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Mirrors > Home > MPE Home > Th. List > indlcim | Structured version Visualization version GIF version |
Description: An independent, spanning family extends to an isomorphism from a free module. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
Ref | Expression |
---|---|
indlcim.f | β’ πΉ = (π freeLMod πΌ) |
indlcim.b | β’ π΅ = (BaseβπΉ) |
indlcim.c | β’ πΆ = (Baseβπ) |
indlcim.v | β’ Β· = ( Β·π βπ) |
indlcim.n | β’ π = (LSpanβπ) |
indlcim.e | β’ πΈ = (π₯ β π΅ β¦ (π Ξ£g (π₯ βf Β· π΄))) |
indlcim.t | β’ (π β π β LMod) |
indlcim.i | β’ (π β πΌ β π) |
indlcim.r | β’ (π β π = (Scalarβπ)) |
indlcim.a | β’ (π β π΄:πΌβontoβπ½) |
indlcim.l | β’ (π β π΄ LIndF π) |
indlcim.s | β’ (π β (πβπ½) = πΆ) |
Ref | Expression |
---|---|
indlcim | β’ (π β πΈ β (πΉ LMIso π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indlcim.f | . . 3 β’ πΉ = (π freeLMod πΌ) | |
2 | indlcim.b | . . 3 β’ π΅ = (BaseβπΉ) | |
3 | indlcim.c | . . 3 β’ πΆ = (Baseβπ) | |
4 | indlcim.v | . . 3 β’ Β· = ( Β·π βπ) | |
5 | indlcim.e | . . 3 β’ πΈ = (π₯ β π΅ β¦ (π Ξ£g (π₯ βf Β· π΄))) | |
6 | indlcim.t | . . 3 β’ (π β π β LMod) | |
7 | indlcim.i | . . 3 β’ (π β πΌ β π) | |
8 | indlcim.r | . . 3 β’ (π β π = (Scalarβπ)) | |
9 | indlcim.a | . . . . 5 β’ (π β π΄:πΌβontoβπ½) | |
10 | fofn 6759 | . . . . 5 β’ (π΄:πΌβontoβπ½ β π΄ Fn πΌ) | |
11 | 9, 10 | syl 17 | . . . 4 β’ (π β π΄ Fn πΌ) |
12 | indlcim.l | . . . . . 6 β’ (π β π΄ LIndF π) | |
13 | 3 | lindff 21237 | . . . . . 6 β’ ((π΄ LIndF π β§ π β LMod) β π΄:dom π΄βΆπΆ) |
14 | 12, 6, 13 | syl2anc 585 | . . . . 5 β’ (π β π΄:dom π΄βΆπΆ) |
15 | 14 | frnd 6677 | . . . 4 β’ (π β ran π΄ β πΆ) |
16 | df-f 6501 | . . . 4 β’ (π΄:πΌβΆπΆ β (π΄ Fn πΌ β§ ran π΄ β πΆ)) | |
17 | 11, 15, 16 | sylanbrc 584 | . . 3 β’ (π β π΄:πΌβΆπΆ) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 17 | frlmup1 21220 | . 2 β’ (π β πΈ β (πΉ LMHom π)) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 17 | islindf5 21261 | . . . 4 β’ (π β (π΄ LIndF π β πΈ:π΅β1-1βπΆ)) |
20 | 12, 19 | mpbid 231 | . . 3 β’ (π β πΈ:π΅β1-1βπΆ) |
21 | indlcim.n | . . . . 5 β’ π = (LSpanβπ) | |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 17, 21 | frlmup3 21222 | . . . 4 β’ (π β ran πΈ = (πβran π΄)) |
23 | forn 6760 | . . . . . 6 β’ (π΄:πΌβontoβπ½ β ran π΄ = π½) | |
24 | 9, 23 | syl 17 | . . . . 5 β’ (π β ran π΄ = π½) |
25 | 24 | fveq2d 6847 | . . . 4 β’ (π β (πβran π΄) = (πβπ½)) |
26 | indlcim.s | . . . 4 β’ (π β (πβπ½) = πΆ) | |
27 | 22, 25, 26 | 3eqtrd 2777 | . . 3 β’ (π β ran πΈ = πΆ) |
28 | dff1o5 6794 | . . 3 β’ (πΈ:π΅β1-1-ontoβπΆ β (πΈ:π΅β1-1βπΆ β§ ran πΈ = πΆ)) | |
29 | 20, 27, 28 | sylanbrc 584 | . 2 β’ (π β πΈ:π΅β1-1-ontoβπΆ) |
30 | 2, 3 | islmim 20538 | . 2 β’ (πΈ β (πΉ LMIso π) β (πΈ β (πΉ LMHom π) β§ πΈ:π΅β1-1-ontoβπΆ)) |
31 | 18, 29, 30 | sylanbrc 584 | 1 β’ (π β πΈ β (πΉ LMIso π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3911 class class class wbr 5106 β¦ cmpt 5189 dom cdm 5634 ran crn 5635 Fn wfn 6492 βΆwf 6493 β1-1βwf1 6494 βontoβwfo 6495 β1-1-ontoβwf1o 6496 βcfv 6497 (class class class)co 7358 βf cof 7616 Basecbs 17088 Scalarcsca 17141 Β·π cvsca 17142 Ξ£g cgsu 17327 LModclmod 20336 LSpanclspn 20447 LMHom clmhm 20495 LMIso clmim 20496 freeLMod cfrlm 21168 LIndF clindf 21226 |
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-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 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-se 5590 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-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-sup 9383 df-oi 9451 df-card 9880 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-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-hom 17162 df-cco 17163 df-0g 17328 df-gsum 17329 df-prds 17334 df-pws 17336 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mulg 18878 df-subg 18930 df-ghm 19011 df-cntz 19102 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-subrg 20234 df-lmod 20338 df-lss 20408 df-lsp 20448 df-lmhm 20498 df-lmim 20499 df-lbs 20551 df-sra 20649 df-rgmod 20650 df-nzr 20744 df-dsmm 21154 df-frlm 21169 df-uvc 21205 df-lindf 21228 |
This theorem is referenced by: lbslcic 21263 |
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