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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem3 | Structured version Visualization version GIF version |
Description: Prior to part 14 in [Baer] p. 49, line 26. (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 |
---|---|
hdmap14lem3 | ⊢ (𝜑 → ∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem1.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap14lem1.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap14lem1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap14lem1.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑈) | |
5 | hdmap14lem3.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
6 | hdmap14lem1.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑈) | |
7 | hdmap14lem1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
8 | hdmap14lem1.z | . . . 4 ⊢ 𝑍 = (0g‘𝑅) | |
9 | hdmap14lem1.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
10 | hdmap14lem2.e | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
11 | hdmap14lem1.l | . . . 4 ⊢ 𝐿 = (LSpan‘𝐶) | |
12 | hdmap14lem2.p | . . . 4 ⊢ 𝑃 = (Scalar‘𝐶) | |
13 | hdmap14lem2.a | . . . 4 ⊢ 𝐴 = (Base‘𝑃) | |
14 | hdmap14lem2.q | . . . 4 ⊢ 𝑄 = (0g‘𝑃) | |
15 | hdmap14lem1.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
16 | hdmap14lem1.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | hdmap14lem3.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
18 | hdmap14lem1.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) | |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | hdmap14lem1 37477 | . . 3 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))})) |
20 | 19 | eqcomd 2657 | . 2 ⊢ (𝜑 → (𝐿‘{(𝑆‘(𝐹 · 𝑋))}) = (𝐿‘{(𝑆‘𝑋)})) |
21 | eqid 2651 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
22 | eqid 2651 | . . 3 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
23 | 1, 9, 16 | lcdlvec 37197 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LVec) |
24 | 1, 2, 16 | dvhlmod 36716 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
25 | 18 | eldifad 3619 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
26 | 17 | eldifad 3619 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
27 | 3, 6, 4, 7 | lmodvscl 18928 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉) |
28 | 24, 25, 26, 27 | syl3anc 1366 | . . . 4 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉) |
29 | 1, 2, 3, 9, 21, 15, 16, 28 | hdmapcl 37439 | . . 3 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) ∈ (Base‘𝐶)) |
30 | 1, 2, 3, 5, 9, 22, 21, 15, 16, 17 | hdmapnzcl 37454 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ ((Base‘𝐶) ∖ {(0g‘𝐶)})) |
31 | 21, 12, 13, 14, 10, 22, 11, 23, 29, 30 | lspsneu 19171 | . 2 ⊢ (𝜑 → ((𝐿‘{(𝑆‘(𝐹 · 𝑋))}) = (𝐿‘{(𝑆‘𝑋)}) ↔ ∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))) |
32 | 20, 31 | mpbid 222 | 1 ⊢ (𝜑 → ∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∃!wreu 2943 ∖ cdif 3604 {csn 4210 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 Scalarcsca 15991 ·𝑠 cvsca 15992 0gc0g 16147 LModclmod 18911 LSpanclspn 19019 HLchlt 34955 LHypclh 35588 DVecHcdvh 36684 LCDualclcd 37192 HDMapchdma 37399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-riotaBAD 34557 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-ot 4219 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-undef 7444 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-0g 16149 df-mre 16293 df-mrc 16294 df-acs 16296 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-cntz 17796 df-oppg 17822 df-lsm 18097 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-drng 18797 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lvec 19151 df-lsatoms 34581 df-lshyp 34582 df-lcv 34624 df-lfl 34663 df-lkr 34691 df-ldual 34729 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-lplanes 35103 df-lvols 35104 df-lines 35105 df-psubsp 35107 df-pmap 35108 df-padd 35400 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 df-trl 35764 df-tgrp 36348 df-tendo 36360 df-edring 36362 df-dveca 36608 df-disoa 36635 df-dvech 36685 df-dib 36745 df-dic 36779 df-dih 36835 df-doch 36954 df-djh 37001 df-lcdual 37193 df-mapd 37231 df-hvmap 37363 df-hdmap1 37400 df-hdmap 37401 |
This theorem is referenced by: hdmap14lem4 37481 |
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