Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdordlem1a | Structured version Visualization version GIF version |
Description: Lemma for mapdord 39691. (Contributed by NM, 27-Jan-2015.) |
Ref | Expression |
---|---|
mapdordlem1a.h | β’ π» = (LHypβπΎ) |
mapdordlem1a.o | β’ π = ((ocHβπΎ)βπ) |
mapdordlem1a.u | β’ π = ((DVecHβπΎ)βπ) |
mapdordlem1a.v | β’ π = (Baseβπ) |
mapdordlem1a.y | β’ π = (LSHypβπ) |
mapdordlem1a.f | β’ πΉ = (LFnlβπ) |
mapdordlem1a.l | β’ πΏ = (LKerβπ) |
mapdordlem1a.t | β’ π = {π β πΉ β£ (πβ(πβ(πΏβπ))) β π} |
mapdordlem1a.c | β’ πΆ = {π β πΉ β£ (πβ(πβ(πΏβπ))) = (πΏβπ)} |
mapdordlem1a.k | β’ (π β (πΎ β HL β§ π β π»)) |
Ref | Expression |
---|---|
mapdordlem1a | β’ (π β (π½ β π β (π½ β πΆ β§ (πβ(πβ(πΏβπ½))) β π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 771 | . . . . . 6 β’ ((π β§ (π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π)) β (πβ(πβ(πΏβπ½))) β π) | |
2 | mapdordlem1a.h | . . . . . . 7 β’ π» = (LHypβπΎ) | |
3 | mapdordlem1a.o | . . . . . . 7 β’ π = ((ocHβπΎ)βπ) | |
4 | mapdordlem1a.u | . . . . . . 7 β’ π = ((DVecHβπΎ)βπ) | |
5 | mapdordlem1a.f | . . . . . . 7 β’ πΉ = (LFnlβπ) | |
6 | mapdordlem1a.y | . . . . . . 7 β’ π = (LSHypβπ) | |
7 | mapdordlem1a.l | . . . . . . 7 β’ πΏ = (LKerβπ) | |
8 | mapdordlem1a.k | . . . . . . . 8 β’ (π β (πΎ β HL β§ π β π»)) | |
9 | 8 | adantr 482 | . . . . . . 7 β’ ((π β§ (π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π)) β (πΎ β HL β§ π β π»)) |
10 | simprl 769 | . . . . . . 7 β’ ((π β§ (π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π)) β π½ β πΉ) | |
11 | 2, 3, 4, 5, 6, 7, 9, 10 | dochlkr 39438 | . . . . . 6 β’ ((π β§ (π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π)) β ((πβ(πβ(πΏβπ½))) β π β ((πβ(πβ(πΏβπ½))) = (πΏβπ½) β§ (πΏβπ½) β π))) |
12 | 1, 11 | mpbid 232 | . . . . 5 β’ ((π β§ (π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π)) β ((πβ(πβ(πΏβπ½))) = (πΏβπ½) β§ (πΏβπ½) β π)) |
13 | 12 | simpld 496 | . . . 4 β’ ((π β§ (π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π)) β (πβ(πβ(πΏβπ½))) = (πΏβπ½)) |
14 | 13 | ex 414 | . . 3 β’ (π β ((π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π) β (πβ(πβ(πΏβπ½))) = (πΏβπ½))) |
15 | 14 | pm4.71rd 564 | . 2 β’ (π β ((π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π) β ((πβ(πβ(πΏβπ½))) = (πΏβπ½) β§ (π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π)))) |
16 | 2fveq3 6805 | . . . . 5 β’ (π = π½ β (πβ(πΏβπ)) = (πβ(πΏβπ½))) | |
17 | 16 | fveq2d 6804 | . . . 4 β’ (π = π½ β (πβ(πβ(πΏβπ))) = (πβ(πβ(πΏβπ½)))) |
18 | 17 | eleq1d 2821 | . . 3 β’ (π = π½ β ((πβ(πβ(πΏβπ))) β π β (πβ(πβ(πΏβπ½))) β π)) |
19 | mapdordlem1a.t | . . 3 β’ π = {π β πΉ β£ (πβ(πβ(πΏβπ))) β π} | |
20 | 18, 19 | elrab2 3632 | . 2 β’ (π½ β π β (π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π)) |
