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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 | ⊢ 𝐵 = (Base‘𝑊) | 
| lindflbs.k | ⊢ 𝐾 = (Base‘𝐹) | 
| lindflbs.r | ⊢ 𝑆 = (Scalar‘𝑊) | 
| lindflbs.t | ⊢ · = ( ·𝑠 ‘𝑊) | 
| lindflbs.z | ⊢ 𝑂 = (0g‘𝑊) | 
| lindflbs.y | ⊢ 0 = (0g‘𝑆) | 
| lindflbs.n | ⊢ 𝑁 = (LSpan‘𝑊) | 
| lindflbs.1 | ⊢ (𝜑 → 𝑊 ∈ LMod) | 
| lindflbs.2 | ⊢ (𝜑 → 𝑆 ∈ NzRing) | 
| lindflbs.3 | ⊢ (𝜑 → 𝐼 ∈ 𝑉) | 
| lindflbs.4 | ⊢ (𝜑 → 𝐹:𝐼–1-1→𝐵) | 
| Ref | Expression | 
|---|---|
| lindflbs | ⊢ (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (𝐹 LIndF 𝑊 ∧ (𝑁‘ran 𝐹) = 𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lindflbs.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
| 2 | eqid 2736 | . . 3 ⊢ (LBasis‘𝑊) = (LBasis‘𝑊) | |
| 3 | lindflbs.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 4 | 1, 2, 3 | islbs4 21853 | . 2 ⊢ (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (ran 𝐹 ∈ (LIndS‘𝑊) ∧ (𝑁‘ran 𝐹) = 𝐵)) | 
| 5 | lindflbs.4 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐼–1-1→𝐵) | |
| 6 | ssv 4007 | . . . . . 6 ⊢ ran 𝐹 ⊆ V | |
| 7 | f1ssr 6809 | . . . . . 6 ⊢ ((𝐹:𝐼–1-1→𝐵 ∧ ran 𝐹 ⊆ V) → 𝐹:𝐼–1-1→V) | |
| 8 | 5, 6, 7 | sylancl 586 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐼–1-1→V) | 
| 9 | f1dm 6807 | . . . . . 6 ⊢ (𝐹:𝐼–1-1→𝐵 → dom 𝐹 = 𝐼) | |
| 10 | f1eq2 6799 | . . . . . 6 ⊢ (dom 𝐹 = 𝐼 → (𝐹:dom 𝐹–1-1→V ↔ 𝐹:𝐼–1-1→V)) | |
| 11 | 5, 9, 10 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹:dom 𝐹–1-1→V ↔ 𝐹:𝐼–1-1→V)) | 
| 12 | 8, 11 | mpbird 257 | . . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹–1-1→V) | 
| 13 | lindflbs.1 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
| 14 | lindflbs.2 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ NzRing) | |
| 15 | lindflbs.r | . . . . . 6 ⊢ 𝑆 = (Scalar‘𝑊) | |
| 16 | 15 | islindf3 21847 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ NzRing) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊)))) | 
| 17 | 13, 14, 16 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊)))) | 
| 18 | 12, 17 | mpbirand 707 | . . 3 ⊢ (𝜑 → (𝐹 LIndF 𝑊 ↔ ran 𝐹 ∈ (LIndS‘𝑊))) | 
| 19 | 18 | anbi1d 631 | . 2 ⊢ (𝜑 → ((𝐹 LIndF 𝑊 ∧ (𝑁‘ran 𝐹) = 𝐵) ↔ (ran 𝐹 ∈ (LIndS‘𝑊) ∧ (𝑁‘ran 𝐹) = 𝐵))) | 
| 20 | 4, 19 | bitr4id 290 | 1 ⊢ (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (𝐹 LIndF 𝑊 ∧ (𝑁‘ran 𝐹) = 𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 class class class wbr 5142 dom cdm 5684 ran crn 5685 –1-1→wf1 6557 ‘cfv 6560 Basecbs 17248 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17485 NzRingcnzr 20513 LModclmod 20859 LSpanclspn 20970 LBasisclbs 21074 LIndF clindf 21825 LIndSclinds 21826 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-mgp 20139 df-ur 20180 df-ring 20233 df-nzr 20514 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lbs 21075 df-lindf 21827 df-linds 21828 | 
| This theorem is referenced by: islbs5 33409 fedgmul 33683 | 
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