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Mirrors > Home > MPE Home > Th. List > Mathboxes > lincreslvec3 | Structured version Visualization version GIF version |
Description: Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.) |
Ref | Expression |
---|---|
lincresunit.b | ⊢ 𝐵 = (Base‘𝑀) |
lincresunit.r | ⊢ 𝑅 = (Scalar‘𝑀) |
lincresunit.e | ⊢ 𝐸 = (Base‘𝑅) |
lincresunit.u | ⊢ 𝑈 = (Unit‘𝑅) |
lincresunit.0 | ⊢ 0 = (0g‘𝑅) |
lincresunit.z | ⊢ 𝑍 = (0g‘𝑀) |
lincresunit.n | ⊢ 𝑁 = (invg‘𝑅) |
lincresunit.i | ⊢ 𝐼 = (invr‘𝑅) |
lincresunit.t | ⊢ · = (.r‘𝑅) |
lincresunit.g | ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) |
Ref | Expression |
---|---|
lincreslvec3 | ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lveclmod 21084 | . . . 4 ⊢ (𝑀 ∈ LVec → 𝑀 ∈ LMod) | |
2 | 1 | 3anim2i 1150 | . . 3 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆)) |
3 | 2 | 3ad2ant1 1130 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆)) |
4 | simp21 1203 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝐹 ∈ (𝐸 ↑m 𝑆)) | |
5 | elmapi 8878 | . . . . . 6 ⊢ (𝐹 ∈ (𝐸 ↑m 𝑆) → 𝐹:𝑆⟶𝐸) | |
6 | 5 | 3ad2ant1 1130 | . . . . 5 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) → 𝐹:𝑆⟶𝐸) |
7 | simp3 1135 | . . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
8 | ffvelcdm 7095 | . . . . 5 ⊢ ((𝐹:𝑆⟶𝐸 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ 𝐸) | |
9 | 6, 7, 8 | syl2anr 595 | . . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝐸) |
10 | simpr2 1192 | . . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ≠ 0 ) | |
11 | lincresunit.r | . . . . . . . 8 ⊢ 𝑅 = (Scalar‘𝑀) | |
12 | 11 | lvecdrng 21083 | . . . . . . 7 ⊢ (𝑀 ∈ LVec → 𝑅 ∈ DivRing) |
13 | 12 | 3ad2ant2 1131 | . . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) → 𝑅 ∈ DivRing) |
14 | 13 | adantr 479 | . . . . 5 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 )) → 𝑅 ∈ DivRing) |
15 | lincresunit.e | . . . . . 6 ⊢ 𝐸 = (Base‘𝑅) | |
16 | lincresunit.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
17 | lincresunit.0 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
18 | 15, 16, 17 | drngunit 20712 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((𝐹‘𝑋) ∈ 𝑈 ↔ ((𝐹‘𝑋) ∈ 𝐸 ∧ (𝐹‘𝑋) ≠ 0 ))) |
19 | 14, 18 | syl 17 | . . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 )) → ((𝐹‘𝑋) ∈ 𝑈 ↔ ((𝐹‘𝑋) ∈ 𝐸 ∧ (𝐹‘𝑋) ≠ 0 ))) |
20 | 9, 10, 19 | mpbir2and 711 | . . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝑈) |
21 | 20 | 3adant3 1129 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐹‘𝑋) ∈ 𝑈) |
22 | simp3 1135 | . . 3 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) → 𝐹 finSupp 0 ) | |
23 | 22 | 3ad2ant2 1131 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝐹 finSupp 0 ) |
24 | simp3 1135 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐹( linC ‘𝑀)𝑆) = 𝑍) | |
25 | lincresunit.b | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
26 | lincresunit.z | . . 3 ⊢ 𝑍 = (0g‘𝑀) | |
27 | lincresunit.n | . . 3 ⊢ 𝑁 = (invg‘𝑅) | |
28 | lincresunit.i | . . 3 ⊢ 𝐼 = (invr‘𝑅) | |
29 | lincresunit.t | . . 3 ⊢ · = (.r‘𝑅) | |
30 | lincresunit.g | . . 3 ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) | |
31 | 25, 11, 15, 16, 17, 26, 27, 28, 29, 30 | lincresunit3 47864 | . 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋) |
32 | 3, 4, 21, 23, 24, 31 | syl131anc 1380 | 1 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3944 𝒫 cpw 4607 {csn 4633 class class class wbr 5153 ↦ cmpt 5236 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ↑m cmap 8855 finSupp cfsupp 9405 Basecbs 17213 .rcmulr 17267 Scalarcsca 17269 0gc0g 17454 invgcminusg 18929 Unitcui 20337 invrcinvr 20369 DivRingcdr 20707 LModclmod 20836 LVecclvec 21080 linC clinc 47787 |
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 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-seq 14022 df-hash 14348 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-0g 17456 df-gsum 17457 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-grp 18931 df-minusg 18932 df-mulg 19062 df-ghm 19207 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-drng 20709 df-lmod 20838 df-lvec 21081 df-linc 47789 |
This theorem is referenced by: isldepslvec2 47868 |
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