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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap11lem1 | Structured version Visualization version GIF version |
Description: Lemma for hdmapadd 39469. (Contributed by NM, 26-May-2015.) |
Ref | Expression |
---|---|
hdmap11.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap11.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap11.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap11.p | ⊢ + = (+g‘𝑈) |
hdmap11.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap11.a | ⊢ ✚ = (+g‘𝐶) |
hdmap11.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap11.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap11.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmap11.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
hdmap11.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
hdmap11.o | ⊢ 0 = (0g‘𝑈) |
hdmap11.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap11.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmap11.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap11.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap11.j | ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) |
hdmap11.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmap11lem0.1a | ⊢ (𝜑 → 𝑧 ∈ 𝑉) |
hdmap11lem0.6 | ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) |
hdmap11lem0.2a | ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸})) |
Ref | Expression |
---|---|
hdmap11lem1 | ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap11.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap11.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap11.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap11.p | . . 3 ⊢ + = (+g‘𝑈) | |
5 | hdmap11.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
6 | hdmap11.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | hdmap11.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmap11.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
9 | hdmap11.a | . . 3 ⊢ ✚ = (+g‘𝐶) | |
10 | hdmap11.l | . . 3 ⊢ 𝐿 = (LSpan‘𝐶) | |
11 | hdmap11.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
12 | hdmap11.i | . . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
13 | hdmap11.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
14 | eqid 2738 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
15 | hdmap11.j | . . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
16 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | eqid 2738 | . . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
18 | hdmap11.e | . . . . . . 7 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
19 | 1, 16, 17, 2, 3, 5, 18, 13 | dvheveccl 38738 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 })) |
20 | 1, 2, 3, 5, 7, 8, 14, 15, 13, 19 | hvmapcl2 39392 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
21 | 20 | eldifad 3853 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
22 | 1, 2, 3, 5, 6, 7, 10, 11, 15, 13, 19 | mapdhvmap 39395 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝐸})) = (𝐿‘{(𝐽‘𝐸)})) |
23 | hdmap11lem0.2a | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸})) | |
24 | 23 | necomd 2989 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑧})) |
25 | hdmap11lem0.1a | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝑉) | |
26 | 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 21, 22, 24, 19, 25 | hdmap1cl 39430 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) ∈ 𝐷) |
27 | eqid 2738 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
28 | 1, 2, 13 | dvhlmod 38736 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
29 | hdmap11.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
30 | hdmap11.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
31 | 3, 27, 6, 28, 29, 30 | lspprcl 19862 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
32 | hdmap11lem0.6 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) | |
33 | 5, 27, 28, 31, 25, 32 | lssneln0 19836 | . . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑉 ∖ { 0 })) |
34 | eqidd 2739 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)) | |
35 | eqid 2738 | . . . . . 6 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
36 | eqid 2738 | . . . . . 6 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
37 | 1, 2, 3, 35, 5, 6, 7, 8, 36, 10, 11, 12, 13, 19, 21, 33, 26, 24, 22 | hdmap1eq 39427 | . . . . 5 ⊢ (𝜑 → ((𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) ↔ ((𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)}) ∧ (𝑀‘(𝑁‘{(𝐸(-g‘𝑈)𝑧)})) = (𝐿‘{((𝐽‘𝐸)(-g‘𝐶)(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉))})))) |
38 | 34, 37 | mpbid 235 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)}) ∧ (𝑀‘(𝑁‘{(𝐸(-g‘𝑈)𝑧)})) = (𝐿‘{((𝐽‘𝐸)(-g‘𝐶)(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉))}))) |
39 | 38 | simpld 498 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)})) |
40 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 26, 33, 29, 30, 32, 39 | hdmap1l6 39447 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), (𝑋 + 𝑌)〉) = ((𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑋〉) ✚ (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑌〉))) |
41 | hdmap11.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
42 | 3, 4 | lmodvacl 19760 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
43 | 28, 29, 30, 42 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
44 | 1, 2, 13 | dvhlvec 38735 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec) |
45 | 19 | eldifad 3853 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
46 | 3, 4, 6, 28, 29, 30 | lspprvacl 19883 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌})) |
47 | 27, 6, 28, 31, 46 | lspsnel5a 19880 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋, 𝑌})) |
48 | 47, 32 | ssneldd 3878 | . . . . 5 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{(𝑋 + 𝑌)})) |
49 | 3, 6, 28, 25, 43, 48 | lspsnne2 20002 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{(𝑋 + 𝑌)})) |
50 | 3, 6, 5, 44, 45, 43, 33, 23, 49 | hdmaplem4 39400 | . . 3 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{(𝑋 + 𝑌)}))) |
51 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 43, 25, 50 | hdmapval2 39458 | . 2 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), (𝑋 + 𝑌)〉)) |
52 | 3, 6, 44, 25, 29, 30, 32 | lspindpi 20016 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑌}))) |
53 | 52 | simpld 498 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})) |
54 | 3, 6, 5, 44, 45, 29, 33, 23, 53 | hdmaplem4 39400 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑋}))) |
55 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 29, 25, 54 | hdmapval2 39458 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑋〉)) |
56 | 52 | simprd 499 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑌})) |
57 | 3, 6, 5, 44, 45, 30, 33, 23, 56 | hdmaplem4 39400 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑌}))) |
58 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 30, 25, 57 | hdmapval2 39458 | . . 3 ⊢ (𝜑 → (𝑆‘𝑌) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑌〉)) |
59 | 55, 58 | oveq12d 7182 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋) ✚ (𝑆‘𝑌)) = ((𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑋〉) ✚ (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑌〉))) |
60 | 40, 51, 59 | 3eqtr4d 2783 | 1 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 {csn 4513 {cpr 4515 〈cop 4519 〈cotp 4521 I cid 5424 ↾ cres 5521 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 +gcplusg 16661 0gc0g 16809 -gcsg 18214 LModclmod 19746 LSubSpclss 19815 LSpanclspn 19855 HLchlt 36976 LHypclh 37610 LTrncltrn 37727 DVecHcdvh 38704 LCDualclcd 39212 mapdcmpd 39250 HVMapchvm 39382 HDMap1chdma1 39417 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-ot 4522 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-of 7419 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-mre 16953 df-mrc 16954 df-acs 16956 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-submnd 18066 df-grp 18215 df-minusg 18216 df-sbg 18217 df-subg 18387 df-cntz 18558 df-oppg 18585 df-lsm 18872 df-cmn 19019 df-abl 19020 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-lsatoms 36602 df-lshyp 36603 df-lcv 36645 df-lfl 36684 df-lkr 36712 df-ldual 36750 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-tgrp 38369 df-tendo 38381 df-edring 38383 df-dveca 38629 df-disoa 38655 df-dvech 38705 df-dib 38765 df-dic 38799 df-dih 38855 df-doch 38974 df-djh 39021 df-lcdual 39213 df-mapd 39251 df-hvmap 39383 df-hdmap1 39419 df-hdmap 39420 |
This theorem is referenced by: hdmap11lem2 39468 |
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