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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh8b | Structured version Visualization version GIF version |
Description: Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-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)) |
mapdh8b.f | ⊢ (𝜑 → 𝐺 ∈ 𝐷) |
mapdh8b.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) |
mapdh8b.a | ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑤〉) = 𝐸) |
mapdh8b.x | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdh8b.y | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
mapdh8b.yz | ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) |
mapdh8b.xt | ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) |
mapdh8b.vw | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) |
mapdh8b.e | ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) |
mapdh8b.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) |
Ref | Expression |
---|---|
mapdh8b | ⊢ (𝜑 → (𝐼‘〈𝑤, 𝐸, 𝑇〉) = (𝐼‘〈𝑌, 𝐺, 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 | mapdh8b.f | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝐷) | |
16 | mapdh8b.mn | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) | |
17 | mapdh8b.a | . 2 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑤〉) = 𝐸) | |
18 | mapdh8b.x | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
19 | mapdh8b.y | . 2 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
20 | mapdh8b.yz | . 2 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) | |
21 | mapdh8b.xt | . 2 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) | |
22 | 1, 2, 14 | dvhlvec 38670 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec) |
23 | 18 | eldifad 3866 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
24 | 19 | eldifad 3866 | . . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
25 | 21 | eldifad 3866 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
26 | mapdh8b.e | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) | |
27 | mapdh8b.xn | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) | |
28 | 3, 6, 22, 23, 24, 25, 26, 27 | lspindp5 39331 | . . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ (𝑁‘{𝑌, 𝑤})) |
29 | prcom 4618 | . . . . . 6 ⊢ {𝑤, 𝑇} = {𝑇, 𝑤} | |
30 | 29 | fveq2i 6654 | . . . . 5 ⊢ (𝑁‘{𝑤, 𝑇}) = (𝑁‘{𝑇, 𝑤}) |
31 | 30 | eleq2i 2842 | . . . 4 ⊢ (𝑌 ∈ (𝑁‘{𝑤, 𝑇}) ↔ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) |
32 | 22 | adantr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑈 ∈ LVec) |
33 | 18 | adantr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
34 | 25 | adantr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑇 ∈ 𝑉) |
35 | 24 | adantr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑤 ∈ 𝑉) |
36 | mapdh8b.vw | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) | |
37 | 36 | adantr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) |
38 | simpr 489 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) | |
39 | 3, 5, 6, 32, 33, 34, 35, 37, 38 | lspexch 19954 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤})) |
40 | 39 | ex 417 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘{𝑇, 𝑤}) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤}))) |
41 | 31, 40 | syl5bi 245 | . . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘{𝑤, 𝑇}) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤}))) |
42 | 28, 41 | mtod 201 | . 2 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑤, 𝑇})) |
43 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 42 | mapdh8a 39336 | 1 ⊢ (𝜑 → (𝐼‘〈𝑤, 𝐸, 𝑇〉) = (𝐼‘〈𝑌, 𝐺, 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ≠ wne 2949 Vcvv 3407 ∖ cdif 3851 ifcif 4413 {csn 4515 {cpr 4517 〈cotp 4523 ↦ cmpt 5105 ‘cfv 6328 ℩crio 7100 (class class class)co 7143 1st c1st 7684 2nd c2nd 7685 Basecbs 16526 0gc0g 16756 -gcsg 18156 LSpanclspn 19796 LVecclvec 19927 HLchlt 36911 LHypclh 37545 DVecHcdvh 38639 LCDualclcd 39147 mapdcmpd 39185 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-riotaBAD 36514 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-ot 4524 df-uni 4792 df-int 4832 df-iun 4878 df-iin 4879 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-of 7398 df-om 7573 df-1st 7686 df-2nd 7687 df-tpos 7895 df-undef 7942 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-n0 11920 df-z 12006 df-uz 12268 df-fz 12925 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-sca 16624 df-vsca 16625 df-0g 16758 df-mre 16900 df-mrc 16901 df-acs 16903 df-proset 17589 df-poset 17607 df-plt 17619 df-lub 17635 df-glb 17636 df-join 17637 df-meet 17638 df-p0 17700 df-p1 17701 df-lat 17707 df-clat 17769 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-submnd 18008 df-grp 18157 df-minusg 18158 df-sbg 18159 df-subg 18328 df-cntz 18499 df-oppg 18526 df-lsm 18813 df-cmn 18960 df-abl 18961 df-mgp 19293 df-ur 19305 df-ring 19352 df-oppr 19429 df-dvdsr 19447 df-unit 19448 df-invr 19478 df-dvr 19489 df-drng 19557 df-lmod 19689 df-lss 19757 df-lsp 19797 df-lvec 19928 df-lsatoms 36537 df-lshyp 36538 df-lcv 36580 df-lfl 36619 df-lkr 36647 df-ldual 36685 df-oposet 36737 df-ol 36739 df-oml 36740 df-covers 36827 df-ats 36828 df-atl 36859 df-cvlat 36883 df-hlat 36912 df-llines 37059 df-lplanes 37060 df-lvols 37061 df-lines 37062 df-psubsp 37064 df-pmap 37065 df-padd 37357 df-lhyp 37549 df-laut 37550 df-ldil 37665 df-ltrn 37666 df-trl 37720 df-tgrp 38304 df-tendo 38316 df-edring 38318 df-dveca 38564 df-disoa 38590 df-dvech 38640 df-dib 38700 df-dic 38734 df-dih 38790 df-doch 38909 df-djh 38956 df-lcdual 39148 df-mapd 39186 |
This theorem is referenced by: mapdh8c 39342 mapdh8d0N 39343 mapdh8d 39344 |
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