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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapglnm2 | Structured version Visualization version GIF version |
Description: g-linear property of second (inner product) argument. Line 19 in [Holland95] p. 14. (Contributed by NM, 10-Jun-2015.) |
Ref | Expression |
---|---|
hdmapglnm2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapglnm2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapglnm2.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapglnm2.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmapglnm2.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmapglnm2.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmapglnm2.m | ⊢ × = (.r‘𝑅) |
hdmapglnm2.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapglnm2.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hdmapglnm2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapglnm2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmapglnm2.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
hdmapglnm2.z | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
Ref | Expression |
---|---|
hdmapglnm2 | ⊢ (𝜑 → ((𝑆‘(𝐴 · 𝑌))‘𝑋) = (((𝑆‘𝑌)‘𝑋) × (𝐺‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapglnm2.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmapglnm2.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmapglnm2.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmapglnm2.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑈) | |
5 | hdmapglnm2.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | hdmapglnm2.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
7 | eqid 2738 | . . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
8 | eqid 2738 | . . . 4 ⊢ ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) | |
9 | hdmapglnm2.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
10 | hdmapglnm2.g | . . . 4 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
11 | hdmapglnm2.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | hdmapglnm2.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
13 | hdmapglnm2.z | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | hgmapvs 39891 | . . 3 ⊢ (𝜑 → (𝑆‘(𝐴 · 𝑌)) = ((𝐺‘𝐴)( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝑌))) |
15 | 14 | fveq1d 6769 | . 2 ⊢ (𝜑 → ((𝑆‘(𝐴 · 𝑌))‘𝑋) = (((𝐺‘𝐴)( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝑌))‘𝑋)) |
16 | hdmapglnm2.m | . . 3 ⊢ × = (.r‘𝑅) | |
17 | eqid 2738 | . . 3 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊)) | |
18 | 1, 2, 5, 6, 10, 11, 13 | hgmapcl 39889 | . . 3 ⊢ (𝜑 → (𝐺‘𝐴) ∈ 𝐵) |
19 | 1, 2, 3, 7, 17, 9, 11, 12 | hdmapcl 39830 | . . 3 ⊢ (𝜑 → (𝑆‘𝑌) ∈ (Base‘((LCDual‘𝐾)‘𝑊))) |
20 | hdmapglnm2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
21 | 1, 2, 3, 5, 6, 16, 7, 17, 8, 11, 18, 19, 20 | lcdvsval 39604 | . 2 ⊢ (𝜑 → (((𝐺‘𝐴)( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝑌))‘𝑋) = (((𝑆‘𝑌)‘𝑋) × (𝐺‘𝐴))) |
22 | 15, 21 | eqtrd 2778 | 1 ⊢ (𝜑 → ((𝑆‘(𝐴 · 𝑌))‘𝑋) = (((𝑆‘𝑌)‘𝑋) × (𝐺‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ‘cfv 6427 (class class class)co 7268 Basecbs 16900 .rcmulr 16951 Scalarcsca 16953 ·𝑠 cvsca 16954 HLchlt 37350 LHypclh 37984 DVecHcdvh 39078 LCDualclcd 39586 HDMapchdma 39792 HGMapchg 39883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-riotaBAD 36953 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-ot 4571 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7704 df-1st 7821 df-2nd 7822 df-tpos 8030 df-undef 8077 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-2 12024 df-3 12025 df-4 12026 df-5 12027 df-6 12028 df-n0 12222 df-z 12308 df-uz 12571 df-fz 13228 df-struct 16836 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-mulr 16964 df-sca 16966 df-vsca 16967 df-0g 17140 df-mre 17283 df-mrc 17284 df-acs 17286 df-proset 18001 df-poset 18019 df-plt 18036 df-lub 18052 df-glb 18053 df-join 18054 df-meet 18055 df-p0 18131 df-p1 18132 df-lat 18138 df-clat 18205 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-submnd 18419 df-grp 18568 df-minusg 18569 df-sbg 18570 df-subg 18740 df-cntz 18911 df-oppg 18938 df-lsm 19229 df-cmn 19376 df-abl 19377 df-mgp 19709 df-ur 19726 df-ring 19773 df-oppr 19850 df-dvdsr 19871 df-unit 19872 df-invr 19902 df-dvr 19913 df-drng 19981 df-lmod 20113 df-lss 20182 df-lsp 20222 df-lvec 20353 df-lsatoms 36976 df-lshyp 36977 df-lcv 37019 df-lfl 37058 df-lkr 37086 df-ldual 37124 df-oposet 37176 df-ol 37178 df-oml 37179 df-covers 37266 df-ats 37267 df-atl 37298 df-cvlat 37322 df-hlat 37351 df-llines 37498 df-lplanes 37499 df-lvols 37500 df-lines 37501 df-psubsp 37503 df-pmap 37504 df-padd 37796 df-lhyp 37988 df-laut 37989 df-ldil 38104 df-ltrn 38105 df-trl 38159 df-tgrp 38743 df-tendo 38755 df-edring 38757 df-dveca 39003 df-disoa 39029 df-dvech 39079 df-dib 39139 df-dic 39173 df-dih 39229 df-doch 39348 df-djh 39395 df-lcdual 39587 df-mapd 39625 df-hvmap 39757 df-hdmap1 39793 df-hdmap 39794 df-hgmap 39884 |
This theorem is referenced by: hdmapgln2 39912 hdmapinvlem3 39920 hdmapinvlem4 39921 |
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