Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilsrng | Structured version Visualization version GIF version |
Description: The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 21-Jun-2015.) |
Ref | Expression |
---|---|
hlhillvec.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhillvec.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhillvec.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hlhildrng.r | ⊢ 𝑅 = (Scalar‘𝑈) |
Ref | Expression |
---|---|
hlhilsrng | ⊢ (𝜑 → 𝑅 ∈ *-Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhillvec.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hlhillvec.u | . 2 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
3 | hlhillvec.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | hlhildrng.r | . 2 ⊢ 𝑅 = (Scalar‘𝑈) | |
5 | eqid 2739 | . 2 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
6 | eqid 2739 | . 2 ⊢ (Scalar‘((DVecH‘𝐾)‘𝑊)) = (Scalar‘((DVecH‘𝐾)‘𝑊)) | |
7 | eqid 2739 | . 2 ⊢ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊))) = (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊))) | |
8 | eqid 2739 | . 2 ⊢ (+g‘(Scalar‘((DVecH‘𝐾)‘𝑊))) = (+g‘(Scalar‘((DVecH‘𝐾)‘𝑊))) | |
9 | eqid 2739 | . 2 ⊢ (.r‘(Scalar‘((DVecH‘𝐾)‘𝑊))) = (.r‘(Scalar‘((DVecH‘𝐾)‘𝑊))) | |
10 | eqid 2739 | . 2 ⊢ ((HGMap‘𝐾)‘𝑊) = ((HGMap‘𝐾)‘𝑊) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | hlhilsrnglem 39736 | 1 ⊢ (𝜑 → 𝑅 ∈ *-Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ‘cfv 6400 Basecbs 16790 +gcplusg 16832 .rcmulr 16833 Scalarcsca 16835 *-Ringcsr 19910 HLchlt 37133 LHypclh 37767 DVecHcdvh 38861 HGMapchg 39666 HLHilchlh 39715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10812 ax-resscn 10813 ax-1cn 10814 ax-icn 10815 ax-addcl 10816 ax-addrcl 10817 ax-mulcl 10818 ax-mulrcl 10819 ax-mulcom 10820 ax-addass 10821 ax-mulass 10822 ax-distr 10823 ax-i2m1 10824 ax-1ne0 10825 ax-1rid 10826 ax-rnegex 10827 ax-rrecex 10828 ax-cnre 10829 ax-pre-lttri 10830 ax-pre-lttrn 10831 ax-pre-ltadd 10832 ax-pre-mulgt0 10833 ax-riotaBAD 36736 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3711 df-csb 3828 df-dif 3885 df-un 3887 df-in 3889 df-ss 3899 df-pss 3901 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4836 df-int 4876 df-iun 4922 df-iin 4923 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-of 7490 df-om 7666 df-1st 7782 df-2nd 7783 df-tpos 7991 df-undef 8038 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-1o 8225 df-er 8414 df-map 8533 df-en 8650 df-dom 8651 df-sdom 8652 df-fin 8653 df-pnf 10896 df-mnf 10897 df-xr 10898 df-ltxr 10899 df-le 10900 df-sub 11091 df-neg 11092 df-nn 11858 df-2 11920 df-3 11921 df-4 11922 df-5 11923 df-6 11924 df-7 11925 df-8 11926 df-n0 12118 df-z 12204 df-uz 12466 df-fz 13123 df-struct 16730 df-sets 16747 df-slot 16765 df-ndx 16775 df-base 16791 df-ress 16815 df-plusg 16845 df-mulr 16846 df-starv 16847 df-sca 16848 df-vsca 16849 df-ip 16850 df-0g 16976 df-mre 17119 df-mrc 17120 df-acs 17122 df-proset 17832 df-poset 17850 df-plt 17866 df-lub 17882 df-glb 17883 df-join 17884 df-meet 17885 df-p0 17961 df-p1 17962 df-lat 17968 df-clat 18035 df-mgm 18144 df-sgrp 18193 df-mnd 18204 df-mhm 18248 df-submnd 18249 df-grp 18398 df-minusg 18399 df-sbg 18400 df-subg 18570 df-ghm 18650 df-cntz 18741 df-oppg 18768 df-lsm 19055 df-cmn 19202 df-abl 19203 df-mgp 19535 df-ur 19547 df-ring 19594 df-oppr 19671 df-dvdsr 19689 df-unit 19690 df-invr 19720 df-dvr 19731 df-rnghom 19765 df-drng 19799 df-staf 19911 df-srng 19912 df-lmod 19931 df-lss 19999 df-lsp 20039 df-lvec 20170 df-lsatoms 36759 df-lshyp 36760 df-lcv 36802 df-lfl 36841 df-lkr 36869 df-ldual 36907 df-oposet 36959 df-ol 36961 df-oml 36962 df-covers 37049 df-ats 37050 df-atl 37081 df-cvlat 37105 df-hlat 37134 df-llines 37281 df-lplanes 37282 df-lvols 37283 df-lines 37284 df-psubsp 37286 df-pmap 37287 df-padd 37579 df-lhyp 37771 df-laut 37772 df-ldil 37887 df-ltrn 37888 df-trl 37942 df-tgrp 38526 df-tendo 38538 df-edring 38540 df-dveca 38786 df-disoa 38812 df-dvech 38862 df-dib 38922 df-dic 38956 df-dih 39012 df-doch 39131 df-djh 39178 df-lcdual 39370 df-mapd 39408 df-hvmap 39540 df-hdmap1 39576 df-hdmap 39577 df-hgmap 39667 df-hlhil 39716 |
This theorem is referenced by: hlhilphllem 39742 |
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