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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 20795 | . . . . 5 ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil) | |
2 | phllvec 20701 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | |
3 | eqid 2818 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | 3 | lvecdrng 19806 | . . . . 5 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing) |
5 | 1, 2, 4 | 3syl 18 | . . . 4 ⊢ (𝐵 ∈ (OBasis‘𝑊) → (Scalar‘𝑊) ∈ DivRing) |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (Scalar‘𝑊) ∈ DivRing) |
7 | eqid 2818 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
8 | eqid 2818 | . . . 4 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
9 | 7, 8 | drngunz 19446 | . . 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 2818 | . . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
12 | 11, 3, 8 | obsipid 20794 | . . . . 5 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴(·𝑖‘𝑊)𝐴) = (1r‘(Scalar‘𝑊))) |
13 | 12 | eqeq1d 2820 | . . . 4 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → ((𝐴(·𝑖‘𝑊)𝐴) = (0g‘(Scalar‘𝑊)) ↔ (1r‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)))) |
14 | eqid 2818 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
15 | 14 | obsss 20796 | . . . . . 6 ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝐵 ⊆ (Base‘𝑊)) |
16 | 15 | sselda 3964 | . . . . 5 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (Base‘𝑊)) |
17 | obsocv.z | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
18 | 3, 11, 14, 7, 17 | ipeq0 20710 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ (Base‘𝑊)) → ((𝐴(·𝑖‘𝑊)𝐴) = (0g‘(Scalar‘𝑊)) ↔ 𝐴 = 0 )) |
19 | 1, 16, 18 | syl2an2r 681 | . . . 4 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → ((𝐴(·𝑖‘𝑊)𝐴) = (0g‘(Scalar‘𝑊)) ↔ 𝐴 = 0 )) |
20 | 13, 19 | bitr3d 282 | . . 3 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → ((1r‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) ↔ 𝐴 = 0 )) |
21 | 20 | necon3bid 3057 | . 2 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → ((1r‘(Scalar‘𝑊)) ≠ (0g‘(Scalar‘𝑊)) ↔ 𝐴 ≠ 0 )) |
22 | 10, 21 | mpbid 233 | 1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Scalarcsca 16556 ·𝑖cip 16558 0gc0g 16701 1rcur 19180 DivRingcdr 19431 LVecclvec 19803 PreHilcphl 20696 OBasiscobs 20774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-ghm 18294 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-drng 19433 df-lmod 19565 df-lmhm 19723 df-lvec 19804 df-sra 19873 df-rgmod 19874 df-phl 20698 df-obs 20777 |
This theorem is referenced by: obselocv 20800 obs2ss 20801 obslbs 20802 |
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