Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > djhj | Structured version Visualization version GIF version |
Description: DVecH vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.) |
Ref | Expression |
---|---|
djhj.k | ⊢ ∨ = (join‘𝐾) |
djhj.h | ⊢ 𝐻 = (LHyp‘𝐾) |
djhj.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
djhj.j | ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊) |
djhj.w | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
djhj.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) |
djhj.y | ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) |
Ref | Expression |
---|---|
djhj | ⊢ (𝜑 → (◡𝐼‘(𝑋𝐽𝑌)) = ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djhj.k | . . . 4 ⊢ ∨ = (join‘𝐾) | |
2 | djhj.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | djhj.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
4 | djhj.j | . . . 4 ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊) | |
5 | djhj.w | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | djhj.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) | |
7 | djhj.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) | |
8 | 1, 2, 3, 4, 5, 6, 7 | djhjlj 39152 | . . 3 ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)))) |
9 | 8 | fveq2d 6718 | . 2 ⊢ (𝜑 → (◡𝐼‘(𝑋𝐽𝑌)) = (◡𝐼‘(𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌))))) |
10 | 5 | simpld 498 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
11 | 10 | hllatd 37113 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat) |
12 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | 12, 2, 3 | dihcnvcl 39020 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
14 | 5, 6, 13 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
15 | 12, 2, 3 | dihcnvcl 39020 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → (◡𝐼‘𝑌) ∈ (Base‘𝐾)) |
16 | 5, 7, 15 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (◡𝐼‘𝑌) ∈ (Base‘𝐾)) |
17 | 12, 1 | latjcl 17942 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (◡𝐼‘𝑋) ∈ (Base‘𝐾) ∧ (◡𝐼‘𝑌) ∈ (Base‘𝐾)) → ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)) ∈ (Base‘𝐾)) |
18 | 11, 14, 16, 17 | syl3anc 1373 | . . 3 ⊢ (𝜑 → ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)) ∈ (Base‘𝐾)) |
19 | 12, 2, 3 | dihcnvid1 39021 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)) ∈ (Base‘𝐾)) → (◡𝐼‘(𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)))) = ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌))) |
20 | 5, 18, 19 | syl2anc 587 | . 2 ⊢ (𝜑 → (◡𝐼‘(𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)))) = ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌))) |
21 | 9, 20 | eqtrd 2777 | 1 ⊢ (𝜑 → (◡𝐼‘(𝑋𝐽𝑌)) = ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ◡ccnv 5547 ran crn 5549 ‘cfv 6377 (class class class)co 7210 Basecbs 16757 joincjn 17815 Latclat 17934 HLchlt 37099 LHypclh 37733 DIsoHcdih 38977 joinHcdjh 39143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-riotaBAD 36702 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-iin 4904 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-tpos 7965 df-undef 8012 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-er 8388 df-map 8507 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-n0 12088 df-z 12174 df-uz 12436 df-fz 13093 df-struct 16697 df-sets 16714 df-slot 16732 df-ndx 16742 df-base 16758 df-ress 16782 df-plusg 16812 df-mulr 16813 df-sca 16815 df-vsca 16816 df-0g 16943 df-proset 17799 df-poset 17817 df-plt 17833 df-lub 17849 df-glb 17850 df-join 17851 df-meet 17852 df-p0 17928 df-p1 17929 df-lat 17935 df-clat 18002 df-mgm 18111 df-sgrp 18160 df-mnd 18171 df-submnd 18216 df-grp 18365 df-minusg 18366 df-sbg 18367 df-subg 18537 df-cntz 18708 df-lsm 19022 df-cmn 19169 df-abl 19170 df-mgp 19502 df-ur 19514 df-ring 19561 df-oppr 19638 df-dvdsr 19656 df-unit 19657 df-invr 19687 df-dvr 19698 df-drng 19766 df-lmod 19898 df-lss 19966 df-lsp 20006 df-lvec 20137 df-lsatoms 36725 df-oposet 36925 df-ol 36927 df-oml 36928 df-covers 37015 df-ats 37016 df-atl 37047 df-cvlat 37071 df-hlat 37100 df-llines 37247 df-lplanes 37248 df-lvols 37249 df-lines 37250 df-psubsp 37252 df-pmap 37253 df-padd 37545 df-lhyp 37737 df-laut 37738 df-ldil 37853 df-ltrn 37854 df-trl 37908 df-tendo 38504 df-edring 38506 df-disoa 38778 df-dvech 38828 df-dib 38888 df-dic 38922 df-dih 38978 df-doch 39097 df-djh 39144 |
This theorem is referenced by: djhcvat42 39164 |
Copyright terms: Public domain | W3C validator |