Mathbox for Norm Megill |
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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 39459 | . . 3 ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)))) |
9 | 8 | fveq2d 6808 | . 2 ⊢ (𝜑 → (◡𝐼‘(𝑋𝐽𝑌)) = (◡𝐼‘(𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌))))) |
10 | 5 | simpld 496 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
11 | 10 | hllatd 37420 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat) |
12 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | 12, 2, 3 | dihcnvcl 39327 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
14 | 5, 6, 13 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
15 | 12, 2, 3 | dihcnvcl 39327 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → (◡𝐼‘𝑌) ∈ (Base‘𝐾)) |
16 | 5, 7, 15 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (◡𝐼‘𝑌) ∈ (Base‘𝐾)) |
17 | 12, 1 | latjcl 18202 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (◡𝐼‘𝑋) ∈ (Base‘𝐾) ∧ (◡𝐼‘𝑌) ∈ (Base‘𝐾)) → ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)) ∈ (Base‘𝐾)) |
18 | 11, 14, 16, 17 | syl3anc 1371 | . . 3 ⊢ (𝜑 → ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)) ∈ (Base‘𝐾)) |
19 | 12, 2, 3 | dihcnvid1 39328 | . . 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 585 | . 2 ⊢ (𝜑 → (◡𝐼‘(𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)))) = ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌))) |
21 | 9, 20 | eqtrd 2776 | 1 ⊢ (𝜑 → (◡𝐼‘(𝑋𝐽𝑌)) = ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1539 ∈ wcel 2104 ◡ccnv 5599 ran crn 5601 ‘cfv 6458 (class class class)co 7307 Basecbs 16957 joincjn 18074 Latclat 18194 HLchlt 37406 LHypclh 38040 DIsoHcdih 39284 joinHcdjh 39450 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-riotaBAD 37009 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-tpos 8073 df-undef 8120 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-sca 17023 df-vsca 17024 df-0g 17197 df-proset 18058 df-poset 18076 df-plt 18093 df-lub 18109 df-glb 18110 df-join 18111 df-meet 18112 df-p0 18188 df-p1 18189 df-lat 18195 df-clat 18262 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-submnd 18476 df-grp 18625 df-minusg 18626 df-sbg 18627 df-subg 18797 df-cntz 18968 df-lsm 19286 df-cmn 19433 df-abl 19434 df-mgp 19766 df-ur 19783 df-ring 19830 df-oppr 19907 df-dvdsr 19928 df-unit 19929 df-invr 19959 df-dvr 19970 df-drng 20038 df-lmod 20170 df-lss 20239 df-lsp 20279 df-lvec 20410 df-lsatoms 37032 df-oposet 37232 df-ol 37234 df-oml 37235 df-covers 37322 df-ats 37323 df-atl 37354 df-cvlat 37378 df-hlat 37407 df-llines 37554 df-lplanes 37555 df-lvols 37556 df-lines 37557 df-psubsp 37559 df-pmap 37560 df-padd 37852 df-lhyp 38044 df-laut 38045 df-ldil 38160 df-ltrn 38161 df-trl 38215 df-tendo 38811 df-edring 38813 df-disoa 39085 df-dvech 39135 df-dib 39195 df-dic 39229 df-dih 39285 df-doch 39404 df-djh 39451 |
This theorem is referenced by: djhcvat42 39471 |
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