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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 | ⊢ 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 2733 | . . 3 ⊢ (LBasis‘𝑊) = (LBasis‘𝑊) | |
3 | lindflbs.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | 1, 2, 3 | islbs4 21371 | . 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 4005 | . . . . . 6 ⊢ ran 𝐹 ⊆ V | |
7 | f1ssr 6791 | . . . . . 6 ⊢ ((𝐹:𝐼–1-1→𝐵 ∧ ran 𝐹 ⊆ V) → 𝐹:𝐼–1-1→V) | |
8 | 5, 6, 7 | sylancl 587 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐼–1-1→V) |
9 | f1dm 6788 | . . . . . 6 ⊢ (𝐹:𝐼–1-1→𝐵 → dom 𝐹 = 𝐼) | |
10 | f1eq2 6780 | . . . . . 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 21365 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ NzRing) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊)))) |
17 | 13, 14, 16 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊)))) |
18 | 12, 17 | mpbirand 706 | . . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3947 class class class wbr 5147 dom cdm 5675 ran crn 5676 –1-1→wf1 6537 ‘cfv 6540 Basecbs 17140 Scalarcsca 17196 ·𝑠 cvsca 17197 0gc0g 17381 NzRingcnzr 20280 LModclmod 20459 LSpanclspn 20570 LBasisclbs 20673 LIndF clindf 21343 LIndSclinds 21344 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-mgp 19980 df-ur 19997 df-ring 20049 df-nzr 20281 df-lmod 20461 df-lss 20531 df-lsp 20571 df-lbs 20674 df-lindf 21345 df-linds 21346 |
This theorem is referenced by: islbs5 32461 fedgmul 32661 |
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