | . . . . 5 ⊢ (join‘𝑂) = (join‘𝑂) | |
8 | 3, 7 | joinfval 17613 | . . . 4 ⊢ (𝑂 ∈ V → (join‘𝑂) = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (lub‘𝑂)𝑐}) |
9 | 2 | fvexi 6686 | . . . . 5 ⊢ 𝐷 ∈ V |
10 | eqid 2823 | . . . . . 6 ⊢ (glb‘𝐷) = (glb‘𝐷) | |
11 | eqid 2823 | . . . . . 6 ⊢ (meet‘𝐷) = (meet‘𝐷) | |
12 | 10, 11 | meetfval 17627 | . . . . 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 2868 | . . 3 ⊢ (𝑂 ∈ V → (join‘𝑂) = (meet‘𝐷)) |
15 | fvprc 6665 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (join‘𝑂) = ∅) | |
16 | fvprc 6665 | . . . . . . 7 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
17 | 2, 16 | syl5eq 2870 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → 𝐷 = ∅) |
18 | 17 | fveq2d 6676 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → (meet‘𝐷) = (meet‘∅)) |
19 | meet0 17749 | . . . . 5 ⊢ (meet‘∅) = ∅ | |
20 | 18, 19 | syl6eq 2874 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (meet‘𝐷) = ∅) |
21 | 15, 20 | eqtr4d 2861 | . . 3 ⊢ (¬ 𝑂 ∈ V → (join‘𝑂) = (meet‘𝐷)) |
22 | 14, 21 | pm2.61i 184 | . 2 ⊢ (join‘𝑂) = (meet‘𝐷) |
23 | 1, 22 | eqtri 2846 | 1 ⊢ ∨ = (meet‘𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 {cpr 4571 class class class wbr 5068 ‘cfv 6357 {coprab 7159 lubclub 17554 glbcglb 17555 joincjn 17556 meetcmee 17557 ODualcodu 17740 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-dec 12102 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ple 16587 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-odu 17741 |
This theorem is referenced by: odulatb 17755 latdisd 17802 odudlatb 17808 dlatjmdi 17809 |
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