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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihjat4 | Structured version Visualization version GIF version |
Description: Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.) |
Ref | Expression |
---|---|
dihjat4.j | ⊢ ∨ = (join‘𝐾) |
dihjat4.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihjat4.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihjat4.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihjat4.s | ⊢ ⊕ = (LSSum‘𝑈) |
dihjat4.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
dihjat4.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihjat4.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) |
dihjat4.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
Ref | Expression |
---|---|
dihjat4 | ⊢ (𝜑 → (𝑋 ⊕ 𝑄) = (𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑄)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2795 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | dihjat4.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | dihjat4.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | eqid 2795 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
5 | dihjat4.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | dihjat4.s | . . 3 ⊢ ⊕ = (LSSum‘𝑈) | |
7 | dihjat4.i | . . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
8 | dihjat4.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | dihjat4.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) | |
10 | 1, 2, 7 | dihcnvcl 37963 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
11 | 8, 9, 10 | syl2anc 584 | . . 3 ⊢ (𝜑 → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
12 | dihjat4.q | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
13 | dihjat4.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
14 | 4, 2, 5, 7, 13 | dihlatat 38029 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → (◡𝐼‘𝑄) ∈ (Atoms‘𝐾)) |
15 | 8, 12, 14 | syl2anc 584 | . . 3 ⊢ (𝜑 → (◡𝐼‘𝑄) ∈ (Atoms‘𝐾)) |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 11, 15 | dihjat3 38124 | . 2 ⊢ (𝜑 → (𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑄))) = ((𝐼‘(◡𝐼‘𝑋)) ⊕ (𝐼‘(◡𝐼‘𝑄)))) |
17 | 2, 7 | dihcnvid2 37965 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝐼‘(◡𝐼‘𝑋)) = 𝑋) |
18 | 8, 9, 17 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘𝑋)) = 𝑋) |
19 | 2, 5, 7, 13 | dih1dimat 38022 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ ran 𝐼) |
20 | 8, 12, 19 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ ran 𝐼) |
21 | 2, 7 | dihcnvid2 37965 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ ran 𝐼) → (𝐼‘(◡𝐼‘𝑄)) = 𝑄) |
22 | 8, 20, 21 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘𝑄)) = 𝑄) |
23 | 18, 22 | oveq12d 7039 | . 2 ⊢ (𝜑 → ((𝐼‘(◡𝐼‘𝑋)) ⊕ (𝐼‘(◡𝐼‘𝑄))) = (𝑋 ⊕ 𝑄)) |
24 | 16, 23 | eqtr2d 2832 | 1 ⊢ (𝜑 → (𝑋 ⊕ 𝑄) = (𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑄)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ◡ccnv 5447 ran crn 5449 ‘cfv 6230 (class class class)co 7021 Basecbs 16317 joincjn 17388 LSSumclsm 18494 LSAtomsclsa 35666 Atomscatm 35955 HLchlt 36042 LHypclh 36676 DVecHcdvh 37770 DIsoHcdih 37920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-riotaBAD 35645 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-iin 4832 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-tpos 7748 df-undef 7795 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-map 8263 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-n0 11751 df-z 11835 df-uz 12099 df-fz 12748 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-sca 16415 df-vsca 16416 df-0g 16549 df-proset 17372 df-poset 17390 df-plt 17402 df-lub 17418 df-glb 17419 df-join 17420 df-meet 17421 df-p0 17483 df-p1 17484 df-lat 17490 df-clat 17552 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-submnd 17780 df-grp 17869 df-minusg 17870 df-sbg 17871 df-subg 18035 df-cntz 18193 df-lsm 18496 df-cmn 18640 df-abl 18641 df-mgp 18935 df-ur 18947 df-ring 18994 df-oppr 19068 df-dvdsr 19086 df-unit 19087 df-invr 19117 df-dvr 19128 df-drng 19199 df-lmod 19331 df-lss 19399 df-lsp 19439 df-lvec 19570 df-lsatoms 35668 df-oposet 35868 df-ol 35870 df-oml 35871 df-covers 35958 df-ats 35959 df-atl 35990 df-cvlat 36014 df-hlat 36043 df-llines 36190 df-lplanes 36191 df-lvols 36192 df-lines 36193 df-psubsp 36195 df-pmap 36196 df-padd 36488 df-lhyp 36680 df-laut 36681 df-ldil 36796 df-ltrn 36797 df-trl 36851 df-tgrp 37435 df-tendo 37447 df-edring 37449 df-dveca 37695 df-disoa 37721 df-dvech 37771 df-dib 37831 df-dic 37865 df-dih 37921 df-doch 38040 df-djh 38087 |
This theorem is referenced by: dihjat6 38126 dvh4dimat 38130 |
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