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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 20540 | . . 3 β’ (πΉ β (π LMIso π) β πΉ β (π LMHom π)) | |
2 | eqid 2733 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
3 | eqid 2733 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
4 | 2, 3 | lmimf1o 20539 | . . . 4 β’ (πΉ β (π LMIso π) β πΉ:(Baseβπ)β1-1-ontoβ(Baseβπ)) |
5 | f1of1 6784 | . . . 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 21255 | . . . 4 β’ π½ β (LIndSβπ) |
9 | 8 | sseli 3941 | . . 3 β’ (π΅ β π½ β π΅ β (LIndSβπ)) |
10 | 2, 3 | lindsmm2 21251 | . . 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 1423 | . 2 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β π΅) β (LIndSβπ)) |
12 | eqid 2733 | . . . . . 6 β’ (LSpanβπ) = (LSpanβπ) | |
13 | 2, 7, 12 | lbssp 20555 | . . . . 5 β’ (π΅ β π½ β ((LSpanβπ)βπ΅) = (Baseβπ)) |
14 | 13 | adantl 483 | . . . 4 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β ((LSpanβπ)βπ΅) = (Baseβπ)) |
15 | 14 | imaeq2d 6014 | . . 3 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β ((LSpanβπ)βπ΅)) = (πΉ β (Baseβπ))) |
16 | 2, 7 | lbsss 20553 | . . . 4 β’ (π΅ β π½ β π΅ β (Baseβπ)) |
17 | eqid 2733 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
18 | 2, 12, 17 | lmhmlsp 20525 | . . . 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 597 | . . 3 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β ((LSpanβπ)βπ΅)) = ((LSpanβπ)β(πΉ β π΅))) |
20 | 4 | adantr 482 | . . . 4 β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β πΉ:(Baseβπ)β1-1-ontoβ(Baseβπ)) |
21 | f1ofo 6792 | . . . 4 β’ (πΉ:(Baseβπ)β1-1-ontoβ(Baseβπ) β πΉ:(Baseβπ)βontoβ(Baseβπ)) | |
22 | foima 6762 | . . . 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 2781 | . 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 21254 | . 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 584 | 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 397 = wceq 1542 β wcel 2107 β wss 3911 β cima 5637 β1-1βwf1 6494 βontoβwfo 6495 β1-1-ontoβwf1o 6496 βcfv 6497 (class class class)co 7358 Basecbs 17088 LSpanclspn 20447 LMHom clmhm 20495 LMIso clmim 20496 LBasisclbs 20550 LIndSclinds 21227 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-ghm 19011 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lss 20408 df-lsp 20448 df-lmhm 20498 df-lmim 20499 df-lbs 20551 df-lindf 21228 df-linds 21229 |
This theorem is referenced by: lmiclbs 21259 dimkerim 32379 |
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