![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 2793 | . . . 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 38209 | . . . 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 19428 | . . . 4 ⊢ (𝜑 → (𝐽‘{𝐺, 𝐸}) ∈ (LSubSp‘𝐶)) |
12 | mapdindp.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
13 | 2, 3, 4, 8, 11, 12 | lspsnel5 19445 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐽‘{𝐺, 𝐸}) ↔ (𝐽‘{𝐹}) ⊆ (𝐽‘{𝐺, 𝐸}))) |
14 | mapdindp.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
15 | eqid 2793 | . . . . 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 37727 | . . . . 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 19428 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈)) |
22 | mapdindp.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
23 | 14, 15, 16, 18, 21, 22 | lspsnel5 19445 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
24 | mapdindp.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
25 | 14, 15, 16 | lspsncl 19427 | . . . . . 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 38255 | . . . 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 2793 | . . . . . . . . 9 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈) | |
30 | 14, 16, 29, 18, 19, 20 | lsmpr 19539 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍}))) |
31 | 30 | fveq2d 6534 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌, 𝑍})) = (𝑀‘((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})))) |
32 | eqid 2793 | . . . . . . . 8 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶) | |
33 | 14, 15, 16 | lspsncl 19427 | . . . . . . . . 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 19427 | . . . . . . . . 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 38281 | . . . . . . 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 7025 | . . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) = ((𝐽‘{𝐺})(LSSum‘𝐶)(𝐽‘{𝐸}))) |
41 | 31, 37, 40 | 3eqtrd 2833 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌, 𝑍})) = ((𝐽‘{𝐺})(LSSum‘𝐶)(𝐽‘{𝐸}))) |
42 | 2, 4, 32, 8, 9, 10 | lsmpr 19539 | . . . . . 6 ⊢ (𝜑 → (𝐽‘{𝐺, 𝐸}) = ((𝐽‘{𝐺})(LSSum‘𝐶)(𝐽‘{𝐸}))) |
43 | 41, 42 | eqtr4d 2832 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌, 𝑍})) = (𝐽‘{𝐺, 𝐸})) |
44 | 28, 43 | sseq12d 3916 | . . . 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 1520 ∈ wcel 2079 ⊆ wss 3854 {csn 4466 {cpr 4468 ‘cfv 6217 (class class class)co 7007 Basecbs 16300 LSSumclsm 18477 LModclmod 19312 LSubSpclss 19381 LSpanclspn 19421 HLchlt 35967 LHypclh 36601 DVecHcdvh 37695 LCDualclcd 38203 mapdcmpd 38241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-riotaBAD 35570 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-iin 4822 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-om 7428 df-1st 7536 df-2nd 7537 df-tpos 7734 df-undef 7781 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-n0 11735 df-z 11819 df-uz 12083 df-fz 12732 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-sca 16398 df-vsca 16399 df-0g 16532 df-mre 16674 df-mrc 16675 df-acs 16677 df-proset 17355 df-poset 17373 df-plt 17385 df-lub 17401 df-glb 17402 df-join 17403 df-meet 17404 df-p0 17466 df-p1 17467 df-lat 17473 df-clat 17535 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-submnd 17763 df-grp 17852 df-minusg 17853 df-sbg 17854 df-subg 18018 df-cntz 18176 df-oppg 18203 df-lsm 18479 df-cmn 18623 df-abl 18624 df-mgp 18918 df-ur 18930 df-ring 18977 df-oppr 19051 df-dvdsr 19069 df-unit 19070 df-invr 19100 df-dvr 19111 df-drng 19182 df-lmod 19314 df-lss 19382 df-lsp 19422 df-lvec 19553 df-lsatoms 35593 df-lshyp 35594 df-lcv 35636 df-lfl 35675 df-lkr 35703 df-ldual 35741 df-oposet 35793 df-ol 35795 df-oml 35796 df-covers 35883 df-ats 35884 df-atl 35915 df-cvlat 35939 df-hlat 35968 df-llines 36115 df-lplanes 36116 df-lvols 36117 df-lines 36118 df-psubsp 36120 df-pmap 36121 df-padd 36413 df-lhyp 36605 df-laut 36606 df-ldil 36721 df-ltrn 36722 df-trl 36776 df-tgrp 37360 df-tendo 37372 df-edring 37374 df-dveca 37620 df-disoa 37646 df-dvech 37696 df-dib 37756 df-dic 37790 df-dih 37846 df-doch 37965 df-djh 38012 df-lcdual 38204 df-mapd 38242 |
This theorem is referenced by: mapdheq4lem 38348 mapdh6lem1N 38350 mapdh6lem2N 38351 hdmap1l6lem1 38424 hdmap1l6lem2 38425 |
Copyright terms: Public domain | W3C validator |