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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh8ac | 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)) |
mapdh8ac.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh8ac.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdh8ac.eg | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
mapdh8ac.ee | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) |
mapdh8ac.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdh8ac.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdh8ac.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
mapdh8ac.t | ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) |
mapdh8ac.yn | ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) |
mapdh8ac.ew | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑤〉) = 𝐵) |
mapdh8ac.w | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
mapdh8ac.yw | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) |
mapdh8ac.xy | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) |
mapdh8ac.wz | ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑍})) |
mapdh8ac.xz | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) |
Ref | Expression |
---|---|
mapdh8ac | ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 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 | mapdh8ac.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
16 | mapdh8ac.mn | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
17 | mapdh8ac.eg | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | |
18 | mapdh8ac.ew | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑤〉) = 𝐵) | |
19 | mapdh8ac.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
20 | mapdh8ac.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
21 | mapdh8ac.w | . . 3 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
22 | mapdh8ac.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) | |
23 | mapdh8ac.yw | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) | |
24 | mapdh8ac.xy | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) | |
25 | mapdh8ac.yn | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) | |
26 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 | mapdh8ab 40171 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑤, 𝐵, 𝑇〉)) |
27 | mapdh8ac.ee | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) | |
28 | mapdh8ac.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
29 | mapdh8ac.wz | . . 3 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑍})) | |
30 | mapdh8ac.xz | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) | |
31 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 27, 19, 21, 28, 22, 29, 30, 25 | mapdh8ab 40171 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑤, 𝐵, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 𝑇〉)) |
32 | 26, 31 | eqtrd 2778 | 1 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 Vcvv 3444 ∖ cdif 3906 ifcif 4485 {csn 4585 {cpr 4587 〈cotp 4593 ↦ cmpt 5187 ‘cfv 6492 ℩crio 7305 (class class class)co 7350 1st c1st 7910 2nd c2nd 7911 Basecbs 17019 0gc0g 17257 -gcsg 18686 LSpanclspn 20361 HLchlt 37743 LHypclh 38378 DVecHcdvh 39472 LCDualclcd 39980 mapdcmpd 40018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-riotaBAD 37346 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-of 7608 df-om 7794 df-1st 7912 df-2nd 7913 df-tpos 8125 df-undef 8172 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-n0 12348 df-z 12434 df-uz 12698 df-fz 13355 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-mulr 17083 df-sca 17085 df-vsca 17086 df-0g 17259 df-mre 17402 df-mrc 17403 df-acs 17405 df-proset 18120 df-poset 18138 df-plt 18155 df-lub 18171 df-glb 18172 df-join 18173 df-meet 18174 df-p0 18250 df-p1 18251 df-lat 18257 df-clat 18324 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-submnd 18538 df-grp 18687 df-minusg 18688 df-sbg 18689 df-subg 18860 df-cntz 19032 df-oppg 19059 df-lsm 19353 df-cmn 19499 df-abl 19500 df-mgp 19832 df-ur 19849 df-ring 19896 df-oppr 19978 df-dvdsr 19999 df-unit 20000 df-invr 20030 df-dvr 20041 df-drng 20116 df-lmod 20253 df-lss 20322 df-lsp 20362 df-lvec 20493 df-lsatoms 37369 df-lshyp 37370 df-lcv 37412 df-lfl 37451 df-lkr 37479 df-ldual 37517 df-oposet 37569 df-ol 37571 df-oml 37572 df-covers 37659 df-ats 37660 df-atl 37691 df-cvlat 37715 df-hlat 37744 df-llines 37892 df-lplanes 37893 df-lvols 37894 df-lines 37895 df-psubsp 37897 df-pmap 37898 df-padd 38190 df-lhyp 38382 df-laut 38383 df-ldil 38498 df-ltrn 38499 df-trl 38553 df-tgrp 39137 df-tendo 39149 df-edring 39151 df-dveca 39397 df-disoa 39423 df-dvech 39473 df-dib 39533 df-dic 39567 df-dih 39623 df-doch 39742 df-djh 39789 df-lcdual 39981 df-mapd 40019 |
This theorem is referenced by: mapdh8ad 40173 |
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