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Mirrors > Home > MPE Home > Th. List > obsne0 | Structured version Visualization version GIF version |
Description: A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
obsocv.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
obsne0 | ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | obsrcl 20392 | . . . . 5 ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil) | |
2 | phllvec 20298 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | |
3 | eqid 2799 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | 3 | lvecdrng 19426 | . . . . 5 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing) |
5 | 1, 2, 4 | 3syl 18 | . . . 4 ⊢ (𝐵 ∈ (OBasis‘𝑊) → (Scalar‘𝑊) ∈ DivRing) |
6 | 5 | adantr 473 | . . 3 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (Scalar‘𝑊) ∈ DivRing) |
7 | eqid 2799 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
8 | eqid 2799 | . . . 4 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
9 | 7, 8 | drngunz 19080 | . . 3 ⊢ ((Scalar‘𝑊) ∈ DivRing → (1r‘(Scalar‘𝑊)) ≠ (0g‘(Scalar‘𝑊))) |
10 | 6, 9 | syl 17 | . 2 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (1r‘(Scalar‘𝑊)) ≠ (0g‘(Scalar‘𝑊))) |
11 | eqid 2799 | . . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
12 | 11, 3, 8 | obsipid 20391 | . . . . 5 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴(·𝑖‘𝑊)𝐴) = (1r‘(Scalar‘𝑊))) |
13 | 12 | eqeq1d 2801 | . . . 4 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → ((𝐴(·𝑖‘𝑊)𝐴) = (0g‘(Scalar‘𝑊)) ↔ (1r‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)))) |
14 | 1 | adantr 473 | . . . . 5 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝑊 ∈ PreHil) |
15 | eqid 2799 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
16 | 15 | obsss 20393 | . . . . . 6 ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝐵 ⊆ (Base‘𝑊)) |
17 | 16 | sselda 3798 | . . . . 5 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (Base‘𝑊)) |
18 | obsocv.z | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
19 | 3, 11, 15, 7, 18 | ipeq0 20307 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ (Base‘𝑊)) → ((𝐴(·𝑖‘𝑊)𝐴) = (0g‘(Scalar‘𝑊)) ↔ 𝐴 = 0 )) |
20 | 14, 17, 19 | syl2anc 580 | . . . 4 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → ((𝐴(·𝑖‘𝑊)𝐴) = (0g‘(Scalar‘𝑊)) ↔ 𝐴 = 0 )) |
21 | 13, 20 | bitr3d 273 | . . 3 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → ((1r‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) ↔ 𝐴 = 0 )) |
22 | 21 | necon3bid 3015 | . 2 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → ((1r‘(Scalar‘𝑊)) ≠ (0g‘(Scalar‘𝑊)) ↔ 𝐴 ≠ 0 )) |
23 | 10, 22 | mpbid 224 | 1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 Scalarcsca 16270 ·𝑖cip 16272 0gc0g 16415 1rcur 18817 DivRingcdr 19065 LVecclvec 19423 PreHilcphl 20293 OBasiscobs 20371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-tpos 7590 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-ip 16285 df-0g 16417 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-grp 17741 df-ghm 17971 df-mgp 18806 df-ur 18818 df-ring 18865 df-oppr 18939 df-dvdsr 18957 df-unit 18958 df-drng 19067 df-lmod 19183 df-lmhm 19343 df-lvec 19424 df-sra 19495 df-rgmod 19496 df-phl 20295 df-obs 20374 |
This theorem is referenced by: obselocv 20397 obs2ss 20398 obslbs 20399 |
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