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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihsmsprn | Structured version Visualization version GIF version |
Description: Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.) |
Ref | Expression |
---|---|
dihsmsprn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihsmsprn.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihsmsprn.v | ⊢ 𝑉 = (Base‘𝑈) |
dihsmsprn.p | ⊢ ⊕ = (LSSum‘𝑈) |
dihsmsprn.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dihsmsprn.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihsmsprn.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihsmsprn.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) |
dihsmsprn.t | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
Ref | Expression |
---|---|
dihsmsprn | ⊢ (𝜑 → (𝑋 ⊕ (𝑁‘{𝑇})) ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihsmsprn.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dihsmsprn.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | dihsmsprn.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | dihsmsprn.p | . . 3 ⊢ ⊕ = (LSSum‘𝑈) | |
5 | dihsmsprn.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
6 | dihsmsprn.i | . . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
7 | eqid 2725 | . . 3 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊) | |
8 | dihsmsprn.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | dihsmsprn.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) | |
10 | dihsmsprn.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | dihjat1 41052 | . 2 ⊢ (𝜑 → (𝑋((joinH‘𝐾)‘𝑊)(𝑁‘{𝑇})) = (𝑋 ⊕ (𝑁‘{𝑇}))) |
12 | 1, 2, 6, 3 | dihrnss 40901 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ 𝑉) |
13 | 8, 9, 12 | syl2anc 582 | . . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
14 | 1, 2, 8 | dvhlmod 40733 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
15 | 10 | snssd 4814 | . . . 4 ⊢ (𝜑 → {𝑇} ⊆ 𝑉) |
16 | 3, 5 | lspssv 20896 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ {𝑇} ⊆ 𝑉) → (𝑁‘{𝑇}) ⊆ 𝑉) |
17 | 14, 15, 16 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ 𝑉) |
18 | 1, 6, 2, 3, 7 | djhcl 41023 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ (𝑁‘{𝑇}) ⊆ 𝑉)) → (𝑋((joinH‘𝐾)‘𝑊)(𝑁‘{𝑇})) ∈ ran 𝐼) |
19 | 8, 13, 17, 18 | syl12anc 835 | . 2 ⊢ (𝜑 → (𝑋((joinH‘𝐾)‘𝑊)(𝑁‘{𝑇})) ∈ ran 𝐼) |
20 | 11, 19 | eqeltrrd 2826 | 1 ⊢ (𝜑 → (𝑋 ⊕ (𝑁‘{𝑇})) ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 {csn 4630 ran crn 5679 ‘cfv 6549 (class class class)co 7419 Basecbs 17199 LSSumclsm 19618 LModclmod 20772 LSpanclspn 20884 HLchlt 38972 LHypclh 39607 DVecHcdvh 40701 DIsoHcdih 40851 joinHcdjh 41017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-riotaBAD 38575 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-sca 17268 df-vsca 17269 df-0g 17442 df-proset 18306 df-poset 18324 df-plt 18341 df-lub 18357 df-glb 18358 df-join 18359 df-meet 18360 df-p0 18436 df-p1 18437 df-lat 18443 df-clat 18510 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-submnd 18760 df-grp 18917 df-minusg 18918 df-sbg 18919 df-subg 19103 df-cntz 19297 df-lsm 19620 df-cmn 19766 df-abl 19767 df-mgp 20104 df-rng 20122 df-ur 20151 df-ring 20204 df-oppr 20302 df-dvdsr 20325 df-unit 20326 df-invr 20356 df-dvr 20369 df-drng 20655 df-lmod 20774 df-lss 20845 df-lsp 20885 df-lvec 21017 df-lsatoms 38598 df-oposet 38798 df-ol 38800 df-oml 38801 df-covers 38888 df-ats 38889 df-atl 38920 df-cvlat 38944 df-hlat 38973 df-llines 39121 df-lplanes 39122 df-lvols 39123 df-lines 39124 df-psubsp 39126 df-pmap 39127 df-padd 39419 df-lhyp 39611 df-laut 39612 df-ldil 39727 df-ltrn 39728 df-trl 39782 df-tgrp 40366 df-tendo 40378 df-edring 40380 df-dveca 40626 df-disoa 40652 df-dvech 40702 df-dib 40762 df-dic 40796 df-dih 40852 df-doch 40971 df-djh 41018 |
This theorem is referenced by: dihsmsnrn 41058 lclkrlem2d 41133 |
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