Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem9 | Structured version Visualization version GIF version |
Description: Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 1-Jun-2015.) |
Ref | Expression |
---|---|
hdmap14lem8.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap14lem8.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap14lem8.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap14lem8.q | ⊢ + = (+g‘𝑈) |
hdmap14lem8.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmap14lem8.o | ⊢ 0 = (0g‘𝑈) |
hdmap14lem8.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap14lem8.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmap14lem8.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmap14lem8.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap14lem8.d | ⊢ ✚ = (+g‘𝐶) |
hdmap14lem8.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hdmap14lem8.p | ⊢ 𝑃 = (Scalar‘𝐶) |
hdmap14lem8.a | ⊢ 𝐴 = (Base‘𝑃) |
hdmap14lem8.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap14lem8.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap14lem8.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem8.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem8.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
hdmap14lem8.g | ⊢ (𝜑 → 𝐺 ∈ 𝐴) |
hdmap14lem8.i | ⊢ (𝜑 → 𝐼 ∈ 𝐴) |
hdmap14lem8.xx | ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) |
hdmap14lem8.yy | ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) |
hdmap14lem8.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
hdmap14lem8.j | ⊢ (𝜑 → 𝐽 ∈ 𝐴) |
hdmap14lem8.xy | ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌)))) |
Ref | Expression |
---|---|
hdmap14lem9 | ⊢ (𝜑 → 𝐺 = 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
2 | hdmap14lem8.d | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
3 | hdmap14lem8.p | . . . 4 ⊢ 𝑃 = (Scalar‘𝐶) | |
4 | hdmap14lem8.a | . . . 4 ⊢ 𝐴 = (Base‘𝑃) | |
5 | hdmap14lem8.e | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
6 | eqid 2738 | . . . 4 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
7 | eqid 2738 | . . . 4 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
8 | hdmap14lem8.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | hdmap14lem8.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
10 | hdmap14lem8.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | 8, 9, 10 | lcdlvec 39602 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec) |
12 | hdmap14lem8.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
13 | hdmap14lem8.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
14 | hdmap14lem8.o | . . . . 5 ⊢ 0 = (0g‘𝑈) | |
15 | hdmap14lem8.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
16 | hdmap14lem8.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
17 | 8, 12, 13, 14, 9, 6, 1, 15, 10, 16 | hdmapnzcl 39856 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ∈ ((Base‘𝐶) ∖ {(0g‘𝐶)})) |
18 | hdmap14lem8.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
19 | 8, 12, 13, 14, 9, 6, 1, 15, 10, 18 | hdmapnzcl 39856 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑌) ∈ ((Base‘𝐶) ∖ {(0g‘𝐶)})) |
20 | hdmap14lem8.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ 𝐴) | |
21 | hdmap14lem8.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐴) | |
22 | hdmap14lem8.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝐴) | |
23 | hdmap14lem8.ne | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
24 | eqid 2738 | . . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
25 | eqid 2738 | . . . . . . . 8 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊) | |
26 | 8, 12, 10 | dvhlmod 39121 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
27 | 16 | eldifad 3900 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
28 | hdmap14lem8.n | . . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈) | |
29 | 13, 24, 28 | lspsncl 20237 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
30 | 26, 27, 29 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
31 | 18 | eldifad 3900 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
32 | 13, 24, 28 | lspsncl 20237 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
33 | 26, 31, 32 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
34 | 8, 12, 24, 25, 10, 30, 33 | mapd11 39650 | . . . . . . 7 ⊢ (𝜑 → ((((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑋})) = (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
35 | 34 | necon3bid 2988 | . . . . . 6 ⊢ (𝜑 → ((((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑋})) ≠ (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))) |
36 | 23, 35 | mpbird 256 | . . . . 5 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑋})) ≠ (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑌}))) |
37 | 8, 12, 13, 28, 9, 7, 25, 15, 10, 27 | hdmap10 39851 | . . . . 5 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑋})) = ((LSpan‘𝐶)‘{(𝑆‘𝑋)})) |
38 | 8, 12, 13, 28, 9, 7, 25, 15, 10, 31 | hdmap10 39851 | . . . . 5 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑌})) = ((LSpan‘𝐶)‘{(𝑆‘𝑌)})) |
39 | 36, 37, 38 | 3netr3d 3020 | . . . 4 ⊢ (𝜑 → ((LSpan‘𝐶)‘{(𝑆‘𝑋)}) ≠ ((LSpan‘𝐶)‘{(𝑆‘𝑌)})) |
40 | hdmap14lem8.q | . . . . 5 ⊢ + = (+g‘𝑈) | |
41 | hdmap14lem8.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑈) | |
42 | hdmap14lem8.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑈) | |
43 | hdmap14lem8.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
44 | hdmap14lem8.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
45 | hdmap14lem8.xx | . . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) | |
46 | hdmap14lem8.yy | . . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) | |
47 | hdmap14lem8.xy | . . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌)))) | |
48 | 8, 12, 13, 40, 41, 14, 28, 42, 43, 9, 2, 5, 3, 4, 15, 10, 16, 18, 44, 21, 22, 45, 46, 23, 20, 47 | hdmap14lem8 39886 | . . . 4 ⊢ (𝜑 → ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌)))) |
49 | 1, 2, 3, 4, 5, 6, 7, 11, 17, 19, 20, 20, 21, 22, 39, 48 | lvecindp2 20399 | . . 3 ⊢ (𝜑 → (𝐽 = 𝐺 ∧ 𝐽 = 𝐼)) |
50 | 49 | simpld 495 | . 2 ⊢ (𝜑 → 𝐽 = 𝐺) |
51 | 49 | simprd 496 | . 2 ⊢ (𝜑 → 𝐽 = 𝐼) |
52 | 50, 51 | eqtr3d 2780 | 1 ⊢ (𝜑 → 𝐺 = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3885 {csn 4563 ‘cfv 6435 (class class class)co 7277 Basecbs 16910 +gcplusg 16960 Scalarcsca 16963 ·𝑠 cvsca 16964 0gc0g 17148 LModclmod 20121 LSubSpclss 20191 LSpanclspn 20231 HLchlt 37361 LHypclh 37995 DVecHcdvh 39089 LCDualclcd 39597 mapdcmpd 39635 HDMapchdma 39803 |
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 |
This theorem is referenced by: hdmap14lem10 39888 |
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