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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh8 | Structured version Visualization version GIF version |
Description: Part (8) in [Baer] p. 48. Given a reference vector 𝑋, the value of function 𝐼 at a vector 𝑇 is independent of the choice of auxiliary vectors 𝑌 and 𝑍. Unlike Baer's, our version does not require 𝑋, 𝑌, and 𝑍 to be independent, and also is defined for all 𝑌 and 𝑍 that are not colinear with 𝑋 or 𝑇. We do this to make the definition of Baer's sigma function more straightforward. (This part eliminates 𝑇 ≠ 0.) (Contributed by NM, 13-May-2015.) |
Ref | Expression |
---|---|
mapdh8a.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdh8a.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdh8a.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdh8a.s | ⊢ − = (-g‘𝑈) |
mapdh8a.o | ⊢ 0 = (0g‘𝑈) |
mapdh8a.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdh8a.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdh8a.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdh8a.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdh8a.q | ⊢ 𝑄 = (0g‘𝐶) |
mapdh8a.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdh8a.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdh8a.i | ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) |
mapdh8a.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdh8h.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh8h.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdh8i.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdh8i.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdh8i.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
mapdh8i.xy | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdh8i.xz | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
mapdh8i.yt | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) |
mapdh8i.zt | ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) |
mapdh8.t | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
Ref | Expression |
---|---|
mapdh8 | ⊢ (𝜑 → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh8a.q | . . . . . 6 ⊢ 𝑄 = (0g‘𝐶) | |
2 | mapdh8a.i | . . . . . 6 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) | |
3 | mapdh8a.o | . . . . . 6 ⊢ 0 = (0g‘𝑈) | |
4 | mapdh8i.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
5 | fvexd 6203 | . . . . . 6 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ V) | |
6 | 1, 2, 3, 4, 5 | mapdhval0 37014 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉) = 𝑄) |
7 | mapdh8i.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
8 | fvexd 6203 | . . . . . 6 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ V) | |
9 | 1, 2, 3, 7, 8 | mapdhval0 37014 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉) = 𝑄) |
10 | 6, 9 | eqtr4d 2659 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉)) |
11 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉)) |
12 | oteq3 4413 | . . . . 5 ⊢ (𝑇 = 0 → 〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉 = 〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉) | |
13 | 12 | fveq2d 6195 | . . . 4 ⊢ (𝑇 = 0 → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉) = (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉)) |
14 | 13 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉) = (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉)) |
15 | oteq3 4413 | . . . . 5 ⊢ (𝑇 = 0 → 〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉 = 〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉) | |
16 | 15 | fveq2d 6195 | . . . 4 ⊢ (𝑇 = 0 → (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉)) |
17 | 16 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉)) |
18 | 11, 14, 17 | 3eqtr4d 2666 | . 2 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉)) |
19 | mapdh8a.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
20 | mapdh8a.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
21 | mapdh8a.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
22 | mapdh8a.s | . . 3 ⊢ − = (-g‘𝑈) | |
23 | mapdh8a.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
24 | mapdh8a.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
25 | mapdh8a.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
26 | mapdh8a.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
27 | mapdh8a.j | . . 3 ⊢ 𝐽 = (LSpan‘𝐶) | |
28 | mapdh8a.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
29 | mapdh8a.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
30 | 29 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
31 | mapdh8h.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
32 | 31 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝐹 ∈ 𝐷) |
33 | mapdh8h.mn | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
34 | 33 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
35 | mapdh8i.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
36 | 35 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
37 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
38 | 7 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑍 ∈ (𝑉 ∖ { 0 })) |
39 | mapdh8i.xy | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
40 | 39 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
41 | mapdh8i.xz | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) | |
42 | 41 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
43 | mapdh8i.yt | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) | |
44 | 43 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) |
45 | mapdh8i.zt | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) | |
46 | 45 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) |
47 | mapdh8.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
48 | 47 | anim1i 592 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 )) |
49 | eldifsn 4317 | . . . 4 ⊢ (𝑇 ∈ (𝑉 ∖ { 0 }) ↔ (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 )) | |
50 | 48, 49 | sylibr 224 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑇 ∈ (𝑉 ∖ { 0 })) |
51 | 19, 20, 21, 22, 3, 23, 24, 25, 26, 1, 27, 28, 2, 30, 32, 34, 36, 37, 38, 40, 42, 44, 46, 50 | mapdh8j 37077 | . 2 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉)) |
52 | 18, 51 | pm2.61dane 2881 | 1 ⊢ (𝜑 → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∖ cdif 3571 ifcif 4086 {csn 4177 〈cotp 4185 ↦ cmpt 4729 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 Basecbs 15857 0gc0g 16100 -gcsg 17424 LSpanclspn 18971 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 LCDualclcd 36875 mapdcmpd 36913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914 |
This theorem is referenced by: mapdh9a 37079 mapdh9aOLDN 37080 |
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