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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem1 | Structured version Visualization version GIF version |
Description: Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.) |
Ref | Expression |
---|---|
hdmap14lem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap14lem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap14lem1.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap14lem1.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmap14lem3.o | ⊢ 0 = (0g‘𝑈) |
hdmap14lem1.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmap14lem1.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmap14lem1.z | ⊢ 𝑍 = (0g‘𝑅) |
hdmap14lem1.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap14lem2.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hdmap14lem1.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap14lem2.p | ⊢ 𝑃 = (Scalar‘𝐶) |
hdmap14lem2.a | ⊢ 𝐴 = (Base‘𝑃) |
hdmap14lem2.q | ⊢ 𝑄 = (0g‘𝑃) |
hdmap14lem1.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap14lem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap14lem3.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem1.f | ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) |
Ref | Expression |
---|---|
hdmap14lem1 | ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem1.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap14lem1.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap14lem1.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap14lem1.t | . 2 ⊢ · = ( ·𝑠 ‘𝑈) | |
5 | hdmap14lem1.r | . 2 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | hdmap14lem1.b | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
7 | hdmap14lem1.c | . 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmap14lem2.e | . 2 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
9 | hdmap14lem1.l | . 2 ⊢ 𝐿 = (LSpan‘𝐶) | |
10 | hdmap14lem2.p | . 2 ⊢ 𝑃 = (Scalar‘𝐶) | |
11 | hdmap14lem2.a | . 2 ⊢ 𝐴 = (Base‘𝑃) | |
12 | hdmap14lem1.s | . 2 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
13 | hdmap14lem1.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
14 | hdmap14lem3.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
15 | 14 | eldifad 3810 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
16 | hdmap14lem1.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) | |
17 | 16 | eldifad 3810 | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
18 | hdmap14lem1.z | . 2 ⊢ 𝑍 = (0g‘𝑅) | |
19 | eldifsni 4542 | . . 3 ⊢ (𝐹 ∈ (𝐵 ∖ {𝑍}) → 𝐹 ≠ 𝑍) | |
20 | 16, 19 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 ≠ 𝑍) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 18, 20 | hdmap14lem1a 37940 | 1 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∖ cdif 3795 {csn 4399 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 Scalarcsca 16315 ·𝑠 cvsca 16316 0gc0g 16460 LSpanclspn 19337 HLchlt 35424 LHypclh 36058 DVecHcdvh 37152 LCDualclcd 37660 HDMapchdma 37866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-riotaBAD 35027 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-ot 4408 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-undef 7669 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-0g 16462 df-mre 16606 df-mrc 16607 df-acs 16609 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-cntz 18107 df-oppg 18133 df-lsm 18409 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-lmod 19228 df-lss 19296 df-lsp 19338 df-lvec 19469 df-lsatoms 35050 df-lshyp 35051 df-lcv 35093 df-lfl 35132 df-lkr 35160 df-ldual 35198 df-oposet 35250 df-ol 35252 df-oml 35253 df-covers 35340 df-ats 35341 df-atl 35372 df-cvlat 35396 df-hlat 35425 df-llines 35572 df-lplanes 35573 df-lvols 35574 df-lines 35575 df-psubsp 35577 df-pmap 35578 df-padd 35870 df-lhyp 36062 df-laut 36063 df-ldil 36178 df-ltrn 36179 df-trl 36233 df-tgrp 36817 df-tendo 36829 df-edring 36831 df-dveca 37077 df-disoa 37103 df-dvech 37153 df-dib 37213 df-dic 37247 df-dih 37303 df-doch 37422 df-djh 37469 df-lcdual 37661 df-mapd 37699 df-hvmap 37831 df-hdmap1 37867 df-hdmap 37868 |
This theorem is referenced by: hdmap14lem2N 37943 hdmap14lem3 37944 |
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