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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdindp | Structured version Visualization version GIF version |
Description: Transfer (part of) vector independence condition from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.) |
Ref | Expression |
---|---|
mapdindp.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdindp.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdindp.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdindp.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdindp.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdindp.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdindp.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdindp.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdindp.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdindp.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdindp.mx | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdindp.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
mapdindp.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
mapdindp.g | ⊢ (𝜑 → 𝐺 ∈ 𝐷) |
mapdindp.my | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) |
mapdindp.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
mapdindp.e | ⊢ (𝜑 → 𝐸 ∈ 𝐷) |
mapdindp.mg | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸})) |
mapdindp.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
Ref | Expression |
---|---|
mapdindp | ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐽‘{𝐺, 𝐸})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdindp.xn | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
2 | mapdindp.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
3 | eqid 2818 | . . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
4 | mapdindp.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
5 | mapdindp.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | mapdindp.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
7 | mapdindp.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | 5, 6, 7 | lcdlmod 38608 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
9 | mapdindp.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐷) | |
10 | mapdindp.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐷) | |
11 | 2, 3, 4, 8, 9, 10 | lspprcl 19679 | . . . 4 ⊢ (𝜑 → (𝐽‘{𝐺, 𝐸}) ∈ (LSubSp‘𝐶)) |
12 | mapdindp.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
13 | 2, 3, 4, 8, 11, 12 | lspsnel5 19696 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐽‘{𝐺, 𝐸}) ↔ (𝐽‘{𝐹}) ⊆ (𝐽‘{𝐺, 𝐸}))) |
14 | mapdindp.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
15 | eqid 2818 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
16 | mapdindp.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
17 | mapdindp.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
18 | 5, 17, 7 | dvhlmod 38126 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
19 | mapdindp.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
20 | mapdindp.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
21 | 14, 15, 16, 18, 19, 20 | lspprcl 19679 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈)) |
22 | mapdindp.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
23 | 14, 15, 16, 18, 21, 22 | lspsnel5 19696 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
24 | mapdindp.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
25 | 14, 15, 16 | lspsncl 19678 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
26 | 18, 22, 25 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
27 | 5, 17, 15, 24, 7, 26, 21 | mapdord 38654 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ⊆ (𝑀‘(𝑁‘{𝑌, 𝑍})) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
28 | mapdindp.mx | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
29 | eqid 2818 | . . . . . . . . 9 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈) | |
30 | 14, 16, 29, 18, 19, 20 | lsmpr 19790 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍}))) |
31 | 30 | fveq2d 6667 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌, 𝑍})) = (𝑀‘((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})))) |
32 | eqid 2818 | . . . . . . . 8 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶) | |
33 | 14, 15, 16 | lspsncl 19678 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
34 | 18, 19, 33 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
35 | 14, 15, 16 | lspsncl 19678 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈)) |
36 | 18, 20, 35 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈)) |
37 | 5, 24, 17, 15, 29, 6, 32, 7, 34, 36 | mapdlsm 38680 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍}))) = ((𝑀‘(𝑁‘{𝑌}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍})))) |
38 | mapdindp.my | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) | |
39 | mapdindp.mg | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸})) | |
40 | 38, 39 | oveq12d 7163 | . . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) = ((𝐽‘{𝐺})(LSSum‘𝐶)(𝐽‘{𝐸}))) |
41 | 31, 37, 40 | 3eqtrd 2857 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌, 𝑍})) = ((𝐽‘{𝐺})(LSSum‘𝐶)(𝐽‘{𝐸}))) |
42 | 2, 4, 32, 8, 9, 10 | lsmpr 19790 | . . . . . 6 ⊢ (𝜑 → (𝐽‘{𝐺, 𝐸}) = ((𝐽‘{𝐺})(LSSum‘𝐶)(𝐽‘{𝐸}))) |
43 | 41, 42 | eqtr4d 2856 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌, 𝑍})) = (𝐽‘{𝐺, 𝐸})) |
44 | 28, 43 | sseq12d 3997 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ⊆ (𝑀‘(𝑁‘{𝑌, 𝑍})) ↔ (𝐽‘{𝐹}) ⊆ (𝐽‘{𝐺, 𝐸}))) |
45 | 23, 27, 44 | 3bitr2rd 309 | . . 3 ⊢ (𝜑 → ((𝐽‘{𝐹}) ⊆ (𝐽‘{𝐺, 𝐸}) ↔ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))) |
46 | 13, 45 | bitrd 280 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝐽‘{𝐺, 𝐸}) ↔ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))) |
47 | 1, 46 | mtbird 326 | 1 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐽‘{𝐺, 𝐸})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 {csn 4557 {cpr 4559 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 LSSumclsm 18688 LModclmod 19563 LSubSpclss 19632 LSpanclspn 19672 HLchlt 36366 LHypclh 37000 DVecHcdvh 38094 LCDualclcd 38602 mapdcmpd 38640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-mre 16845 df-mrc 16846 df-acs 16848 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-oppg 18412 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lsatoms 35992 df-lshyp 35993 df-lcv 36035 df-lfl 36074 df-lkr 36102 df-ldual 36140 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tgrp 37759 df-tendo 37771 df-edring 37773 df-dveca 38019 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 df-dih 38245 df-doch 38364 df-djh 38411 df-lcdual 38603 df-mapd 38641 |
This theorem is referenced by: mapdheq4lem 38747 mapdh6lem1N 38749 mapdh6lem2N 38750 hdmap1l6lem1 38823 hdmap1l6lem2 38824 |
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