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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 6683 | . . . . . 6 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ V) | |
6 | 1, 2, 3, 4, 5 | mapdhval0 39351 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉) = 𝑄) |
7 | mapdh8i.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
8 | fvexd 6683 | . . . . . 6 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ V) | |
9 | 1, 2, 3, 7, 8 | mapdhval0 39351 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉) = 𝑄) |
10 | 6, 9 | eqtr4d 2776 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉)) |
11 | 10 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉)) |
12 | oteq3 4769 | . . . . 5 ⊢ (𝑇 = 0 → 〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉 = 〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉) | |
13 | 12 | fveq2d 6672 | . . . 4 ⊢ (𝑇 = 0 → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉) = (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉)) |
14 | 13 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉) = (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 0 〉)) |
15 | oteq3 4769 | . . . . 5 ⊢ (𝑇 = 0 → 〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉 = 〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉) | |
16 | 15 | fveq2d 6672 | . . . 4 ⊢ (𝑇 = 0 → (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉)) |
17 | 16 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝑇 = 0 ) → (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 0 〉)) |
18 | 11, 14, 17 | 3eqtr4d 2783 | . 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 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
31 | mapdh8h.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
32 | 31 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝐹 ∈ 𝐷) |
33 | mapdh8h.mn | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
34 | 33 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
35 | mapdh8i.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
36 | 35 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
37 | 4 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
38 | 7 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → 𝑍 ∈ (𝑉 ∖ { 0 })) |
39 | mapdh8i.xy | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
40 | 39 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
41 | mapdh8i.xz | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) | |
42 | 41 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
43 | mapdh8i.yt | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) | |
44 | 43 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) |
45 | mapdh8i.zt | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) | |
46 | 45 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) |
47 | mapdh8.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
48 | 47 | anim1i 618 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 )) |
49 | eldifsn 4672 | . . . 4 ⊢ (𝑇 ∈ (𝑉 ∖ { 0 }) ↔ (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 )) | |
50 | 48, 49 | sylibr 237 | . . 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 39413 | . 2 ⊢ ((𝜑 ∧ 𝑇 ≠ 0 ) → (𝐼‘〈𝑌, (𝐼‘〈𝑋, 𝐹, 𝑌〉), 𝑇〉) = (𝐼‘〈𝑍, (𝐼‘〈𝑋, 𝐹, 𝑍〉), 𝑇〉)) |
52 | 18, 51 | pm2.61dane 3021 | 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 399 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 Vcvv 3397 ∖ cdif 3838 ifcif 4411 {csn 4513 〈cotp 4521 ↦ cmpt 5107 ‘cfv 6333 ℩crio 7120 (class class class)co 7164 1st c1st 7705 2nd c2nd 7706 Basecbs 16579 0gc0g 16809 -gcsg 18214 LSpanclspn 19855 HLchlt 36976 LHypclh 37610 DVecHcdvh 38704 LCDualclcd 39212 mapdcmpd 39250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-riotaBAD 36579 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-ot 4522 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-undef 7961 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-0g 16811 df-mre 16953 df-mrc 16954 df-acs 16956 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-p1 17759 df-lat 17765 df-clat 17827 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-grp 18215 df-minusg 18216 df-sbg 18217 df-subg 18387 df-cntz 18558 df-oppg 18585 df-lsm 18872 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-dvr 19548 df-drng 19616 df-lmod 19748 df-lss 19816 df-lsp 19856 df-lvec 19987 df-lsatoms 36602 df-lshyp 36603 df-lcv 36645 df-lfl 36684 df-lkr 36712 df-ldual 36750 df-oposet 36802 df-ol 36804 df-oml 36805 df-covers 36892 df-ats 36893 df-atl 36924 df-cvlat 36948 df-hlat 36977 df-llines 37124 df-lplanes 37125 df-lvols 37126 df-lines 37127 df-psubsp 37129 df-pmap 37130 df-padd 37422 df-lhyp 37614 df-laut 37615 df-ldil 37730 df-ltrn 37731 df-trl 37785 df-tgrp 38369 df-tendo 38381 df-edring 38383 df-dveca 38629 df-disoa 38655 df-dvech 38705 df-dib 38765 df-dic 38799 df-dih 38855 df-doch 38974 df-djh 39021 df-lcdual 39213 df-mapd 39251 |
This theorem is referenced by: mapdh9a 39415 mapdh9aOLDN 39416 |
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