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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 20956 | . . 3 β’ (πΉ β (π LMIso π) β πΉ β (π LMHom π)) | |
2 | eqid 2728 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
3 | eqid 2728 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
4 | 2, 3 | lmimf1o 20955 | . . . 4 β’ (πΉ β (π LMIso π) β πΉ:(Baseβπ)β1-1-ontoβ(Baseβπ)) |
5 | f1of1 6843 | . . . 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 21774 | . . . 4 β’ π½ β (LIndSβπ) |
9 | 8 | sseli 3978 | . . 3 β’ (π΅ β π½ β π΅ β (LIndSβπ)) |
10 | 2, 3 | lindsmm2 21770 | . . 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 2728 | . . . . . 6 β’ (LSpanβπ) = (LSpanβπ) | |
13 | 2, 7, 12 | lbssp 20971 | . . . . 5 β’ (π΅ β π½ β ((LSpanβπ)βπ΅) = (Baseβπ)) |
14 | 13 | adantl 480 | . . . 4 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β ((LSpanβπ)βπ΅) = (Baseβπ)) |
15 | 14 | imaeq2d 6068 | . . 3 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β ((LSpanβπ)βπ΅)) = (πΉ β (Baseβπ))) |
16 | 2, 7 | lbsss 20969 | . . . 4 β’ (π΅ β π½ β π΅ β (Baseβπ)) |
17 | eqid 2728 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
18 | 2, 12, 17 | lmhmlsp 20941 | . . . 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 594 | . . 3 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β ((LSpanβπ)βπ΅)) = ((LSpanβπ)β(πΉ β π΅))) |
20 | 4 | adantr 479 | . . . 4 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β πΉ:(Baseβπ)β1-1-ontoβ(Baseβπ)) |
21 | f1ofo 6851 | . . . 4 β’ (πΉ:(Baseβπ)β1-1-ontoβ(Baseβπ) β πΉ:(Baseβπ)βontoβ(Baseβπ)) | |
22 | foima 6821 | . . . 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 2776 | . 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 21773 | . 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 581 | 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 394 = wceq 1533 β wcel 2098 β wss 3949 β cima 5685 β1-1βwf1 6550 βontoβwfo 6551 β1-1-ontoβwf1o 6552 βcfv 6553 (class class class)co 7426 Basecbs 17187 LSpanclspn 20862 LMHom clmhm 20911 LMIso clmim 20912 LBasisclbs 20966 LIndSclinds 21746 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-ghm 19175 df-mgp 20082 df-ur 20129 df-ring 20182 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lmhm 20914 df-lmim 20915 df-lbs 20967 df-lindf 21747 df-linds 21748 |
This theorem is referenced by: lmiclbs 21778 lmimdim 33334 dimkerim 33358 |
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