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Mirrors > Home > MPE Home > Th. List > Mathboxes > djhlsmcl | Structured version Visualization version GIF version |
Description: A closed subspace sum equals subspace join. (shjshseli 29528 analog.) (Contributed by NM, 13-Aug-2014.) |
Ref | Expression |
---|---|
djhlsmcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
djhlsmcl.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
djhlsmcl.v | ⊢ 𝑉 = (Base‘𝑈) |
djhlsmcl.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
djhlsmcl.p | ⊢ ⊕ = (LSSum‘𝑈) |
djhlsmcl.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
djhlsmcl.j | ⊢ ∨ = ((joinH‘𝐾)‘𝑊) |
djhlsmcl.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
djhlsmcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
djhlsmcl.y | ⊢ (𝜑 → 𝑌 ∈ 𝑆) |
Ref | Expression |
---|---|
djhlsmcl | ⊢ (𝜑 → ((𝑋 ⊕ 𝑌) ∈ ran 𝐼 ↔ (𝑋 ⊕ 𝑌) = (𝑋 ∨ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djhlsmcl.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | 1 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
3 | djhlsmcl.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
4 | djhlsmcl.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈) | |
5 | djhlsmcl.s | . . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑈) | |
6 | 4, 5 | lssss 19927 | . . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉) |
7 | 3, 6 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
8 | 7 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → 𝑋 ⊆ 𝑉) |
9 | djhlsmcl.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
10 | 4, 5 | lssss 19927 | . . . . . . 7 ⊢ (𝑌 ∈ 𝑆 → 𝑌 ⊆ 𝑉) |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑉) |
12 | 11 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → 𝑌 ⊆ 𝑉) |
13 | djhlsmcl.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
14 | djhlsmcl.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
15 | eqid 2736 | . . . . . 6 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
16 | djhlsmcl.j | . . . . . 6 ⊢ ∨ = ((joinH‘𝐾)‘𝑊) | |
17 | 13, 14, 4, 15, 16 | djhval2 39099 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (𝑋 ∨ 𝑌) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(𝑋 ∪ 𝑌)))) |
18 | 2, 8, 12, 17 | syl3anc 1373 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (𝑋 ∨ 𝑌) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(𝑋 ∪ 𝑌)))) |
19 | 13, 14, 1 | dvhlmod 38810 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
20 | 19 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → 𝑈 ∈ LMod) |
21 | 3 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → 𝑋 ∈ 𝑆) |
22 | 9 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → 𝑌 ∈ 𝑆) |
23 | eqid 2736 | . . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
24 | djhlsmcl.p | . . . . . . . . 9 ⊢ ⊕ = (LSSum‘𝑈) | |
25 | 5, 23, 24 | lsmsp 20077 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 ⊕ 𝑌) = ((LSpan‘𝑈)‘(𝑋 ∪ 𝑌))) |
26 | 20, 21, 22, 25 | syl3anc 1373 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (𝑋 ⊕ 𝑌) = ((LSpan‘𝑈)‘(𝑋 ∪ 𝑌))) |
27 | 26 | fveq2d 6699 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (((ocH‘𝐾)‘𝑊)‘(𝑋 ⊕ 𝑌)) = (((ocH‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘(𝑋 ∪ 𝑌)))) |
28 | 7, 11 | unssd 4086 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 ∪ 𝑌) ⊆ 𝑉) |
29 | 28 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (𝑋 ∪ 𝑌) ⊆ 𝑉) |
30 | 13, 14, 15, 4, 23, 2, 29 | dochocsp 39079 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (((ocH‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘(𝑋 ∪ 𝑌))) = (((ocH‘𝐾)‘𝑊)‘(𝑋 ∪ 𝑌))) |
31 | 27, 30 | eqtrd 2771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (((ocH‘𝐾)‘𝑊)‘(𝑋 ⊕ 𝑌)) = (((ocH‘𝐾)‘𝑊)‘(𝑋 ∪ 𝑌))) |
32 | 31 | fveq2d 6699 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(𝑋 ⊕ 𝑌))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(𝑋 ∪ 𝑌)))) |
33 | djhlsmcl.i | . . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
34 | 13, 33, 15 | dochoc 39067 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(𝑋 ⊕ 𝑌))) = (𝑋 ⊕ 𝑌)) |
35 | 1, 34 | sylan 583 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(𝑋 ⊕ 𝑌))) = (𝑋 ⊕ 𝑌)) |
36 | 18, 32, 35 | 3eqtr2rd 2778 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (𝑋 ⊕ 𝑌) = (𝑋 ∨ 𝑌)) |
37 | 36 | ex 416 | . 2 ⊢ (𝜑 → ((𝑋 ⊕ 𝑌) ∈ ran 𝐼 → (𝑋 ⊕ 𝑌) = (𝑋 ∨ 𝑌))) |
38 | 13, 33, 14, 4, 16 | djhcl 39100 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋 ∨ 𝑌) ∈ ran 𝐼) |
39 | 1, 7, 11, 38 | syl12anc 837 | . . 3 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ ran 𝐼) |
40 | eleq1a 2826 | . . 3 ⊢ ((𝑋 ∨ 𝑌) ∈ ran 𝐼 → ((𝑋 ⊕ 𝑌) = (𝑋 ∨ 𝑌) → (𝑋 ⊕ 𝑌) ∈ ran 𝐼)) | |
41 | 39, 40 | syl 17 | . 2 ⊢ (𝜑 → ((𝑋 ⊕ 𝑌) = (𝑋 ∨ 𝑌) → (𝑋 ⊕ 𝑌) ∈ ran 𝐼)) |
42 | 37, 41 | impbid 215 | 1 ⊢ (𝜑 → ((𝑋 ⊕ 𝑌) ∈ ran 𝐼 ↔ (𝑋 ⊕ 𝑌) = (𝑋 ∨ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∪ cun 3851 ⊆ wss 3853 ran crn 5537 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 LSSumclsm 18977 LModclmod 19853 LSubSpclss 19922 LSpanclspn 19962 HLchlt 37050 LHypclh 37684 DVecHcdvh 38778 DIsoHcdih 38928 ocHcoch 39047 joinHcdjh 39094 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-riotaBAD 36653 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-tpos 7946 df-undef 7993 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-0g 16900 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-p1 17886 df-lat 17892 df-clat 17959 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-subg 18494 df-cntz 18665 df-lsm 18979 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-invr 19644 df-dvr 19655 df-drng 19723 df-lmod 19855 df-lss 19923 df-lsp 19963 df-lvec 20094 df-lsatoms 36676 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 df-llines 37198 df-lplanes 37199 df-lvols 37200 df-lines 37201 df-psubsp 37203 df-pmap 37204 df-padd 37496 df-lhyp 37688 df-laut 37689 df-ldil 37804 df-ltrn 37805 df-trl 37859 df-tendo 38455 df-edring 38457 df-disoa 38729 df-dvech 38779 df-dib 38839 df-dic 38873 df-dih 38929 df-doch 39048 df-djh 39095 |
This theorem is referenced by: djhlsmat 39127 dihsmatrn 39136 lclkrslem2 39238 |
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