21 | mapdordlem1a.c | . . . . 5 β’ πΆ = {π β πΉ β£ (πβ(πβ(πΏβπ))) = (πΏβπ)} | |
22 | 21 | lcfl1lem 39544 | . . . 4 β’ (π½ β πΆ β (π½ β πΉ β§ (πβ(πβ(πΏβπ½))) = (πΏβπ½))) |
23 | 22 | anbi1i 625 | . . 3 β’ ((π½ β πΆ β§ (πβ(πβ(πΏβπ½))) β π) β ((π½ β πΉ β§ (πβ(πβ(πΏβπ½))) = (πΏβπ½)) β§ (πβ(πβ(πΏβπ½))) β π)) |
24 | anass 470 | . . 3 β’ (((π½ β πΉ β§ (πβ(πβ(πΏβπ½))) = (πΏβπ½)) β§ (πβ(πβ(πΏβπ½))) β π) β (π½ β πΉ β§ ((πβ(πβ(πΏβπ½))) = (πΏβπ½) β§ (πβ(πβ(πΏβπ½))) β π))) | |
25 | an12 643 | . . 3 β’ ((π½ β πΉ β§ ((πβ(πβ(πΏβπ½))) = (πΏβπ½) β§ (πβ(πβ(πΏβπ½))) β π)) β ((πβ(πβ(πΏβπ½))) = (πΏβπ½) β§ (π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π))) | |
26 | 23, 24, 25 | 3bitri 298 | . 2 β’ ((π½ β πΆ β§ (πβ(πβ(πΏβπ½))) β π) β ((πβ(πβ(πΏβπ½))) = (πΏβπ½) β§ (π½ β πΉ β§ (πβ(πβ(πΏβπ½))) β π))) |
27 | 15, 20, 26 | 3bitr4g 315 | 1 β’ (π β (π½ β π β (π½ β πΆ β§ (πβ(πβ(πΏβπ½))) β π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1539 β wcel 2104 {crab 3284 βcfv 6454 Basecbs 16953 LSHypclsh 37028 LFnlclfn 37110 LKerclk 37138 HLchlt 37403 LHypclh 38037 DVecHcdvh 39131 ocHcoch 39400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 ax-riotaBAD 37006 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-1st 7859 df-2nd 7860 df-tpos 8069 df-undef 8116 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-1o 8324 df-er 8525 df-map 8644 df-en 8761 df-dom 8762 df-sdom 8763 df-fin 8764 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-nn 12016 df-2 12078 df-3 12079 df-4 12080 df-5 12081 df-6 12082 df-n0 12276 df-z 12362 df-uz 12625 df-fz 13282 df-struct 16889 df-sets 16906 df-slot 16924 df-ndx 16936 df-base 16954 df-ress 16983 df-plusg 17016 df-mulr 17017 df-sca 17019 df-vsca 17020 df-0g 17193 df-proset 18054 df-poset 18072 df-plt 18089 df-lub 18105 df-glb 18106 df-join 18107 df-meet 18108 df-p0 18184 df-p1 18185 df-lat 18191 df-clat 18258 df-mgm 18367 df-sgrp 18416 df-mnd 18427 df-submnd 18472 df-grp 18621 df-minusg 18622 df-sbg 18623 df-subg 18793 df-cntz 18964 df-lsm 19282 df-cmn 19429 df-abl 19430 df-mgp 19762 df-ur 19779 df-ring 19826 df-oppr 19903 df-dvdsr 19924 df-unit 19925 df-invr 19955 df-dvr 19966 df-drng 20034 df-lmod 20166 df-lss 20235 df-lsp 20275 df-lvec 20406 df-lsatoms 37029 df-lshyp 37030 df-lfl 37111 df-lkr 37139 df-oposet 37229 df-ol 37231 df-oml 37232 df-covers 37319 df-ats 37320 df-atl 37351 df-cvlat 37375 df-hlat 37404 df-llines 37551 df-lplanes 37552 df-lvols 37553 df-lines 37554 df-psubsp 37556 df-pmap 37557 df-padd 37849 df-lhyp 38041 df-laut 38042 df-ldil 38157 df-ltrn 38158 df-trl 38212 df-tendo 38808 df-edring 38810 df-disoa 39082 df-dvech 39132 df-dib 39192 df-dic 39226 df-dih 39282 df-doch 39401 |
This theorem is referenced by: mapdordlem2 39690 |
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