![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 2731 | . . 3 β’ (LBasisβπ) = (LBasisβπ) | |
3 | lindflbs.n | . . 3 β’ π = (LSpanβπ) | |
4 | 1, 2, 3 | islbs4 21607 | . 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 4006 | . . . . . 6 β’ ran πΉ β V | |
7 | f1ssr 6794 | . . . . . 6 β’ ((πΉ:πΌβ1-1βπ΅ β§ ran πΉ β V) β πΉ:πΌβ1-1βV) | |
8 | 5, 6, 7 | sylancl 585 | . . . . 5 β’ (π β πΉ:πΌβ1-1βV) |
9 | f1dm 6791 | . . . . . 6 β’ (πΉ:πΌβ1-1βπ΅ β dom πΉ = πΌ) | |
10 | f1eq2 6783 | . . . . . 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 257 | . . . 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 21601 | . . . . 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 583 | . . . 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 629 | . 2 β’ (π β ((πΉ LIndF π β§ (πβran πΉ) = π΅) β (ran πΉ β (LIndSβπ) β§ (πβran πΉ) = π΅))) |
20 | 4, 19 | bitr4id 290 | 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 395 = wceq 1540 β wcel 2105 Vcvv 3473 β wss 3948 class class class wbr 5148 dom cdm 5676 ran crn 5677 β1-1βwf1 6540 βcfv 6543 Basecbs 17149 Scalarcsca 17205 Β·π cvsca 17206 0gc0g 17390 NzRingcnzr 20404 LModclmod 20615 LSpanclspn 20727 LBasisclbs 20830 LIndF clindf 21579 LIndSclinds 21580 |
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 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-mgp 20030 df-ur 20077 df-ring 20130 df-nzr 20405 df-lmod 20617 df-lss 20688 df-lsp 20728 df-lbs 20831 df-lindf 21581 df-linds 21582 |
This theorem is referenced by: islbs5 32771 fedgmul 33005 |
Copyright terms: Public domain | W3C validator |