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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 2735 | . . 3 ⊢ (LBasis‘𝑊) = (LBasis‘𝑊) | |
3 | lindflbs.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | 1, 2, 3 | islbs4 21870 | . 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 4020 | . . . . . 6 ⊢ ran 𝐹 ⊆ V | |
7 | f1ssr 6811 | . . . . . 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 6809 | . . . . . 6 ⊢ (𝐹:𝐼–1-1→𝐵 → dom 𝐹 = 𝐼) | |
10 | f1eq2 6801 | . . . . . 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 21864 | . . . . 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 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 dom cdm 5689 ran crn 5690 –1-1→wf1 6560 ‘cfv 6563 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 NzRingcnzr 20529 LModclmod 20875 LSpanclspn 20987 LBasisclbs 21091 LIndF clindf 21842 LIndSclinds 21843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-mgp 20153 df-ur 20200 df-ring 20253 df-nzr 20530 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lbs 21092 df-lindf 21844 df-linds 21845 |
This theorem is referenced by: islbs5 33388 fedgmul 33659 |
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