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Mirrors > Home > MPE Home > Th. List > lmimlbs | Structured version Visualization version GIF version |
Description: The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
Ref | Expression |
---|---|
lmimlbs.j | β’ π½ = (LBasisβπ) |
lmimlbs.k | β’ πΎ = (LBasisβπ) |
Ref | Expression |
---|---|
lmimlbs | β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β π΅) β πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmimlmhm 20909 | . . 3 β’ (πΉ β (π LMIso π) β πΉ β (π LMHom π)) | |
2 | eqid 2726 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
3 | eqid 2726 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
4 | 2, 3 | lmimf1o 20908 | . . . 4 β’ (πΉ β (π LMIso π) β πΉ:(Baseβπ)β1-1-ontoβ(Baseβπ)) |
5 | f1of1 6825 | . . . 4 β’ (πΉ:(Baseβπ)β1-1-ontoβ(Baseβπ) β πΉ:(Baseβπ)β1-1β(Baseβπ)) | |
6 | 4, 5 | syl 17 | . . 3 β’ (πΉ β (π LMIso π) β πΉ:(Baseβπ)β1-1β(Baseβπ)) |
7 | lmimlbs.j | . . . . 5 β’ π½ = (LBasisβπ) | |
8 | 7 | lbslinds 21723 | . . . 4 β’ π½ β (LIndSβπ) |
9 | 8 | sseli 3973 | . . 3 β’ (π΅ β π½ β π΅ β (LIndSβπ)) |
10 | 2, 3 | lindsmm2 21719 | . . 3 β’ ((πΉ β (π LMHom π) β§ πΉ:(Baseβπ)β1-1β(Baseβπ) β§ π΅ β (LIndSβπ)) β (πΉ β π΅) β (LIndSβπ)) |
11 | 1, 6, 9, 10 | syl2an3an 1419 | . 2 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β π΅) β (LIndSβπ)) |
12 | eqid 2726 | . . . . . 6 β’ (LSpanβπ) = (LSpanβπ) | |
13 | 2, 7, 12 | lbssp 20924 | . . . . 5 β’ (π΅ β π½ β ((LSpanβπ)βπ΅) = (Baseβπ)) |
14 | 13 | adantl 481 | . . . 4 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β ((LSpanβπ)βπ΅) = (Baseβπ)) |
15 | 14 | imaeq2d 6052 | . . 3 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β ((LSpanβπ)βπ΅)) = (πΉ β (Baseβπ))) |
16 | 2, 7 | lbsss 20922 | . . . 4 β’ (π΅ β π½ β π΅ β (Baseβπ)) |
17 | eqid 2726 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
18 | 2, 12, 17 | lmhmlsp 20894 | . . . 4 β’ ((πΉ β (π LMHom π) β§ π΅ β (Baseβπ)) β (πΉ β ((LSpanβπ)βπ΅)) = ((LSpanβπ)β(πΉ β π΅))) |
19 | 1, 16, 18 | syl2an 595 | . . 3 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β ((LSpanβπ)βπ΅)) = ((LSpanβπ)β(πΉ β π΅))) |
20 | 4 | adantr 480 | . . . 4 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β πΉ:(Baseβπ)β1-1-ontoβ(Baseβπ)) |
21 | f1ofo 6833 | . . . 4 β’ (πΉ:(Baseβπ)β1-1-ontoβ(Baseβπ) β πΉ:(Baseβπ)βontoβ(Baseβπ)) | |
22 | foima 6803 | . . . 4 β’ (πΉ:(Baseβπ)βontoβ(Baseβπ) β (πΉ β (Baseβπ)) = (Baseβπ)) | |
23 | 20, 21, 22 | 3syl 18 | . . 3 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β (Baseβπ)) = (Baseβπ)) |
24 | 15, 19, 23 | 3eqtr3d 2774 | . 2 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β ((LSpanβπ)β(πΉ β π΅)) = (Baseβπ)) |
25 | lmimlbs.k | . . 3 β’ πΎ = (LBasisβπ) | |
26 | 3, 25, 17 | islbs4 21722 | . 2 β’ ((πΉ β π΅) β πΎ β ((πΉ β π΅) β (LIndSβπ) β§ ((LSpanβπ)β(πΉ β π΅)) = (Baseβπ))) |
27 | 11, 24, 26 | sylanbrc 582 | 1 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β π΅) β πΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3943 β cima 5672 β1-1βwf1 6533 βontoβwfo 6534 β1-1-ontoβwf1o 6535 βcfv 6536 (class class class)co 7404 Basecbs 17150 LSpanclspn 20815 LMHom clmhm 20864 LMIso clmim 20865 LBasisclbs 20919 LIndSclinds 21695 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-sbg 18865 df-subg 19047 df-ghm 19136 df-mgp 20037 df-ur 20084 df-ring 20137 df-lmod 20705 df-lss 20776 df-lsp 20816 df-lmhm 20867 df-lmim 20868 df-lbs 20920 df-lindf 21696 df-linds 21697 |
This theorem is referenced by: lmiclbs 21727 lmimdim 33205 dimkerim 33229 |
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