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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 2798 | . . . . . . 7 ⊢ (lub‘𝑂) = (lub‘𝑂) | |
4 | 2, 3 | oduglb 17741 | . . . . . 6 ⊢ (𝑂 ∈ V → (lub‘𝑂) = (glb‘𝐷)) |
5 | 4 | breqd 5041 | . . . . 5 ⊢ (𝑂 ∈ V → ({𝑎, 𝑏} (lub‘𝑂)𝑐 ↔ {𝑎, 𝑏} (glb‘𝐷)𝑐)) |
6 | 5 | oprabbidv 7199 | . . . 4 ⊢ (𝑂 ∈ V → {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (lub‘𝑂)𝑐} = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (glb‘𝐷)𝑐}) |
7 | eqid 2798 | . . . . 5 ⊢ (join‘𝑂) = (join‘𝑂) | |
8 | 3, 7 | joinfval 17603 | . . . 4 ⊢ (𝑂 ∈ V → (join‘𝑂) = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (lub‘𝑂)𝑐}) |
9 | 2 | fvexi 6659 | . . . . 5 ⊢ 𝐷 ∈ V |
10 | eqid 2798 | . . . . . 6 ⊢ (glb‘𝐷) = (glb‘𝐷) | |
11 | eqid 2798 | . . . . . 6 ⊢ (meet‘𝐷) = (meet‘𝐷) | |
12 | 10, 11 | meetfval 17617 | . . . . 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 2843 | . . 3 ⊢ (𝑂 ∈ V → (join‘𝑂) = (meet‘𝐷)) |
15 | fvprc 6638 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (join‘𝑂) = ∅) | |
16 | fvprc 6638 | . . . . . . 7 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
17 | 2, 16 | syl5eq 2845 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → 𝐷 = ∅) |
18 | 17 | fveq2d 6649 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → (meet‘𝐷) = (meet‘∅)) |
19 | meet0 17739 | . . . . 5 ⊢ (meet‘∅) = ∅ | |
20 | 18, 19 | eqtrdi 2849 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (meet‘𝐷) = ∅) |
21 | 15, 20 | eqtr4d 2836 | . . 3 ⊢ (¬ 𝑂 ∈ V → (join‘𝑂) = (meet‘𝐷)) |
22 | 14, 21 | pm2.61i 185 | . 2 ⊢ (join‘𝑂) = (meet‘𝐷) |
23 | 1, 22 | eqtri 2821 | 1 ⊢ ∨ = (meet‘𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {cpr 4527 class class class wbr 5030 ‘cfv 6324 {coprab 7136 lubclub 17544 glbcglb 17545 joincjn 17546 meetcmee 17547 ODualcodu 17730 |
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 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 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-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 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-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 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-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 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-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ple 16577 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-odu 17731 |
This theorem is referenced by: odulatb 17745 latdisd 17792 odudlatb 17798 dlatjmdi 17799 |
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