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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem9 | Structured version Visualization version GIF version |
Description: Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 1-Jun-2015.) |
Ref | Expression |
---|---|
hdmap14lem8.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap14lem8.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap14lem8.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap14lem8.q | ⊢ + = (+g‘𝑈) |
hdmap14lem8.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmap14lem8.o | ⊢ 0 = (0g‘𝑈) |
hdmap14lem8.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap14lem8.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmap14lem8.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmap14lem8.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap14lem8.d | ⊢ ✚ = (+g‘𝐶) |
hdmap14lem8.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hdmap14lem8.p | ⊢ 𝑃 = (Scalar‘𝐶) |
hdmap14lem8.a | ⊢ 𝐴 = (Base‘𝑃) |
hdmap14lem8.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap14lem8.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap14lem8.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem8.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem8.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
hdmap14lem8.g | ⊢ (𝜑 → 𝐺 ∈ 𝐴) |
hdmap14lem8.i | ⊢ (𝜑 → 𝐼 ∈ 𝐴) |
hdmap14lem8.xx | ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) |
hdmap14lem8.yy | ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) |
hdmap14lem8.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
hdmap14lem8.j | ⊢ (𝜑 → 𝐽 ∈ 𝐴) |
hdmap14lem8.xy | ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌)))) |
Ref | Expression |
---|---|
hdmap14lem9 | ⊢ (𝜑 → 𝐺 = 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
2 | hdmap14lem8.d | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
3 | hdmap14lem8.p | . . . 4 ⊢ 𝑃 = (Scalar‘𝐶) | |
4 | hdmap14lem8.a | . . . 4 ⊢ 𝐴 = (Base‘𝑃) | |
5 | hdmap14lem8.e | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
6 | eqid 2821 | . . . 4 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
7 | eqid 2821 | . . . 4 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
8 | hdmap14lem8.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | hdmap14lem8.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
10 | hdmap14lem8.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | 8, 9, 10 | lcdlvec 38726 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec) |
12 | hdmap14lem8.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
13 | hdmap14lem8.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
14 | hdmap14lem8.o | . . . . 5 ⊢ 0 = (0g‘𝑈) | |
15 | hdmap14lem8.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
16 | hdmap14lem8.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
17 | 8, 12, 13, 14, 9, 6, 1, 15, 10, 16 | hdmapnzcl 38980 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ∈ ((Base‘𝐶) ∖ {(0g‘𝐶)})) |
18 | hdmap14lem8.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
19 | 8, 12, 13, 14, 9, 6, 1, 15, 10, 18 | hdmapnzcl 38980 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑌) ∈ ((Base‘𝐶) ∖ {(0g‘𝐶)})) |
20 | hdmap14lem8.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ 𝐴) | |
21 | hdmap14lem8.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐴) | |
22 | hdmap14lem8.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝐴) | |
23 | hdmap14lem8.ne | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
24 | eqid 2821 | . . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
25 | eqid 2821 | . . . . . . . 8 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊) | |
26 | 8, 12, 10 | dvhlmod 38245 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
27 | 16 | eldifad 3947 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
28 | hdmap14lem8.n | . . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈) | |
29 | 13, 24, 28 | lspsncl 19748 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
30 | 26, 27, 29 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
31 | 18 | eldifad 3947 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
32 | 13, 24, 28 | lspsncl 19748 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
33 | 26, 31, 32 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
34 | 8, 12, 24, 25, 10, 30, 33 | mapd11 38774 | . . . . . . 7 ⊢ (𝜑 → ((((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑋})) = (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
35 | 34 | necon3bid 3060 | . . . . . 6 ⊢ (𝜑 → ((((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑋})) ≠ (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))) |
36 | 23, 35 | mpbird 259 | . . . . 5 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑋})) ≠ (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑌}))) |
37 | 8, 12, 13, 28, 9, 7, 25, 15, 10, 27 | hdmap10 38975 | . . . . 5 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑋})) = ((LSpan‘𝐶)‘{(𝑆‘𝑋)})) |
38 | 8, 12, 13, 28, 9, 7, 25, 15, 10, 31 | hdmap10 38975 | . . . . 5 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑌})) = ((LSpan‘𝐶)‘{(𝑆‘𝑌)})) |
39 | 36, 37, 38 | 3netr3d 3092 | . . . 4 ⊢ (𝜑 → ((LSpan‘𝐶)‘{(𝑆‘𝑋)}) ≠ ((LSpan‘𝐶)‘{(𝑆‘𝑌)})) |
40 | hdmap14lem8.q | . . . . 5 ⊢ + = (+g‘𝑈) | |
41 | hdmap14lem8.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑈) | |
42 | hdmap14lem8.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑈) | |
43 | hdmap14lem8.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
44 | hdmap14lem8.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
45 | hdmap14lem8.xx | . . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) | |
46 | hdmap14lem8.yy | . . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) | |
47 | hdmap14lem8.xy | . . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌)))) | |
48 | 8, 12, 13, 40, 41, 14, 28, 42, 43, 9, 2, 5, 3, 4, 15, 10, 16, 18, 44, 21, 22, 45, 46, 23, 20, 47 | hdmap14lem8 39010 | . . . 4 ⊢ (𝜑 → ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌)))) |
49 | 1, 2, 3, 4, 5, 6, 7, 11, 17, 19, 20, 20, 21, 22, 39, 48 | lvecindp2 19910 | . . 3 ⊢ (𝜑 → (𝐽 = 𝐺 ∧ 𝐽 = 𝐼)) |
50 | 49 | simpld 497 | . 2 ⊢ (𝜑 → 𝐽 = 𝐺) |
51 | 49 | simprd 498 | . 2 ⊢ (𝜑 → 𝐽 = 𝐼) |
52 | 50, 51 | eqtr3d 2858 | 1 ⊢ (𝜑 → 𝐺 = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3932 {csn 4566 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 Scalarcsca 16567 ·𝑠 cvsca 16568 0gc0g 16712 LModclmod 19633 LSubSpclss 19702 LSpanclspn 19742 HLchlt 36485 LHypclh 37119 DVecHcdvh 38213 LCDualclcd 38721 mapdcmpd 38759 HDMapchdma 38927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-riotaBAD 36088 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-tpos 7891 df-undef 7938 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-0g 16714 df-mre 16856 df-mrc 16857 df-acs 16859 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-subg 18275 df-cntz 18446 df-oppg 18473 df-lsm 18760 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-drng 19503 df-lmod 19635 df-lss 19703 df-lsp 19743 df-lvec 19874 df-lsatoms 36111 df-lshyp 36112 df-lcv 36154 df-lfl 36193 df-lkr 36221 df-ldual 36259 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-llines 36633 df-lplanes 36634 df-lvols 36635 df-lines 36636 df-psubsp 36638 df-pmap 36639 df-padd 36931 df-lhyp 37123 df-laut 37124 df-ldil 37239 df-ltrn 37240 df-trl 37294 df-tgrp 37878 df-tendo 37890 df-edring 37892 df-dveca 38138 df-disoa 38164 df-dvech 38214 df-dib 38274 df-dic 38308 df-dih 38364 df-doch 38483 df-djh 38530 df-lcdual 38722 df-mapd 38760 df-hvmap 38892 df-hdmap1 38928 df-hdmap 38929 |
This theorem is referenced by: hdmap14lem10 39012 |
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