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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 38779 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec) |
23 | 18 | eldifad 3865 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
24 | 19 | eldifad 3865 | . . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
25 | 21 | eldifad 3865 | . . . 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 39440 | . . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ (𝑁‘{𝑌, 𝑤})) |
29 | prcom 4633 | . . . . . 6 ⊢ {𝑤, 𝑇} = {𝑇, 𝑤} | |
30 | 29 | fveq2i 6690 | . . . . 5 ⊢ (𝑁‘{𝑤, 𝑇}) = (𝑁‘{𝑇, 𝑤}) |
31 | 30 | eleq2i 2825 | . . . 4 ⊢ (𝑌 ∈ (𝑁‘{𝑤, 𝑇}) ↔ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) |
32 | 22 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑈 ∈ LVec) |
33 | 18 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
34 | 25 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑇 ∈ 𝑉) |
35 | 24 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑤 ∈ 𝑉) |
36 | mapdh8b.vw | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) | |
37 | 36 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) |
38 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) | |
39 | 3, 5, 6, 32, 33, 34, 35, 37, 38 | lspexch 20033 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤})) |
40 | 39 | ex 416 | . . . 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 39445 | 1 ⊢ (𝜑 → (𝐼‘〈𝑤, 𝐸, 𝑇〉) = (𝐼‘〈𝑌, 𝐺, 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 Vcvv 3400 ∖ cdif 3850 ifcif 4424 {csn 4526 {cpr 4528 〈cotp 4534 ↦ cmpt 5120 ‘cfv 6350 ℩crio 7139 (class class class)co 7183 1st c1st 7725 2nd c2nd 7726 Basecbs 16599 0gc0g 16829 -gcsg 18234 LSpanclspn 19875 LVecclvec 20006 HLchlt 37020 LHypclh 37654 DVecHcdvh 38748 LCDualclcd 39256 mapdcmpd 39294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-riotaBAD 36623 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-ot 4535 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-of 7438 df-om 7613 df-1st 7727 df-2nd 7728 df-tpos 7934 df-undef 7981 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-n0 11990 df-z 12076 df-uz 12338 df-fz 12995 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-sca 16697 df-vsca 16698 df-0g 16831 df-mre 16973 df-mrc 16974 df-acs 16976 df-proset 17667 df-poset 17685 df-plt 17697 df-lub 17713 df-glb 17714 df-join 17715 df-meet 17716 df-p0 17778 df-p1 17779 df-lat 17785 df-clat 17847 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-submnd 18086 df-grp 18235 df-minusg 18236 df-sbg 18237 df-subg 18407 df-cntz 18578 df-oppg 18605 df-lsm 18892 df-cmn 19039 df-abl 19040 df-mgp 19372 df-ur 19384 df-ring 19431 df-oppr 19508 df-dvdsr 19526 df-unit 19527 df-invr 19557 df-dvr 19568 df-drng 19636 df-lmod 19768 df-lss 19836 df-lsp 19876 df-lvec 20007 df-lsatoms 36646 df-lshyp 36647 df-lcv 36689 df-lfl 36728 df-lkr 36756 df-ldual 36794 df-oposet 36846 df-ol 36848 df-oml 36849 df-covers 36936 df-ats 36937 df-atl 36968 df-cvlat 36992 df-hlat 37021 df-llines 37168 df-lplanes 37169 df-lvols 37170 df-lines 37171 df-psubsp 37173 df-pmap 37174 df-padd 37466 df-lhyp 37658 df-laut 37659 df-ldil 37774 df-ltrn 37775 df-trl 37829 df-tgrp 38413 df-tendo 38425 df-edring 38427 df-dveca 38673 df-disoa 38699 df-dvech 38749 df-dib 38809 df-dic 38843 df-dih 38899 df-doch 39018 df-djh 39065 df-lcdual 39257 df-mapd 39295 |
This theorem is referenced by: mapdh8c 39451 mapdh8d0N 39452 mapdh8d 39453 |
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