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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapip0 | Structured version Visualization version GIF version |
Description: Zero property that will be used for inner product. (Contributed by NM, 9-Jun-2015.) |
Ref | Expression |
---|---|
hdmapip0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapip0.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapip0.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapip0.o | ⊢ 0 = (0g‘𝑈) |
hdmapip0.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmapip0.z | ⊢ 𝑍 = (0g‘𝑅) |
hdmapip0.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapip0.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapip0.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
hdmapip0 | ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 ↔ 𝑋 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapip0.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2726 | . . . . . . . 8 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
3 | hdmapip0.u | . . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | hdmapip0.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈) | |
5 | hdmapip0.o | . . . . . . . 8 ⊢ 0 = (0g‘𝑈) | |
6 | hdmapip0.k | . . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 6 | adantr 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
8 | hdmapip0.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
9 | 8 | anim1i 613 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) |
10 | eldifsn 4795 | . . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
11 | 9, 10 | sylibr 233 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
12 | 1, 2, 3, 4, 5, 7, 11 | dochnel 41092 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ¬ 𝑋 ∈ (((ocH‘𝐾)‘𝑊)‘{𝑋})) |
13 | hdmapip0.r | . . . . . . . . . . . 12 ⊢ 𝑅 = (Scalar‘𝑈) | |
14 | hdmapip0.z | . . . . . . . . . . . 12 ⊢ 𝑍 = (0g‘𝑅) | |
15 | eqid 2726 | . . . . . . . . . . . 12 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
16 | eqid 2726 | . . . . . . . . . . . 12 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
17 | 1, 3, 6 | dvhlmod 40809 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LMod) |
18 | eqid 2726 | . . . . . . . . . . . . 13 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
19 | eqid 2726 | . . . . . . . . . . . . 13 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊)) | |
20 | hdmapip0.s | . . . . . . . . . . . . . 14 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
21 | 1, 3, 4, 18, 19, 20, 6, 8 | hdmapcl 41529 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘((LCDual‘𝐾)‘𝑊))) |
22 | 1, 18, 19, 3, 15, 6, 21 | lcdvbaselfl 41294 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (LFnl‘𝑈)) |
23 | 4, 13, 14, 15, 16, 17, 22, 8 | ellkr2 38789 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((LKer‘𝑈)‘(𝑆‘𝑋)) ↔ ((𝑆‘𝑋)‘𝑋) = 𝑍)) |
24 | 23 | biimpar 476 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → 𝑋 ∈ ((LKer‘𝑈)‘(𝑆‘𝑋))) |
25 | 1, 2, 3, 4, 15, 16, 20, 6, 8 | hdmaplkr 41612 | . . . . . . . . . . 11 ⊢ (𝜑 → ((LKer‘𝑈)‘(𝑆‘𝑋)) = (((ocH‘𝐾)‘𝑊)‘{𝑋})) |
26 | 25 | adantr 479 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → ((LKer‘𝑈)‘(𝑆‘𝑋)) = (((ocH‘𝐾)‘𝑊)‘{𝑋})) |
27 | 24, 26 | eleqtrd 2828 | . . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → 𝑋 ∈ (((ocH‘𝐾)‘𝑊)‘{𝑋})) |
28 | 27 | ex 411 | . . . . . . . 8 ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 → 𝑋 ∈ (((ocH‘𝐾)‘𝑊)‘{𝑋}))) |
29 | 28 | adantr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (((𝑆‘𝑋)‘𝑋) = 𝑍 → 𝑋 ∈ (((ocH‘𝐾)‘𝑊)‘{𝑋}))) |
30 | 12, 29 | mtod 197 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ¬ ((𝑆‘𝑋)‘𝑋) = 𝑍) |
31 | 30 | neqned 2937 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ((𝑆‘𝑋)‘𝑋) ≠ 𝑍) |
32 | 31 | ex 411 | . . . 4 ⊢ (𝜑 → (𝑋 ≠ 0 → ((𝑆‘𝑋)‘𝑋) ≠ 𝑍)) |
33 | 32 | necon4d 2954 | . . 3 ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 → 𝑋 = 0 )) |
34 | 33 | imp 405 | . 2 ⊢ ((𝜑 ∧ ((𝑆‘𝑋)‘𝑋) = 𝑍) → 𝑋 = 0 ) |
35 | fveq2 6901 | . . 3 ⊢ (𝑋 = 0 → ((𝑆‘𝑋)‘𝑋) = ((𝑆‘𝑋)‘ 0 )) | |
36 | 13, 14, 5, 15 | lfl0 38763 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑆‘𝑋) ∈ (LFnl‘𝑈)) → ((𝑆‘𝑋)‘ 0 ) = 𝑍) |
37 | 17, 22, 36 | syl2anc 582 | . . 3 ⊢ (𝜑 → ((𝑆‘𝑋)‘ 0 ) = 𝑍) |
38 | 35, 37 | sylan9eqr 2788 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ((𝑆‘𝑋)‘𝑋) = 𝑍) |
39 | 34, 38 | impbida 799 | 1 ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3944 {csn 4633 ‘cfv 6554 Basecbs 17213 Scalarcsca 17269 0gc0g 17454 LModclmod 20836 LFnlclfn 38755 LKerclk 38783 HLchlt 39048 LHypclh 39683 DVecHcdvh 40777 ocHcoch 41046 LCDualclcd 41285 HDMapchdma 41491 |
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 ax-riotaBAD 38651 |
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-tp 4638 df-op 4640 df-ot 4642 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-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-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-undef 8288 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-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-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-0g 17456 df-mre 17599 df-mrc 17600 df-acs 17602 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-p1 18451 df-lat 18457 df-clat 18524 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-subg 19117 df-cntz 19311 df-oppg 19340 df-lsm 19634 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-dvr 20383 df-nzr 20495 df-rlreg 20672 df-domn 20673 df-drng 20709 df-lmod 20838 df-lss 20909 df-lsp 20949 df-lvec 21081 df-lsatoms 38674 df-lshyp 38675 df-lcv 38717 df-lfl 38756 df-lkr 38784 df-ldual 38822 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-llines 39197 df-lplanes 39198 df-lvols 39199 df-lines 39200 df-psubsp 39202 df-pmap 39203 df-padd 39495 df-lhyp 39687 df-laut 39688 df-ldil 39803 df-ltrn 39804 df-trl 39858 df-tgrp 40442 df-tendo 40454 df-edring 40456 df-dveca 40702 df-disoa 40728 df-dvech 40778 df-dib 40838 df-dic 40872 df-dih 40928 df-doch 41047 df-djh 41094 df-lcdual 41286 df-mapd 41324 df-hvmap 41456 df-hdmap1 41492 df-hdmap 41493 |
This theorem is referenced by: hdmapip1 41615 hgmapvvlem3 41624 hlhilphllem 41662 |
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