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Mathbox for Norm Megill |
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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 27736 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 480 | . . . . 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 18758 | . . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉) |
7 | 3, 6 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
8 | 7 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → 𝑋 ⊆ 𝑉) |
9 | djhlsmcl.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
10 | 4, 5 | lssss 18758 | . . . . . . 7 ⊢ (𝑌 ∈ 𝑆 → 𝑌 ⊆ 𝑉) |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑉) |
12 | 11 | adantr 480 | . . . . 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 2610 | . . . . . 6 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
16 | djhlsmcl.j | . . . . . 6 ⊢ ∨ = ((joinH‘𝐾)‘𝑊) | |
17 | 13, 14, 4, 15, 16 | djhval2 35706 | . . . . 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 1318 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (𝑋 ∨ 𝑌) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(𝑋 ∪ 𝑌)))) |
19 | 13, 14, 1 | dvhlmod 35417 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
20 | 19 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → 𝑈 ∈ LMod) |
21 | 3 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → 𝑋 ∈ 𝑆) |
22 | 9 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → 𝑌 ∈ 𝑆) |
23 | eqid 2610 | . . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
24 | djhlsmcl.p | . . . . . . . . 9 ⊢ ⊕ = (LSSum‘𝑈) | |
25 | 5, 23, 24 | lsmsp 18907 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 ⊕ 𝑌) = ((LSpan‘𝑈)‘(𝑋 ∪ 𝑌))) |
26 | 20, 21, 22, 25 | syl3anc 1318 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (𝑋 ⊕ 𝑌) = ((LSpan‘𝑈)‘(𝑋 ∪ 𝑌))) |
27 | 26 | fveq2d 6107 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (((ocH‘𝐾)‘𝑊)‘(𝑋 ⊕ 𝑌)) = (((ocH‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘(𝑋 ∪ 𝑌)))) |
28 | 7, 11 | unssd 3751 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 ∪ 𝑌) ⊆ 𝑉) |
29 | 28 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (𝑋 ∪ 𝑌) ⊆ 𝑉) |
30 | 13, 14, 15, 4, 23, 2, 29 | dochocsp 35686 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (((ocH‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘(𝑋 ∪ 𝑌))) = (((ocH‘𝐾)‘𝑊)‘(𝑋 ∪ 𝑌))) |
31 | 27, 30 | eqtrd 2644 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (((ocH‘𝐾)‘𝑊)‘(𝑋 ⊕ 𝑌)) = (((ocH‘𝐾)‘𝑊)‘(𝑋 ∪ 𝑌))) |
32 | 31 | fveq2d 6107 | . . . 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 35674 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(𝑋 ⊕ 𝑌))) = (𝑋 ⊕ 𝑌)) |
35 | 1, 34 | sylan 487 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(𝑋 ⊕ 𝑌))) = (𝑋 ⊕ 𝑌)) |
36 | 18, 32, 35 | 3eqtr2rd 2651 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ⊕ 𝑌) ∈ ran 𝐼) → (𝑋 ⊕ 𝑌) = (𝑋 ∨ 𝑌)) |
37 | 36 | ex 449 | . 2 ⊢ (𝜑 → ((𝑋 ⊕ 𝑌) ∈ ran 𝐼 → (𝑋 ⊕ 𝑌) = (𝑋 ∨ 𝑌))) |
38 | 13, 33, 14, 4, 16 | djhcl 35707 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋 ∨ 𝑌) ∈ ran 𝐼) |
39 | 1, 7, 11, 38 | syl12anc 1316 | . . 3 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ ran 𝐼) |
40 | eleq1a 2683 | . . 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 201 | 1 ⊢ (𝜑 → ((𝑋 ⊕ 𝑌) ∈ ran 𝐼 ↔ (𝑋 ⊕ 𝑌) = (𝑋 ∨ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ⊆ wss 3540 ran crn 5039 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 LSSumclsm 17872 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 HLchlt 33655 LHypclh 34288 DVecHcdvh 35385 DIsoHcdih 35535 ocHcoch 35654 joinHcdjh 35701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tendo 35061 df-edring 35063 df-disoa 35336 df-dvech 35386 df-dib 35446 df-dic 35480 df-dih 35536 df-doch 35655 df-djh 35702 |
This theorem is referenced by: djhlsmat 35734 dihsmatrn 35743 lclkrslem2 35845 |
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