Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaprnlem4N | Structured version Visualization version GIF version |
Description: Part of proof of part 12 in [Baer] p. 49 line 19. (T* =) (Ft)* = Gs. (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hdmaprnlem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmaprnlem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmaprnlem1.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmaprnlem1.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmaprnlem1.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmaprnlem1.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmaprnlem1.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmaprnlem1.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmaprnlem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmaprnlem1.se | ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) |
hdmaprnlem1.ve | ⊢ (𝜑 → 𝑣 ∈ 𝑉) |
hdmaprnlem1.e | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) |
hdmaprnlem1.ue | ⊢ (𝜑 → 𝑢 ∈ 𝑉) |
hdmaprnlem1.un | ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) |
hdmaprnlem1.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmaprnlem1.q | ⊢ 𝑄 = (0g‘𝐶) |
hdmaprnlem1.o | ⊢ 0 = (0g‘𝑈) |
hdmaprnlem1.a | ⊢ ✚ = (+g‘𝐶) |
hdmaprnlem1.t2 | ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) |
Ref | Expression |
---|---|
hdmaprnlem4N | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
2 | hdmaprnlem1.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
3 | hdmaprnlem1.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | hdmaprnlem1.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | hdmaprnlem1.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 3, 4, 5 | dvhlmod 38736 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
7 | hdmaprnlem1.ve | . . . . . 6 ⊢ (𝜑 → 𝑣 ∈ 𝑉) | |
8 | hdmaprnlem1.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
9 | 8, 1, 2 | lspsncl 19861 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
10 | 6, 7, 9 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
11 | hdmaprnlem1.t2 | . . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) | |
12 | 11 | eldifad 3853 | . . . . 5 ⊢ (𝜑 → 𝑡 ∈ (𝑁‘{𝑣})) |
13 | 1, 2, 6, 10, 12 | lspsnel5a 19880 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑡}) ⊆ (𝑁‘{𝑣})) |
14 | hdmaprnlem1.o | . . . . 5 ⊢ 0 = (0g‘𝑈) | |
15 | 3, 4, 5 | dvhlvec 38735 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec) |
16 | 8, 1 | lss1 19822 | . . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑉 ∈ (LSubSp‘𝑈)) |
17 | 6, 16 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (LSubSp‘𝑈)) |
18 | 1, 2, 6, 17, 7 | lspsnel5a 19880 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑣}) ⊆ 𝑉) |
19 | 18 | ssdifd 4029 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑣}) ∖ { 0 }) ⊆ (𝑉 ∖ { 0 })) |
20 | 19, 11 | sseldd 3876 | . . . . 5 ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 })) |
21 | 8, 14, 2, 15, 20, 7 | lspsncmp 20000 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑡}) ⊆ (𝑁‘{𝑣}) ↔ (𝑁‘{𝑡}) = (𝑁‘{𝑣}))) |
22 | 13, 21 | mpbid 235 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑡}) = (𝑁‘{𝑣})) |
23 | 22 | fveq2d 6672 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝑀‘(𝑁‘{𝑣}))) |
24 | hdmaprnlem1.e | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) | |
25 | 23, 24 | eqtrd 2773 | 1 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ∖ cdif 3838 ⊆ wss 3841 {csn 4513 ‘cfv 6333 Basecbs 16579 +gcplusg 16661 0gc0g 16809 LModclmod 19746 LSubSpclss 19815 LSpanclspn 19855 HLchlt 36976 LHypclh 37610 DVecHcdvh 38704 LCDualclcd 39212 mapdcmpd 39250 HDMapchdma 39418 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-riotaBAD 36579 |
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 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-undef 7961 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-0g 16811 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-p1 17759 df-lat 17765 df-clat 17827 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-grp 18215 df-minusg 18216 df-sbg 18217 df-mgp 19352 df-ur 19364 df-ring 19411 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-dvr 19548 df-drng 19616 df-lmod 19748 df-lss 19816 df-lsp 19856 df-lvec 19987 df-oposet 36802 df-ol 36804 df-oml 36805 df-covers 36892 df-ats 36893 df-atl 36924 df-cvlat 36948 df-hlat 36977 df-llines 37124 df-lplanes 37125 df-lvols 37126 df-lines 37127 df-psubsp 37129 df-pmap 37130 df-padd 37422 df-lhyp 37614 df-laut 37615 df-ldil 37730 df-ltrn 37731 df-trl 37785 df-tendo 38381 df-edring 38383 df-dvech 38705 |
This theorem is referenced by: hdmaprnlem8N 39482 hdmaprnlem9N 39483 |
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