Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lindflbs | Structured version Visualization version GIF version |
Description: Conditions for an independent family to be a basis. (Contributed by Thierry Arnoux, 21-Jul-2023.) |
Ref | Expression |
---|---|
lindflbs.b | β’ π΅ = (Baseβπ) |
lindflbs.k | β’ πΎ = (BaseβπΉ) |
lindflbs.r | β’ π = (Scalarβπ) |
lindflbs.t | β’ Β· = ( Β·π βπ) |
lindflbs.z | β’ π = (0gβπ) |
lindflbs.y | β’ 0 = (0gβπ) |
lindflbs.n | β’ π = (LSpanβπ) |
lindflbs.1 | β’ (π β π β LMod) |
lindflbs.2 | β’ (π β π β NzRing) |
lindflbs.3 | β’ (π β πΌ β π) |
lindflbs.4 | β’ (π β πΉ:πΌβ1-1βπ΅) |
Ref | Expression |
---|---|
lindflbs | β’ (π β (ran πΉ β (LBasisβπ) β (πΉ LIndF π β§ (πβran πΉ) = π΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lindflbs.b | . . 3 β’ π΅ = (Baseβπ) | |
2 | eqid 2736 | . . 3 β’ (LBasisβπ) = (LBasisβπ) | |
3 | lindflbs.n | . . 3 β’ π = (LSpanβπ) | |
4 | 1, 2, 3 | islbs4 21145 | . 2 β’ (ran πΉ β (LBasisβπ) β (ran πΉ β (LIndSβπ) β§ (πβran πΉ) = π΅)) |
5 | lindflbs.4 | . . . . . 6 β’ (π β πΉ:πΌβ1-1βπ΅) | |
6 | ssv 3956 | . . . . . 6 β’ ran πΉ β V | |
7 | f1ssr 6728 | . . . . . 6 β’ ((πΉ:πΌβ1-1βπ΅ β§ ran πΉ β V) β πΉ:πΌβ1-1βV) | |
8 | 5, 6, 7 | sylancl 586 | . . . . 5 β’ (π β πΉ:πΌβ1-1βV) |
9 | f1dm 6725 | . . . . . 6 β’ (πΉ:πΌβ1-1βπ΅ β dom πΉ = πΌ) | |
10 | f1eq2 6717 | . . . . . 6 β’ (dom πΉ = πΌ β (πΉ:dom πΉβ1-1βV β πΉ:πΌβ1-1βV)) | |
11 | 5, 9, 10 | 3syl 18 | . . . . 5 β’ (π β (πΉ:dom πΉβ1-1βV β πΉ:πΌβ1-1βV)) |
12 | 8, 11 | mpbird 256 | . . . 4 β’ (π β πΉ:dom πΉβ1-1βV) |
13 | lindflbs.1 | . . . . 5 β’ (π β π β LMod) | |
14 | lindflbs.2 | . . . . 5 β’ (π β π β NzRing) | |
15 | lindflbs.r | . . . . . 6 β’ π = (Scalarβπ) | |
16 | 15 | islindf3 21139 | . . . . 5 β’ ((π β LMod β§ π β NzRing) β (πΉ LIndF π β (πΉ:dom πΉβ1-1βV β§ ran πΉ β (LIndSβπ)))) |
17 | 13, 14, 16 | syl2anc 584 | . . . 4 β’ (π β (πΉ LIndF π β (πΉ:dom πΉβ1-1βV β§ ran πΉ β (LIndSβπ)))) |
18 | 12, 17 | mpbirand 704 | . . 3 β’ (π β (πΉ LIndF π β ran πΉ β (LIndSβπ))) |
19 | 18 | anbi1d 630 | . 2 β’ (π β ((πΉ LIndF π β§ (πβran πΉ) = π΅) β (ran πΉ β (LIndSβπ) β§ (πβran πΉ) = π΅))) |
20 | 4, 19 | bitr4id 289 | 1 β’ (π β (ran πΉ β (LBasisβπ) β (πΉ LIndF π β§ (πβran πΉ) = π΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1540 β wcel 2105 Vcvv 3441 β wss 3898 class class class wbr 5092 dom cdm 5620 ran crn 5621 β1-1βwf1 6476 βcfv 6479 Basecbs 17009 Scalarcsca 17062 Β·π cvsca 17063 0gc0g 17247 LModclmod 20229 LSpanclspn 20339 LBasisclbs 20442 NzRingcnzr 20634 LIndF clindf 21117 LIndSclinds 21118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-plusg 17072 df-0g 17249 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-grp 18676 df-mgp 19816 df-ur 19833 df-ring 19880 df-lmod 20231 df-lss 20300 df-lsp 20340 df-lbs 20443 df-nzr 20635 df-lindf 21119 df-linds 21120 |
This theorem is referenced by: fedgmul 32010 |
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