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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islbs5 | Structured version Visualization version GIF version |
Description: An equivalent formulation of the basis predicate in a vector space, using a function 𝐹 for generating the base. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
Ref | Expression |
---|---|
islbs5.b | ⊢ 𝐵 = (Base‘𝑊) |
islbs5.k | ⊢ 𝐾 = (Base‘𝑆) |
islbs5.r | ⊢ 𝑆 = (Scalar‘𝑊) |
islbs5.t | ⊢ · = ( ·𝑠 ‘𝑊) |
islbs5.z | ⊢ 𝑂 = (0g‘𝑊) |
islbs5.y | ⊢ 0 = (0g‘𝑆) |
islbs5.j | ⊢ 𝐽 = (LBasis‘𝑊) |
islbs5.n | ⊢ 𝑁 = (LSpan‘𝑊) |
islbs5.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
islbs5.s | ⊢ (𝜑 → 𝑆 ∈ NzRing) |
islbs5.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
islbs5.f | ⊢ (𝜑 → 𝐹:𝐼–1-1→𝐵) |
Ref | Expression |
---|---|
islbs5 | ⊢ (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (∀𝑎 ∈ (𝐾 ↑m 𝐼)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })) ∧ (𝑁‘ran 𝐹) = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islbs5.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
2 | eqid 2728 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
3 | islbs5.r | . . 3 ⊢ 𝑆 = (Scalar‘𝑊) | |
4 | islbs5.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
5 | islbs5.z | . . 3 ⊢ 𝑂 = (0g‘𝑊) | |
6 | islbs5.y | . . 3 ⊢ 0 = (0g‘𝑆) | |
7 | islbs5.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
8 | islbs5.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
9 | islbs5.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ NzRing) | |
10 | islbs5.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
11 | islbs5.f | . . 3 ⊢ (𝜑 → 𝐹:𝐼–1-1→𝐵) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | lindflbs 33094 | . 2 ⊢ (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (𝐹 LIndF 𝑊 ∧ (𝑁‘ran 𝐹) = 𝐵))) |
13 | f1f 6793 | . . . . . 6 ⊢ (𝐹:𝐼–1-1→𝐵 → 𝐹:𝐼⟶𝐵) | |
14 | 11, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) |
15 | eqid 2728 | . . . . . 6 ⊢ (Base‘(𝑆 freeLMod 𝐼)) = (Base‘(𝑆 freeLMod 𝐼)) | |
16 | 1, 3, 4, 5, 6, 15 | islindf4 21771 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑎 ∈ (Base‘(𝑆 freeLMod 𝐼))((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 })))) |
17 | 8, 10, 14, 16 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → (𝐹 LIndF 𝑊 ↔ ∀𝑎 ∈ (Base‘(𝑆 freeLMod 𝐼))((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 })))) |
18 | 9 | elexd 3492 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ V) |
19 | eqid 2728 | . . . . . . . . 9 ⊢ (𝑆 freeLMod 𝐼) = (𝑆 freeLMod 𝐼) | |
20 | islbs5.k | . . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑆) | |
21 | 19, 20, 6, 15 | frlmelbas 21689 | . . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐼 ∈ 𝑉) → (𝑎 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↔ (𝑎 ∈ (𝐾 ↑m 𝐼) ∧ 𝑎 finSupp 0 ))) |
22 | 18, 10, 21 | syl2anc 583 | . . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↔ (𝑎 ∈ (𝐾 ↑m 𝐼) ∧ 𝑎 finSupp 0 ))) |
23 | 22 | imbi1d 341 | . . . . . 6 ⊢ (𝜑 → ((𝑎 ∈ (Base‘(𝑆 freeLMod 𝐼)) → ((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 }))) ↔ ((𝑎 ∈ (𝐾 ↑m 𝐼) ∧ 𝑎 finSupp 0 ) → ((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 }))))) |
24 | impexp 450 | . . . . . . 7 ⊢ (((𝑎 ∈ (𝐾 ↑m 𝐼) ∧ 𝑎 finSupp 0 ) → ((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 }))) ↔ (𝑎 ∈ (𝐾 ↑m 𝐼) → (𝑎 finSupp 0 → ((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 }))))) | |
25 | impexp 450 | . . . . . . . . . 10 ⊢ (((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })) ↔ (𝑎 finSupp 0 → ((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 })))) | |
26 | 25 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → (((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })) ↔ (𝑎 finSupp 0 → ((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 }))))) |
27 | 26 | bicomd 222 | . . . . . . . 8 ⊢ (𝜑 → ((𝑎 finSupp 0 → ((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 }))) ↔ ((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })))) |
28 | 27 | imbi2d 340 | . . . . . . 7 ⊢ (𝜑 → ((𝑎 ∈ (𝐾 ↑m 𝐼) → (𝑎 finSupp 0 → ((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 })))) ↔ (𝑎 ∈ (𝐾 ↑m 𝐼) → ((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 }))))) |
29 | 24, 28 | bitrid 283 | . . . . . 6 ⊢ (𝜑 → (((𝑎 ∈ (𝐾 ↑m 𝐼) ∧ 𝑎 finSupp 0 ) → ((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 }))) ↔ (𝑎 ∈ (𝐾 ↑m 𝐼) → ((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 }))))) |
30 | 23, 29 | bitrd 279 | . . . . 5 ⊢ (𝜑 → ((𝑎 ∈ (Base‘(𝑆 freeLMod 𝐼)) → ((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 }))) ↔ (𝑎 ∈ (𝐾 ↑m 𝐼) → ((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 }))))) |
31 | 30 | ralbidv2 3170 | . . . 4 ⊢ (𝜑 → (∀𝑎 ∈ (Base‘(𝑆 freeLMod 𝐼))((𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂 → 𝑎 = (𝐼 × { 0 })) ↔ ∀𝑎 ∈ (𝐾 ↑m 𝐼)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })))) |
32 | 17, 31 | bitrd 279 | . . 3 ⊢ (𝜑 → (𝐹 LIndF 𝑊 ↔ ∀𝑎 ∈ (𝐾 ↑m 𝐼)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })))) |
33 | 32 | anbi1d 630 | . 2 ⊢ (𝜑 → ((𝐹 LIndF 𝑊 ∧ (𝑁‘ran 𝐹) = 𝐵) ↔ (∀𝑎 ∈ (𝐾 ↑m 𝐼)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })) ∧ (𝑁‘ran 𝐹) = 𝐵))) |
34 | 12, 33 | bitrd 279 | 1 ⊢ (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (∀𝑎 ∈ (𝐾 ↑m 𝐼)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })) ∧ (𝑁‘ran 𝐹) = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3058 Vcvv 3471 {csn 4629 class class class wbr 5148 × cxp 5676 ran crn 5679 ⟶wf 6544 –1-1→wf1 6545 ‘cfv 6548 (class class class)co 7420 ∘f cof 7683 ↑m cmap 8844 finSupp cfsupp 9385 Basecbs 17179 Scalarcsca 17235 ·𝑠 cvsca 17236 0gc0g 17420 Σg cgsu 17421 NzRingcnzr 20450 LModclmod 20742 LSpanclspn 20854 LBasisclbs 20958 freeLMod cfrlm 21679 LIndF clindf 21737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-nzr 20451 df-subrg 20507 df-lmod 20744 df-lss 20815 df-lsp 20855 df-lmhm 20906 df-lbs 20959 df-sra 21057 df-rgmod 21058 df-dsmm 21665 df-frlm 21680 df-uvc 21716 df-lindf 21739 df-linds 21740 |
This theorem is referenced by: ply1degltdimlem 33316 |
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