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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh8ab | Structured version Visualization version GIF version |
Description: Part of Part (8) in [Baer] p. 48. (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)) |
mapdh8ab.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh8ab.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdh8ab.eg | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
mapdh8ab.ee | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) |
mapdh8ab.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdh8ab.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdh8ab.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
mapdh8ab.t | ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) |
mapdh8ab.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
mapdh8ab.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
mapdh8ab.yn | ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) |
Ref | Expression |
---|---|
mapdh8ab | ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 𝑇〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh8a.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdh8a.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | mapdh8a.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
4 | mapdh8a.s | . 2 ⊢ − = (-g‘𝑈) | |
5 | mapdh8a.o | . 2 ⊢ 0 = (0g‘𝑈) | |
6 | mapdh8a.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | mapdh8a.c | . 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | mapdh8a.d | . 2 ⊢ 𝐷 = (Base‘𝐶) | |
9 | mapdh8a.r | . 2 ⊢ 𝑅 = (-g‘𝐶) | |
10 | mapdh8a.q | . 2 ⊢ 𝑄 = (0g‘𝐶) | |
11 | mapdh8a.j | . 2 ⊢ 𝐽 = (LSpan‘𝐶) | |
12 | mapdh8a.m | . 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
13 | mapdh8a.i | . 2 ⊢ 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)}))))) | |
14 | mapdh8a.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | mapdh8ab.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
16 | mapdh8ab.mn | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
17 | mapdh8ab.eg | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | |
18 | mapdh8ab.ee | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) | |
19 | mapdh8ab.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
20 | mapdh8ab.y | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
21 | mapdh8ab.z | . 2 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
22 | 1, 2, 14 | dvhlvec 38125 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
23 | 19 | eldifad 3945 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
24 | 20 | eldifad 3945 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
25 | 21 | eldifad 3945 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
26 | mapdh8ab.xn | . . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
27 | 3, 6, 22, 23, 24, 25, 26 | lspindpi 19833 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
28 | 27 | simprd 496 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
29 | 28 | necomd 3068 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑋})) |
30 | mapdh8ab.yn | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) | |
31 | 29, 30 | neeqtrd 3082 | . 2 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) |
32 | mapdh8ab.t | . 2 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) | |
33 | 30 | sseq1d 3995 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
34 | eqid 2818 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
35 | 1, 2, 14 | dvhlmod 38126 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
36 | 3, 34, 6, 35, 24, 25 | lspprcl 19679 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈)) |
37 | 3, 34, 6, 35, 36, 23 | lspsnel5 19696 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
38 | 32 | eldifad 3945 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
39 | 3, 34, 6, 35, 36, 38 | lspsnel5 19696 | . . . . 5 ⊢ (𝜑 → (𝑇 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
40 | 33, 37, 39 | 3bitr4d 312 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ 𝑇 ∈ (𝑁‘{𝑌, 𝑍}))) |
41 | 26, 40 | mtbid 325 | . . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ (𝑁‘{𝑌, 𝑍})) |
42 | 22 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑈 ∈ LVec) |
43 | 20 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
44 | 38 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ 𝑉) |
45 | 25 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑍 ∈ 𝑉) |
46 | mapdh8ab.yz | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) | |
47 | 46 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
48 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) | |
49 | prcom 4660 | . . . . . 6 ⊢ {𝑍, 𝑇} = {𝑇, 𝑍} | |
50 | 49 | fveq2i 6666 | . . . . 5 ⊢ (𝑁‘{𝑍, 𝑇}) = (𝑁‘{𝑇, 𝑍}) |
51 | 48, 50 | eleqtrdi 2920 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑇, 𝑍})) |
52 | 3, 5, 6, 42, 43, 44, 45, 47, 51 | lspexch 19830 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ (𝑁‘{𝑌, 𝑍})) |
53 | 41, 52 | mtand 812 | . 2 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) |
54 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 31, 32, 53, 26 | mapdh8aa 38792 | 1 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 ifcif 4463 {csn 4557 {cpr 4559 〈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 LSubSpclss 19632 LSpanclspn 19672 LVecclvec 19803 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: mapdh8ac 38794 |
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