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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem18 | Structured version Visualization version GIF version | ||
| Description: Lemma for lcfr 41086. (Contributed by NM, 24-Feb-2015.) |
| Ref | Expression |
|---|---|
| lcfrlem17.h | β’ π» = (LHypβπΎ) |
| lcfrlem17.o | β’ β₯ = ((ocHβπΎ)βπ) |
| lcfrlem17.u | β’ π = ((DVecHβπΎ)βπ) |
| lcfrlem17.v | β’ π = (Baseβπ) |
| lcfrlem17.p | β’ + = (+gβπ) |
| lcfrlem17.z | β’ 0 = (0gβπ) |
| lcfrlem17.n | β’ π = (LSpanβπ) |
| lcfrlem17.a | β’ π΄ = (LSAtomsβπ) |
| lcfrlem17.k | β’ (π β (πΎ β HL β§ π β π»)) |
| lcfrlem17.x | β’ (π β π β (π β { 0 })) |
| lcfrlem17.y | β’ (π β π β (π β { 0 })) |
| lcfrlem17.ne | β’ (π β (πβ{π}) β (πβ{π})) |
| Ref | Expression |
|---|---|
| lcfrlem18 | β’ (π β ( β₯ β{π, π}) = (( β₯ β{π}) β© ( β₯ β{π}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-pr 4625 | . . 3 β’ {π, π} = ({π} βͺ {π}) | |
| 2 | 1 | fveq2i 6893 | . 2 β’ ( β₯ β{π, π}) = ( β₯ β({π} βͺ {π})) |
| 3 | lcfrlem17.k | . . 3 β’ (π β (πΎ β HL β§ π β π»)) | |
| 4 | lcfrlem17.x | . . . . 5 β’ (π β π β (π β { 0 })) | |
| 5 | 4 | eldifad 3951 | . . . 4 β’ (π β π β π) |
| 6 | 5 | snssd 4806 | . . 3 β’ (π β {π} β π) |
| 7 | lcfrlem17.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
| 8 | 7 | eldifad 3951 | . . . 4 β’ (π β π β π) |
| 9 | 8 | snssd 4806 | . . 3 β’ (π β {π} β π) |
| 10 | lcfrlem17.h | . . . 4 β’ π» = (LHypβπΎ) | |
| 11 | lcfrlem17.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
| 12 | lcfrlem17.v | . . . 4 β’ π = (Baseβπ) | |
| 13 | lcfrlem17.o | . . . 4 β’ β₯ = ((ocHβπΎ)βπ) | |
| 14 | 10, 11, 12, 13 | dochdmj1 40891 | . . 3 β’ (((πΎ β HL β§ π β π») β§ {π} β π β§ {π} β π) β ( β₯ β({π} βͺ {π})) = (( β₯ β{π}) β© ( β₯ β{π}))) |
| 15 | 3, 6, 9, 14 | syl3anc 1368 | . 2 β’ (π β ( β₯ β({π} βͺ {π})) = (( β₯ β{π}) β© ( β₯ β{π}))) |
| 16 | 2, 15 | eqtrid 2777 | 1 β’ (π β ( β₯ β{π, π}) = (( β₯ β{π}) β© ( β₯ β{π}))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 β cdif 3936 βͺ cun 3937 β© cin 3938 β wss 3939 {csn 4622 {cpr 4624 βcfv 6541 Basecbs 17177 +gcplusg 17230 0gc0g 17418 LSpanclspn 20857 LSAtomsclsa 38474 HLchlt 38850 LHypclh 39485 DVecHcdvh 40579 ocHcoch 40848 |
| 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 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-riotaBAD 38453 |
| 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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-tpos 8228 df-undef 8275 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-0g 17420 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-p1 18415 df-lat 18421 df-clat 18488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-cntz 19270 df-lsm 19593 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-drng 20628 df-lmod 20747 df-lss 20818 df-lsp 20858 df-lvec 20990 df-lsatoms 38476 df-oposet 38676 df-ol 38678 df-oml 38679 df-covers 38766 df-ats 38767 df-atl 38798 df-cvlat 38822 df-hlat 38851 df-llines 38999 df-lplanes 39000 df-lvols 39001 df-lines 39002 df-psubsp 39004 df-pmap 39005 df-padd 39297 df-lhyp 39489 df-laut 39490 df-ldil 39605 df-ltrn 39606 df-trl 39660 df-tendo 40256 df-edring 40258 df-disoa 40530 df-dvech 40580 df-dib 40640 df-dic 40674 df-dih 40730 df-doch 40849 |
| This theorem is referenced by: lcfrlem24 41067 |
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