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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 2821 | . . . . . . 7 ⊢ (glb‘𝑂) = (glb‘𝑂) | |
4 | 2, 3 | odulub 17751 | . . . . . 6 ⊢ (𝑂 ∈ V → (glb‘𝑂) = (lub‘𝐷)) |
5 | 4 | breqd 5077 | . . . . 5 ⊢ (𝑂 ∈ V → ({𝑎, 𝑏} (glb‘𝑂)𝑐 ↔ {𝑎, 𝑏} (lub‘𝐷)𝑐)) |
6 | 5 | oprabbidv 7220 | . . . 4 ⊢ (𝑂 ∈ V → {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (glb‘𝑂)𝑐} = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (lub‘𝐷)𝑐}) |
7 | eqid 2821 | . . . . 5 ⊢ (meet‘𝑂) = (meet‘𝑂) | |
8 | 3, 7 | meetfval 17625 | . . . 4 ⊢ (𝑂 ∈ V → (meet‘𝑂) = {〈〈𝑎, 𝑏〉, 𝑐〉 ∣ {𝑎, 𝑏} (glb‘𝑂)𝑐}) |
9 | 2 | fvexi 6684 | . . . . 5 ⊢ 𝐷 ∈ V |
10 | eqid 2821 | . . . . . 6 ⊢ (lub‘𝐷) = (lub‘𝐷) | |
11 | eqid 2821 | . . . . . 6 ⊢ (join‘𝐷) = (join‘𝐷) | |
12 | 10, 11 | joinfval 17611 | . . . . 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 2866 | . . 3 ⊢ (𝑂 ∈ V → (meet‘𝑂) = (join‘𝐷)) |
15 | fvprc 6663 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (meet‘𝑂) = ∅) | |
16 | fvprc 6663 | . . . . . . 7 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
17 | 2, 16 | syl5eq 2868 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → 𝐷 = ∅) |
18 | 17 | fveq2d 6674 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → (join‘𝐷) = (join‘∅)) |
19 | join0 17748 | . . . . 5 ⊢ (join‘∅) = ∅ | |
20 | 18, 19 | syl6eq 2872 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (join‘𝐷) = ∅) |
21 | 15, 20 | eqtr4d 2859 | . . 3 ⊢ (¬ 𝑂 ∈ V → (meet‘𝑂) = (join‘𝐷)) |
22 | 14, 21 | pm2.61i 184 | . 2 ⊢ (meet‘𝑂) = (join‘𝐷) |
23 | 1, 22 | eqtri 2844 | 1 ⊢ ∧ = (join‘𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 {cpr 4569 class class class wbr 5066 ‘cfv 6355 {coprab 7157 lubclub 17552 glbcglb 17553 joincjn 17554 meetcmee 17555 ODualcodu 17738 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-dec 12100 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ple 16585 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-odu 17739 |
This theorem is referenced by: odulatb 17753 latmass 17798 latdisd 17800 odudlatb 17806 dlatjmdi 17807 |
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