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.4 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐼–1-1→𝐵) | |
2 | ssv 3991 | . . . . . 6 ⊢ ran 𝐹 ⊆ V | |
3 | f1ssr 6581 | . . . . . 6 ⊢ ((𝐹:𝐼–1-1→𝐵 ∧ ran 𝐹 ⊆ V) → 𝐹:𝐼–1-1→V) | |
4 | 1, 2, 3 | sylancl 588 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐼–1-1→V) |
5 | f1dm 6579 | . . . . . 6 ⊢ (𝐹:𝐼–1-1→𝐵 → dom 𝐹 = 𝐼) | |
6 | f1eq2 6571 | . . . . . 6 ⊢ (dom 𝐹 = 𝐼 → (𝐹:dom 𝐹–1-1→V ↔ 𝐹:𝐼–1-1→V)) | |
7 | 1, 5, 6 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹:dom 𝐹–1-1→V ↔ 𝐹:𝐼–1-1→V)) |
8 | 4, 7 | mpbird 259 | . . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹–1-1→V) |
9 | lindflbs.1 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
10 | lindflbs.2 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ NzRing) | |
11 | lindflbs.r | . . . . . 6 ⊢ 𝑆 = (Scalar‘𝑊) | |
12 | 11 | islindf3 20970 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ NzRing) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊)))) |
13 | 9, 10, 12 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊)))) |
14 | 8, 13 | mpbirand 705 | . . 3 ⊢ (𝜑 → (𝐹 LIndF 𝑊 ↔ ran 𝐹 ∈ (LIndS‘𝑊))) |
15 | 14 | anbi1d 631 | . 2 ⊢ (𝜑 → ((𝐹 LIndF 𝑊 ∧ (𝑁‘ran 𝐹) = 𝐵) ↔ (ran 𝐹 ∈ (LIndS‘𝑊) ∧ (𝑁‘ran 𝐹) = 𝐵))) |
16 | lindflbs.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
17 | eqid 2821 | . . 3 ⊢ (LBasis‘𝑊) = (LBasis‘𝑊) | |
18 | lindflbs.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
19 | 16, 17, 18 | islbs4 20976 | . 2 ⊢ (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (ran 𝐹 ∈ (LIndS‘𝑊) ∧ (𝑁‘ran 𝐹) = 𝐵)) |
20 | 15, 19 | syl6rbbr 292 | 1 ⊢ (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (𝐹 LIndF 𝑊 ∧ (𝑁‘ran 𝐹) = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 ran crn 5556 –1-1→wf1 6352 ‘cfv 6355 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 LModclmod 19634 LSpanclspn 19743 LBasisclbs 19846 NzRingcnzr 20030 LIndF clindf 20948 LIndSclinds 20949 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lbs 19847 df-nzr 20031 df-lindf 20950 df-linds 20951 |
This theorem is referenced by: fedgmul 31027 |
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