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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh8d0N | Structured version Visualization version GIF version |
Description: Part of Part (8) in [Baer] p. 48. (Contributed by NM, 10-May-2015.) (New usage is discouraged.) |
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)) |
mapdh8d.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh8d.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdh8b.eg | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
mapdh8d.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdh8d.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdh8d.xt | ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) |
mapdh8d.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) |
mapdh8d.w | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
mapdh8d.wt | ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) |
mapdh8d.ut | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) |
mapdh8d.vw | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) |
mapdh8d.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) |
mapdh8d0.e | ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) |
Ref | Expression |
---|---|
mapdh8d0N | ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh8a.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdh8a.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | mapdh8a.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | mapdh8a.s | . . 3 ⊢ − = (-g‘𝑈) | |
5 | mapdh8a.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
6 | mapdh8a.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | mapdh8a.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | mapdh8a.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
9 | mapdh8a.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
10 | mapdh8a.q | . . 3 ⊢ 𝑄 = (0g‘𝐶) | |
11 | mapdh8a.j | . . 3 ⊢ 𝐽 = (LSpan‘𝐶) | |
12 | mapdh8a.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
13 | mapdh8a.i | . . 3 ⊢ 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 | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | mapdh8b.eg | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | |
16 | mapdh8d.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
17 | mapdh8d.mn | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
18 | mapdh8d.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
19 | mapdh8d.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
20 | 19 | eldifad 3959 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
21 | 1, 2, 14 | dvhlvec 40810 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec) |
22 | 18 | eldifad 3959 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
23 | mapdh8d.w | . . . . . . . 8 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
24 | 23 | eldifad 3959 | . . . . . . 7 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
25 | mapdh8d.xn | . . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) | |
26 | 3, 6, 21, 22, 20, 24, 25 | lspindpi 21115 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤}))) |
27 | 26 | simpld 493 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
28 | 10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 16, 17, 18, 20, 27 | mapdhcl 41428 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) |
29 | 15, 28 | eqeltrrd 2827 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷) |
30 | 10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 16, 17, 18, 19, 29, 27 | mapdheq 41429 | . . . . 5 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))) |
31 | 15, 30 | mpbid 231 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) |
32 | 31 | simpld 493 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) |
33 | mapdh8d.vw | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) | |
34 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 15, 18, 19, 33, 23, 25 | mapdh8a 41476 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑤〉) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) |
35 | mapdh8d.wt | . . 3 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) | |
36 | mapdh8d.xt | . . 3 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) | |
37 | mapdh8d0.e | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) | |
38 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 29, 32, 34, 19, 23, 35, 36, 33, 37, 25 | mapdh8b 41481 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉) = (𝐼‘〈𝑌, 𝐺, 𝑇〉)) |
39 | eqidd 2727 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑤〉) = (𝐼‘〈𝑋, 𝐹, 𝑤〉)) | |
40 | mapdh8d.yz | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) | |
41 | mapdh8d.ut | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) | |
42 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 39, 18, 19, 36, 40, 23, 35, 41, 33, 37, 25 | mapdh8c 41482 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑤, (𝐼‘〈𝑋, 𝐹, 𝑤〉), 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) |
43 | 38, 42 | eqtr3d 2768 | 1 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 ∖ cdif 3944 ifcif 4533 {csn 4633 {cpr 4635 〈cotp 4641 ↦ cmpt 5238 ‘cfv 6556 ℩crio 7381 (class class class)co 7426 1st c1st 8003 2nd c2nd 8004 Basecbs 17215 0gc0g 17456 -gcsg 18932 LSpanclspn 20950 HLchlt 39050 LHypclh 39685 DVecHcdvh 40779 LCDualclcd 41287 mapdcmpd 41325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-riotaBAD 38653 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-ot 4642 df-uni 4916 df-int 4957 df-iun 5005 df-iin 5006 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8005 df-2nd 8006 df-tpos 8243 df-undef 8290 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-er 8736 df-map 8859 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12613 df-uz 12877 df-fz 13541 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-sca 17284 df-vsca 17285 df-0g 17458 df-mre 17601 df-mrc 17602 df-acs 17604 df-proset 18322 df-poset 18340 df-plt 18357 df-lub 18373 df-glb 18374 df-join 18375 df-meet 18376 df-p0 18452 df-p1 18453 df-lat 18459 df-clat 18526 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-submnd 18776 df-grp 18933 df-minusg 18934 df-sbg 18935 df-subg 19119 df-cntz 19313 df-oppg 19342 df-lsm 19636 df-cmn 19782 df-abl 19783 df-mgp 20120 df-rng 20138 df-ur 20167 df-ring 20220 df-oppr 20318 df-dvdsr 20341 df-unit 20342 df-invr 20372 df-dvr 20385 df-nzr 20497 df-rlreg 20674 df-domn 20675 df-drng 20711 df-lmod 20840 df-lss 20911 df-lsp 20951 df-lvec 21083 df-lsatoms 38676 df-lshyp 38677 df-lcv 38719 df-lfl 38758 df-lkr 38786 df-ldual 38824 df-oposet 38876 df-ol 38878 df-oml 38879 df-covers 38966 df-ats 38967 df-atl 38998 df-cvlat 39022 df-hlat 39051 df-llines 39199 df-lplanes 39200 df-lvols 39201 df-lines 39202 df-psubsp 39204 df-pmap 39205 df-padd 39497 df-lhyp 39689 df-laut 39690 df-ldil 39805 df-ltrn 39806 df-trl 39860 df-tgrp 40444 df-tendo 40456 df-edring 40458 df-dveca 40704 df-disoa 40730 df-dvech 40780 df-dib 40840 df-dic 40874 df-dih 40930 df-doch 41049 df-djh 41096 df-lcdual 41288 df-mapd 41326 |
This theorem is referenced by: (None) |
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