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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihglblem5 | Structured version Visualization version GIF version |
Description: Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.) |
Ref | Expression |
---|---|
dihglblem5.b | β’ π΅ = (BaseβπΎ) |
dihglblem5.g | β’ πΊ = (glbβπΎ) |
dihglblem5.h | β’ π» = (LHypβπΎ) |
dihglblem5.u | β’ π = ((DVecHβπΎ)βπ) |
dihglblem5.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
dihglblem5.s | β’ π = (LSubSpβπ) |
Ref | Expression |
---|---|
dihglblem5 | β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β β© π₯ β π (πΌβπ₯) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6859 | . . 3 β’ (πΌβπ₯) β V | |
2 | 1 | dfiin2 4998 | . 2 β’ β© π₯ β π (πΌβπ₯) = β© {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} |
3 | dihglblem5.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | dihglblem5.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
5 | simpl 484 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β (πΎ β HL β§ π β π»)) | |
6 | 3, 4, 5 | dvhlmod 39623 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β π β LMod) |
7 | simpll 766 | . . . . . 6 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β (πΎ β HL β§ π β π»)) | |
8 | simplrl 776 | . . . . . . 7 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β π β π΅) | |
9 | simpr 486 | . . . . . . 7 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β π₯ β π) | |
10 | 8, 9 | sseldd 3949 | . . . . . 6 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β π₯ β π΅) |
11 | dihglblem5.b | . . . . . . 7 β’ π΅ = (BaseβπΎ) | |
12 | dihglblem5.i | . . . . . . 7 β’ πΌ = ((DIsoHβπΎ)βπ) | |
13 | dihglblem5.s | . . . . . . 7 β’ π = (LSubSpβπ) | |
14 | 11, 3, 12, 4, 13 | dihlss 39763 | . . . . . 6 β’ (((πΎ β HL β§ π β π») β§ π₯ β π΅) β (πΌβπ₯) β π) |
15 | 7, 10, 14 | syl2anc 585 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β (πΌβπ₯) β π) |
16 | 15 | ralrimiva 3140 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β βπ₯ β π (πΌβπ₯) β π) |
17 | uniiunlem 4048 | . . . . 5 β’ (βπ₯ β π (πΌβπ₯) β π β (βπ₯ β π (πΌβπ₯) β π β {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} β π)) | |
18 | 16, 17 | syl 17 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β (βπ₯ β π (πΌβπ₯) β π β {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} β π)) |
19 | 16, 18 | mpbid 231 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} β π) |
20 | simprr 772 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β π β β ) | |
21 | n0 4310 | . . . . 5 β’ (π β β β βπ₯ π₯ β π) | |
22 | 20, 21 | sylib 217 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β βπ₯ π₯ β π) |
23 | nfre1 3267 | . . . . . . 7 β’ β²π₯βπ₯ β π π¦ = (πΌβπ₯) | |
24 | 23 | nfab 2910 | . . . . . 6 β’ β²π₯{π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} |
25 | nfcv 2904 | . . . . . 6 β’ β²π₯β | |
26 | 24, 25 | nfne 3042 | . . . . 5 β’ β²π₯{π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} β β |
27 | 1 | elabrex 7194 | . . . . . 6 β’ (π₯ β π β (πΌβπ₯) β {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)}) |
28 | 27 | ne0d 4299 | . . . . 5 β’ (π₯ β π β {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} β β ) |
29 | 26, 28 | exlimi 2211 | . . . 4 β’ (βπ₯ π₯ β π β {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} β β ) |
30 | 22, 29 | syl 17 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} β β ) |
31 | 13 | lssintcl 20469 | . . 3 β’ ((π β LMod β§ {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} β π β§ {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} β β ) β β© {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} β π) |
32 | 6, 19, 30, 31 | syl3anc 1372 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β β© {π¦ β£ βπ₯ β π π¦ = (πΌβπ₯)} β π) |
33 | 2, 32 | eqeltrid 2838 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β β© π₯ β π (πΌβπ₯) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 βwex 1782 β wcel 2107 {cab 2710 β wne 2940 βwral 3061 βwrex 3070 β wss 3914 β c0 4286 β© cint 4911 β© ciin 4959 βcfv 6500 Basecbs 17091 glbcglb 18207 LModclmod 20365 LSubSpclss 20436 HLchlt 37862 LHypclh 38497 DVecHcdvh 39591 DIsoHcdih 39741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-riotaBAD 37465 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-undef 8208 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-0g 17331 df-proset 18192 df-poset 18210 df-plt 18227 df-lub 18243 df-glb 18244 df-join 18245 df-meet 18246 df-p0 18322 df-p1 18323 df-lat 18329 df-clat 18396 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-subg 18933 df-cntz 19105 df-lsm 19426 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-drng 20221 df-lmod 20367 df-lss 20437 df-lsp 20477 df-lvec 20608 df-oposet 37688 df-ol 37690 df-oml 37691 df-covers 37778 df-ats 37779 df-atl 37810 df-cvlat 37834 df-hlat 37863 df-llines 38011 df-lplanes 38012 df-lvols 38013 df-lines 38014 df-psubsp 38016 df-pmap 38017 df-padd 38309 df-lhyp 38501 df-laut 38502 df-ldil 38617 df-ltrn 38618 df-trl 38672 df-tendo 39268 df-edring 39270 df-disoa 39542 df-dvech 39592 df-dib 39652 df-dic 39686 df-dih 39742 |
This theorem is referenced by: dihglblem6 39853 |
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