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Mirrors > Home > MPE Home > Th. List > Mathboxes > prjspval2 | Structured version Visualization version GIF version |
Description: Alternate definition of projective space. (Contributed by Steven Nguyen, 7-Jun-2023.) |
Ref | Expression |
---|---|
prjspval2.0 | β’ 0 = (0gβπ) |
prjspval2.b | β’ π΅ = ((Baseβπ) β { 0 }) |
prjspval2.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
prjspval2 | β’ (π β LVec β (βπ£π πβπ) = βͺ π§ β π΅ {((πβ{π§}) β { 0 })}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prjspval2.b | . . . 4 β’ π΅ = ((Baseβπ) β { 0 }) | |
2 | prjspval2.0 | . . . . . 6 β’ 0 = (0gβπ) | |
3 | 2 | sneqi 4638 | . . . . 5 β’ { 0 } = {(0gβπ)} |
4 | 3 | difeq2i 4118 | . . . 4 β’ ((Baseβπ) β { 0 }) = ((Baseβπ) β {(0gβπ)}) |
5 | 1, 4 | eqtri 2758 | . . 3 β’ π΅ = ((Baseβπ) β {(0gβπ)}) |
6 | eqid 2730 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
7 | eqid 2730 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
8 | eqid 2730 | . . 3 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
9 | 5, 6, 7, 8 | prjspval 41647 | . 2 β’ (π β LVec β (βπ£π πβπ) = (π΅ / {β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β (Baseβ(Scalarβπ))π₯ = (π( Β·π βπ)π¦))})) |
10 | dfqs3 41366 | . . 3 β’ (π΅ / {β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β (Baseβ(Scalarβπ))π₯ = (π( Β·π βπ)π¦))}) = βͺ π§ β π΅ {[π§]{β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β (Baseβ(Scalarβπ))π₯ = (π( Β·π βπ)π¦))}} | |
11 | 10 | a1i 11 | . 2 β’ (π β LVec β (π΅ / {β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β (Baseβ(Scalarβπ))π₯ = (π( Β·π βπ)π¦))}) = βͺ π§ β π΅ {[π§]{β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β (Baseβ(Scalarβπ))π₯ = (π( Β·π βπ)π¦))}}) |
12 | eqid 2730 | . . . . . 6 β’ {β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β (Baseβ(Scalarβπ))π₯ = (π( Β·π βπ)π¦))} = {β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β (Baseβ(Scalarβπ))π₯ = (π( Β·π βπ)π¦))} | |
13 | prjspval2.n | . . . . . 6 β’ π = (LSpanβπ) | |
14 | 12, 5, 7, 6, 8, 13 | prjspeclsp 41656 | . . . . 5 β’ ((π β LVec β§ π§ β π΅) β [π§]{β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β (Baseβ(Scalarβπ))π₯ = (π( Β·π βπ)π¦))} = ((πβ{π§}) β {(0gβπ)})) |
15 | 3 | difeq2i 4118 | . . . . 5 β’ ((πβ{π§}) β { 0 }) = ((πβ{π§}) β {(0gβπ)}) |
16 | 14, 15 | eqtr4di 2788 | . . . 4 β’ ((π β LVec β§ π§ β π΅) β [π§]{β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β (Baseβ(Scalarβπ))π₯ = (π( Β·π βπ)π¦))} = ((πβ{π§}) β { 0 })) |
17 | 16 | sneqd 4639 | . . 3 β’ ((π β LVec β§ π§ β π΅) β {[π§]{β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β (Baseβ(Scalarβπ))π₯ = (π( Β·π βπ)π¦))}} = {((πβ{π§}) β { 0 })}) |
18 | 17 | iuneq2dv 5020 | . 2 β’ (π β LVec β βͺ π§ β π΅ {[π§]{β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β (Baseβ(Scalarβπ))π₯ = (π( Β·π βπ)π¦))}} = βͺ π§ β π΅ {((πβ{π§}) β { 0 })}) |
19 | 9, 11, 18 | 3eqtrd 2774 | 1 β’ (π β LVec β (βπ£π πβπ) = βͺ π§ β π΅ {((πβ{π§}) β { 0 })}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 βwrex 3068 β cdif 3944 {csn 4627 βͺ ciun 4996 {copab 5209 βcfv 6542 (class class class)co 7411 [cec 8703 / cqs 8704 Basecbs 17148 Scalarcsca 17204 Β·π cvsca 17205 0gc0g 17389 LSpanclspn 20726 LVecclvec 20857 βπ£π πcprjsp 41645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-ec 8707 df-qs 8711 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-sbg 18860 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-drng 20502 df-lmod 20616 df-lss 20687 df-lsp 20727 df-lvec 20858 df-prjsp 41646 |
This theorem is referenced by: (None) |
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