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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 6678 | . . . . . 6 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ V) | |
6 | 1, 2, 3, 4, 5 | mapdhval0 38741 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉) = 𝑄) |
7 | mapdh8i.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
8 | fvexd 6678 | . . . . . 6 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ V) | |
9 | 1, 2, 3, 7, 8 | mapdhval0 38741 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉) = 𝑄) |
10 | 6, 9 | eqtr4d 2856 | . . . 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 4806 | . . . . 5 ⊢ (𝑇 = 0 → 〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉 = 〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉) | |
13 | 12 | fveq2d 6667 | . . . 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 4806 | . . . . 5 ⊢ (𝑇 = 0 → 〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉 = 〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉) | |
16 | 15 | fveq2d 6667 | . . . 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 2863 | . 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 614 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 )) |
49 | eldifsn 4711 | . . . 4 ⊢ (𝑇 ∈ (𝑉 ∖ { 0 }) ↔ (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 )) | |
50 | 48, 49 | sylibr 235 | . . 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 38803 | . 2 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉)) |
52 | 18, 51 | pm2.61dane 3101 | 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 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∖ cdif 3930 ifcif 4463 {csn 4557 〈cotp 4565 ↦ cmpt 5137 ‘cfv 6348 ℩crio 7102 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 Basecbs 16471 0gc0g 16701 -gcsg 18043 LSpanclspn 19672 HLchlt 36366 LHypclh 37000 DVecHcdvh 38094 LCDualclcd 38602 mapdcmpd 38640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-mre 16845 df-mrc 16846 df-acs 16848 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-oppg 18412 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lsatoms 35992 df-lshyp 35993 df-lcv 36035 df-lfl 36074 df-lkr 36102 df-ldual 36140 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tgrp 37759 df-tendo 37771 df-edring 37773 df-dveca 38019 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 df-dih 38245 df-doch 38364 df-djh 38411 df-lcdual 38603 df-mapd 38641 |
This theorem is referenced by: mapdh9a 38805 mapdh9aOLDN 38806 |
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