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| Mirrors > Home > MPE Home > Th. List > Mathboxes > djhcom | Structured version Visualization version GIF version | ||
| Description: Subspace join commutes. (Contributed by NM, 8-Aug-2014.) |
| Ref | Expression |
|---|---|
| djhcom.h | β’ π» = (LHypβπΎ) |
| djhcom.u | β’ π = ((DVecHβπΎ)βπ) |
| djhcom.v | β’ π = (Baseβπ) |
| djhcom.j | β’ β¨ = ((joinHβπΎ)βπ) |
| djhcom.k | β’ (π β (πΎ β HL β§ π β π»)) |
| djhcom.x | β’ (π β π β π) |
| djhcom.y | β’ (π β π β π) |
| Ref | Expression |
|---|---|
| djhcom | β’ (π β (π β¨ π) = (π β¨ π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uncom 4153 | . . . 4 β’ (π βͺ π) = (π βͺ π) | |
| 2 | 1 | fveq2i 6894 | . . 3 β’ (((ocHβπΎ)βπ)β(π βͺ π)) = (((ocHβπΎ)βπ)β(π βͺ π)) |
| 3 | 2 | fveq2i 6894 | . 2 β’ (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π βͺ π))) = (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π βͺ π))) |
| 4 | djhcom.k | . . 3 β’ (π β (πΎ β HL β§ π β π»)) | |
| 5 | djhcom.x | . . 3 β’ (π β π β π) | |
| 6 | djhcom.y | . . 3 β’ (π β π β π) | |
| 7 | djhcom.h | . . . 4 β’ π» = (LHypβπΎ) | |
| 8 | djhcom.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
| 9 | djhcom.v | . . . 4 β’ π = (Baseβπ) | |
| 10 | eqid 2732 | . . . 4 β’ ((ocHβπΎ)βπ) = ((ocHβπΎ)βπ) | |
| 11 | djhcom.j | . . . 4 β’ β¨ = ((joinHβπΎ)βπ) | |
| 12 | 7, 8, 9, 10, 11 | djhval2 40573 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π β π β§ π β π) β (π β¨ π) = (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π βͺ π)))) |
| 13 | 4, 5, 6, 12 | syl3anc 1371 | . 2 β’ (π β (π β¨ π) = (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π βͺ π)))) |
| 14 | 7, 8, 9, 10, 11 | djhval2 40573 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π β π β§ π β π) β (π β¨ π) = (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π βͺ π)))) |
| 15 | 4, 6, 5, 14 | syl3anc 1371 | . 2 β’ (π β (π β¨ π) = (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π βͺ π)))) |
| 16 | 3, 13, 15 | 3eqtr4a 2798 | 1 β’ (π β (π β¨ π) = (π β¨ π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βͺ cun 3946 β wss 3948 βcfv 6543 (class class class)co 7411 Basecbs 17148 HLchlt 38523 LHypclh 39158 DVecHcdvh 40252 ocHcoch 40521 joinHcdjh 40568 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38126 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-undef 8260 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-0g 17391 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19039 df-cntz 19222 df-lsm 19545 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-drng 20502 df-lmod 20616 df-lss 20687 df-lsp 20727 df-lvec 20858 df-lsatoms 38149 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-llines 38672 df-lplanes 38673 df-lvols 38674 df-lines 38675 df-psubsp 38677 df-pmap 38678 df-padd 38970 df-lhyp 39162 df-laut 39163 df-ldil 39278 df-ltrn 39279 df-trl 39333 df-tendo 39929 df-edring 39931 df-disoa 40203 df-dvech 40253 df-dib 40313 df-dic 40347 df-dih 40403 df-doch 40522 df-djh 40569 |
| This theorem is referenced by: djh02 40587 dochsatshp 40625 |
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