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Mirrors > Home > MPE Home > Th. List > odujoin | Structured version Visualization version GIF version |
Description: Joins in a dual order are meets in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
oduglb.d | ⊢ 𝐷 = (ODual‘𝑂) |
odujoin.m | ⊢ ∧ = (meet‘𝑂) |
Ref | Expression |
---|---|
odujoin | ⊢ ∧ = (join‘𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odujoin.m | . 2 ⊢ ∧ = (meet‘𝑂) | |
2 | oduglb.d | . . . . . . 7 ⊢ 𝐷 = (ODual‘𝑂) | |
3 | eqid 2824 | . . . . . . 7 ⊢ (glb‘𝑂) = (glb‘𝑂) | |
4 | 2, 3 | odulub 17754 | . . . . . 6 ⊢ (𝑂 ∈ V → (glb‘𝑂) = (lub‘𝐷)) |
5 | 4 | breqd 5080 | . . . . 5 ⊢ (𝑂 ∈ V → ({𝑎, 𝑏} (glb‘𝑂)𝑐 ↔ {𝑎, 𝑏} (lub‘𝐷)𝑐)) |
6 | 5 | oprabbidv 7223 | . . . 4 ⊢ (𝑂 ∈ V → {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (glb‘𝑂)𝑐} = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (lub‘𝐷)𝑐}) |
7 | eqid 2824 | . . . . 5 ⊢ (meet‘𝑂) = (meet‘𝑂) | |
8 | 3, 7 | meetfval 17628 | . . . 4 ⊢ (𝑂 ∈ V → (meet‘𝑂) = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (glb‘𝑂)𝑐}) |
9 | 2 | fvexi 6687 | . . . . 5 ⊢ 𝐷 ∈ V |
10 | eqid 2824 | . . . . . 6 ⊢ (lub‘𝐷) = (lub‘𝐷) | |
11 | eqid 2824 | . . . . . 6 ⊢ (join‘𝐷) = (join‘𝐷) | |
12 | 10, 11 | joinfval 17614 | . . . . 5 ⊢ (𝐷 ∈ V → (join‘𝐷) = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (lub‘𝐷)𝑐}) |
13 | 9, 12 | mp1i 13 | . . . 4 ⊢ (𝑂 ∈ V → (join‘𝐷) = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (lub‘𝐷)𝑐}) |
14 | 6, 8, 13 | 3eqtr4d 2869 | . . 3 ⊢ (𝑂 ∈ V → (meet‘𝑂) = (join‘𝐷)) |
15 | fvprc 6666 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (meet‘𝑂) = ∅) | |
16 | fvprc 6666 | . . . . . . 7 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
17 | 2, 16 | syl5eq 2871 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → 𝐷 = ∅) |
18 | 17 | fveq2d 6677 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → (join‘𝐷) = (join‘∅)) |
19 | join0 17751 | . . . . 5 ⊢ (join‘∅) = ∅ | |
20 | 18, 19 | syl6eq 2875 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (join‘𝐷) = ∅) |
21 | 15, 20 | eqtr4d 2862 | . . 3 ⊢ (¬ 𝑂 ∈ V → (meet‘𝑂) = (join‘𝐷)) |
22 | 14, 21 | pm2.61i 184 | . 2 ⊢ (meet‘𝑂) = (join‘𝐷) |
23 | 1, 22 | eqtri 2847 | 1 ⊢ ∧ = (join‘𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 {cpr 4572 class class class wbr 5069 ‘cfv 6358 {coprab 7160 lubclub 17555 glbcglb 17556 joincjn 17557 meetcmee 17558 ODualcodu 17741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 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 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ple 16588 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-odu 17742 |
This theorem is referenced by: odulatb 17756 latmass 17801 latdisd 17803 odudlatb 17809 dlatjmdi 17810 |
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