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Mirrors > Home > MPE Home > Th. List > oduleval | Structured version Visualization version GIF version |
Description: Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
oduval.d | ⊢ 𝐷 = (ODual‘𝑂) |
oduval.l | ⊢ ≤ = (le‘𝑂) |
Ref | Expression |
---|---|
oduleval | ⊢ ◡ ≤ = (le‘𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6685 | . . . . 5 ⊢ (le‘𝑂) ∈ V | |
2 | 1 | cnvex 7632 | . . . 4 ⊢ ◡(le‘𝑂) ∈ V |
3 | pleid 16669 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
4 | 3 | setsid 16540 | . . . 4 ⊢ ((𝑂 ∈ V ∧ ◡(le‘𝑂) ∈ V) → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
5 | 2, 4 | mpan2 689 | . . 3 ⊢ (𝑂 ∈ V → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
6 | 3 | str0 16537 | . . . 4 ⊢ ∅ = (le‘∅) |
7 | fvprc 6665 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → (le‘𝑂) = ∅) | |
8 | 7 | cnveqd 5748 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = ◡∅) |
9 | cnv0 6001 | . . . . 5 ⊢ ◡∅ = ∅ | |
10 | 8, 9 | syl6eq 2874 | . . . 4 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = ∅) |
11 | reldmsets 16513 | . . . . . 6 ⊢ Rel dom sSet | |
12 | 11 | ovprc1 7197 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) = ∅) |
13 | 12 | fveq2d 6676 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) = (le‘∅)) |
14 | 6, 10, 13 | 3eqtr4a 2884 | . . 3 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
15 | 5, 14 | pm2.61i 184 | . 2 ⊢ ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
16 | oduval.l | . . 3 ⊢ ≤ = (le‘𝑂) | |
17 | 16 | cnveqi 5747 | . 2 ⊢ ◡ ≤ = ◡(le‘𝑂) |
18 | oduval.d | . . . 4 ⊢ 𝐷 = (ODual‘𝑂) | |
19 | eqid 2823 | . . . 4 ⊢ (le‘𝑂) = (le‘𝑂) | |
20 | 18, 19 | oduval 17742 | . . 3 ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) |
21 | 20 | fveq2i 6675 | . 2 ⊢ (le‘𝐷) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
22 | 15, 17, 21 | 3eqtr4i 2856 | 1 ⊢ ◡ ≤ = (le‘𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 〈cop 4575 ◡ccnv 5556 ‘cfv 6357 (class class class)co 7158 ndxcnx 16482 sSet csts 16483 lecple 16574 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-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 |
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-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-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-ltxr 10682 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-sets 16492 df-ple 16587 df-odu 17741 |
This theorem is referenced by: oduleg 17744 odupos 17747 oduposb 17748 oduglb 17751 odulub 17753 posglbd 17762 oduprs 30645 odutos 30652 ordtcnvNEW 31165 ordtrest2NEW 31168 |
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