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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 40474 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
23 | 19 | eldifad 3953 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
24 | 20 | eldifad 3953 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
25 | 21 | eldifad 3953 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
26 | mapdh8ab.xn | . . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
27 | 3, 6, 22, 23, 24, 25, 26 | lspindpi 20975 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
28 | 27 | simprd 495 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
29 | 28 | necomd 2988 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑋})) |
30 | mapdh8ab.yn | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) | |
31 | 29, 30 | neeqtrd 3002 | . 2 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) |
32 | mapdh8ab.t | . 2 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) | |
33 | 30 | sseq1d 4006 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
34 | eqid 2724 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
35 | 1, 2, 14 | dvhlmod 40475 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
36 | 3, 34, 6, 35, 24, 25 | lspprcl 20817 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈)) |
37 | 3, 34, 6, 35, 36, 23 | lspsnel5 20834 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
38 | 32 | eldifad 3953 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
39 | 3, 34, 6, 35, 36, 38 | lspsnel5 20834 | . . . . 5 ⊢ (𝜑 → (𝑇 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
40 | 33, 37, 39 | 3bitr4d 311 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ 𝑇 ∈ (𝑁‘{𝑌, 𝑍}))) |
41 | 26, 40 | mtbid 324 | . . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ (𝑁‘{𝑌, 𝑍})) |
42 | 22 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑈 ∈ LVec) |
43 | 20 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
44 | 38 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ 𝑉) |
45 | 25 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑍 ∈ 𝑉) |
46 | mapdh8ab.yz | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) | |
47 | 46 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
48 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) | |
49 | prcom 4729 | . . . . . 6 ⊢ {𝑍, 𝑇} = {𝑇, 𝑍} | |
50 | 49 | fveq2i 6885 | . . . . 5 ⊢ (𝑁‘{𝑍, 𝑇}) = (𝑁‘{𝑇, 𝑍}) |
51 | 48, 50 | eleqtrdi 2835 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑇, 𝑍})) |
52 | 3, 5, 6, 42, 43, 44, 45, 47, 51 | lspexch 20972 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ (𝑁‘{𝑌, 𝑍})) |
53 | 41, 52 | mtand 813 | . 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 41141 | 1 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ∖ cdif 3938 ⊆ wss 3941 ifcif 4521 {csn 4621 {cpr 4623 〈cotp 4629 ↦ cmpt 5222 ‘cfv 6534 ℩crio 7357 (class class class)co 7402 1st c1st 7967 2nd c2nd 7968 Basecbs 17145 0gc0g 17386 -gcsg 18857 LSubSpclss 20770 LSpanclspn 20810 LVecclvec 20942 HLchlt 38714 LHypclh 39349 DVecHcdvh 40443 LCDualclcd 40951 mapdcmpd 40989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 38317 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-ot 4630 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-0g 17388 df-mre 17531 df-mrc 17532 df-acs 17534 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-cntz 19225 df-oppg 19254 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-oppr 20228 df-dvdsr 20251 df-unit 20252 df-invr 20282 df-dvr 20295 df-drng 20581 df-lmod 20700 df-lss 20771 df-lsp 20811 df-lvec 20943 df-lsatoms 38340 df-lshyp 38341 df-lcv 38383 df-lfl 38422 df-lkr 38450 df-ldual 38488 df-oposet 38540 df-ol 38542 df-oml 38543 df-covers 38630 df-ats 38631 df-atl 38662 df-cvlat 38686 df-hlat 38715 df-llines 38863 df-lplanes 38864 df-lvols 38865 df-lines 38866 df-psubsp 38868 df-pmap 38869 df-padd 39161 df-lhyp 39353 df-laut 39354 df-ldil 39469 df-ltrn 39470 df-trl 39524 df-tgrp 40108 df-tendo 40120 df-edring 40122 df-dveca 40368 df-disoa 40394 df-dvech 40444 df-dib 40504 df-dic 40538 df-dih 40594 df-doch 40713 df-djh 40760 df-lcdual 40952 df-mapd 40990 |
This theorem is referenced by: mapdh8ac 41143 |
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