Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmaprnN | Structured version Visualization version GIF version |
Description: Part of proof of part 16 in [Baer] p. 50 line 23, Fs=G, except that we use the original vector space scalars for the range. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hgmaprn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hgmaprn.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hgmaprn.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hgmaprn.b | ⊢ 𝐵 = (Base‘𝑅) |
hgmaprn.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hgmaprn.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
hgmaprnN | ⊢ (𝜑 → ran 𝐺 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hgmaprn.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hgmaprn.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hgmaprn.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑈) | |
4 | hgmaprn.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
5 | hgmaprn.g | . . . . 5 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
6 | hgmaprn.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 1, 2, 3, 4, 5, 6 | hgmapfnN 39899 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) |
8 | eqid 2738 | . . . . . 6 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
9 | eqid 2738 | . . . . . 6 ⊢ (Scalar‘((LCDual‘𝐾)‘𝑊)) = (Scalar‘((LCDual‘𝐾)‘𝑊)) | |
10 | eqid 2738 | . . . . . 6 ⊢ (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊))) = (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊))) | |
11 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
12 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵) | |
13 | 1, 2, 3, 4, 8, 9, 10, 5, 11, 12 | hgmapdcl 39901 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) ∈ (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊)))) |
14 | 13 | ralrimiva 3103 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 (𝐺‘𝑧) ∈ (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊)))) |
15 | fnfvrnss 6996 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ ∀𝑧 ∈ 𝐵 (𝐺‘𝑧) ∈ (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊)))) → ran 𝐺 ⊆ (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊)))) | |
16 | 7, 14, 15 | syl2anc 584 | . . 3 ⊢ (𝜑 → ran 𝐺 ⊆ (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊)))) |
17 | eqid 2738 | . . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
18 | eqid 2738 | . . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
19 | eqid 2738 | . . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
20 | eqid 2738 | . . . 4 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊)) | |
21 | eqid 2738 | . . . 4 ⊢ ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) | |
22 | eqid 2738 | . . . 4 ⊢ (0g‘((LCDual‘𝐾)‘𝑊)) = (0g‘((LCDual‘𝐾)‘𝑊)) | |
23 | eqid 2738 | . . . 4 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊) | |
24 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
25 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊)))) → 𝑧 ∈ (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊)))) | |
26 | 1, 2, 17, 3, 4, 18, 19, 8, 20, 9, 10, 21, 22, 23, 5, 24, 25 | hgmaprnlem5N 39911 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊)))) → 𝑧 ∈ ran 𝐺) |
27 | 16, 26 | eqelssd 3943 | . 2 ⊢ (𝜑 → ran 𝐺 = (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊)))) |
28 | 1, 2, 3, 4, 8, 9, 10, 6 | lcdsbase 39611 | . 2 ⊢ (𝜑 → (Base‘(Scalar‘((LCDual‘𝐾)‘𝑊))) = 𝐵) |
29 | 27, 28 | eqtrd 2778 | 1 ⊢ (𝜑 → ran 𝐺 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3888 ran crn 5592 Fn wfn 6430 ‘cfv 6435 Basecbs 16910 Scalarcsca 16963 ·𝑠 cvsca 16964 0gc0g 17148 HLchlt 37361 LHypclh 37995 DVecHcdvh 39089 LCDualclcd 39597 HDMapchdma 39803 HGMapchg 39894 |
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 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-riotaBAD 36964 |
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-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-ot 4572 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8040 df-undef 8087 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-map 8615 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-n0 12232 df-z 12318 df-uz 12581 df-fz 13238 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-0g 17150 df-mre 17293 df-mrc 17294 df-acs 17296 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-oppg 18948 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 df-lsatoms 36987 df-lshyp 36988 df-lcv 37030 df-lfl 37069 df-lkr 37097 df-ldual 37135 df-oposet 37187 df-ol 37189 df-oml 37190 df-covers 37277 df-ats 37278 df-atl 37309 df-cvlat 37333 df-hlat 37362 df-llines 37509 df-lplanes 37510 df-lvols 37511 df-lines 37512 df-psubsp 37514 df-pmap 37515 df-padd 37807 df-lhyp 37999 df-laut 38000 df-ldil 38115 df-ltrn 38116 df-trl 38170 df-tgrp 38754 df-tendo 38766 df-edring 38768 df-dveca 39014 df-disoa 39040 df-dvech 39090 df-dib 39150 df-dic 39184 df-dih 39240 df-doch 39359 df-djh 39406 df-lcdual 39598 df-mapd 39636 df-hvmap 39768 df-hdmap1 39804 df-hdmap 39805 df-hgmap 39895 |
This theorem is referenced by: hgmapf1oN 39914 |
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