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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpg | Structured version Visualization version GIF version |
Description: Part 1 of proof of the first fundamental theorem of projective geometry. Part (1) in [Baer] p. 44. Our notation corresponds to Baer's as follows: π for *, πβ{} for F(), π½β{} for G(), π for x, πΊ for x', π for y, β for y'. TODO: Rename variables per mapdhval 41197. (Contributed by NM, 22-Mar-2015.) |
Ref | Expression |
---|---|
mapdpg.h | β’ π» = (LHypβπΎ) |
mapdpg.m | β’ π = ((mapdβπΎ)βπ) |
mapdpg.u | β’ π = ((DVecHβπΎ)βπ) |
mapdpg.v | β’ π = (Baseβπ) |
mapdpg.s | β’ β = (-gβπ) |
mapdpg.z | β’ 0 = (0gβπ) |
mapdpg.n | β’ π = (LSpanβπ) |
mapdpg.c | β’ πΆ = ((LCDualβπΎ)βπ) |
mapdpg.f | β’ πΉ = (BaseβπΆ) |
mapdpg.r | β’ π = (-gβπΆ) |
mapdpg.j | β’ π½ = (LSpanβπΆ) |
mapdpg.k | β’ (π β (πΎ β HL β§ π β π»)) |
mapdpg.x | β’ (π β π β (π β { 0 })) |
mapdpg.y | β’ (π β π β (π β { 0 })) |
mapdpg.g | β’ (π β πΊ β πΉ) |
mapdpg.ne | β’ (π β (πβ{π}) β (πβ{π})) |
mapdpg.e | β’ (π β (πβ(πβ{π})) = (π½β{πΊ})) |
Ref | Expression |
---|---|
mapdpg | β’ (π β β!β β πΉ ((πβ(πβ{π})) = (π½β{β}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ β)}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpg.h | . . 3 β’ π» = (LHypβπΎ) | |
2 | mapdpg.m | . . 3 β’ π = ((mapdβπΎ)βπ) | |
3 | mapdpg.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
4 | mapdpg.v | . . 3 β’ π = (Baseβπ) | |
5 | mapdpg.s | . . 3 β’ β = (-gβπ) | |
6 | mapdpg.z | . . 3 β’ 0 = (0gβπ) | |
7 | mapdpg.n | . . 3 β’ π = (LSpanβπ) | |
8 | mapdpg.c | . . 3 β’ πΆ = ((LCDualβπΎ)βπ) | |
9 | mapdpg.f | . . 3 β’ πΉ = (BaseβπΆ) | |
10 | mapdpg.r | . . 3 β’ π = (-gβπΆ) | |
11 | mapdpg.j | . . 3 β’ π½ = (LSpanβπΆ) | |
12 | mapdpg.k | . . 3 β’ (π β (πΎ β HL β§ π β π»)) | |
13 | mapdpg.x | . . 3 β’ (π β π β (π β { 0 })) | |
14 | mapdpg.y | . . 3 β’ (π β π β (π β { 0 })) | |
15 | mapdpg.g | . . 3 β’ (π β πΊ β πΉ) | |
16 | mapdpg.ne | . . 3 β’ (π β (πβ{π}) β (πβ{π})) | |
17 | mapdpg.e | . . 3 β’ (π β (πβ(πβ{π})) = (π½β{πΊ})) | |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | mapdpglem24 41177 | . 2 β’ (π β ββ β πΉ ((πβ(πβ{π})) = (π½β{β}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ β)}))) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | mapdpglem32 41178 | . . . 4 β’ ((π β§ (β β πΉ β§ π β πΉ) β§ (((πβ(πβ{π})) = (π½β{β}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ β)})) β§ ((πβ(πβ{π})) = (π½β{π}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ π)})))) β β = π) |
20 | 19 | 3exp 1117 | . . 3 β’ (π β ((β β πΉ β§ π β πΉ) β ((((πβ(πβ{π})) = (π½β{β}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ β)})) β§ ((πβ(πβ{π})) = (π½β{π}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ π)}))) β β = π))) |
21 | 20 | ralrimivv 3195 | . 2 β’ (π β ββ β πΉ βπ β πΉ ((((πβ(πβ{π})) = (π½β{β}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ β)})) β§ ((πβ(πβ{π})) = (π½β{π}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ π)}))) β β = π)) |
22 | sneq 4639 | . . . . . 6 β’ (β = π β {β} = {π}) | |
23 | 22 | fveq2d 6901 | . . . . 5 β’ (β = π β (π½β{β}) = (π½β{π})) |
24 | 23 | eqeq2d 2739 | . . . 4 β’ (β = π β ((πβ(πβ{π})) = (π½β{β}) β (πβ(πβ{π})) = (π½β{π}))) |
25 | oveq2 7428 | . . . . . . 7 β’ (β = π β (πΊπ β) = (πΊπ π)) | |
26 | 25 | sneqd 4641 | . . . . . 6 β’ (β = π β {(πΊπ β)} = {(πΊπ π)}) |
27 | 26 | fveq2d 6901 | . . . . 5 β’ (β = π β (π½β{(πΊπ β)}) = (π½β{(πΊπ π)})) |
28 | 27 | eqeq2d 2739 | . . . 4 β’ (β = π β ((πβ(πβ{(π β π)})) = (π½β{(πΊπ β)}) β (πβ(πβ{(π β π)})) = (π½β{(πΊπ π)}))) |
29 | 24, 28 | anbi12d 631 | . . 3 β’ (β = π β (((πβ(πβ{π})) = (π½β{β}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ β)})) β ((πβ(πβ{π})) = (π½β{π}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ π)})))) |
30 | 29 | reu4 3726 | . 2 β’ (β!β β πΉ ((πβ(πβ{π})) = (π½β{β}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ β)})) β (ββ β πΉ ((πβ(πβ{π})) = (π½β{β}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ β)})) β§ ββ β πΉ βπ β πΉ ((((πβ(πβ{π})) = (π½β{β}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ β)})) β§ ((πβ(πβ{π})) = (π½β{π}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ π)}))) β β = π))) |
31 | 18, 21, 30 | sylanbrc 582 | 1 β’ (π β β!β β πΉ ((πβ(πβ{π})) = (π½β{β}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΊπ β)}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2937 βwral 3058 βwrex 3067 β!wreu 3371 β cdif 3944 {csn 4629 βcfv 6548 (class class class)co 7420 Basecbs 17180 0gc0g 17421 -gcsg 18892 LSpanclspn 20855 HLchlt 38822 LHypclh 39457 DVecHcdvh 40551 LCDualclcd 41059 mapdcmpd 41097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-riotaBAD 38425 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-0g 17423 df-mre 17566 df-mrc 17567 df-acs 17569 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-cntz 19268 df-oppg 19297 df-lsm 19591 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-drng 20626 df-lmod 20745 df-lss 20816 df-lsp 20856 df-lvec 20988 df-lsatoms 38448 df-lshyp 38449 df-lcv 38491 df-lfl 38530 df-lkr 38558 df-ldual 38596 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-tgrp 40216 df-tendo 40228 df-edring 40230 df-dveca 40476 df-disoa 40502 df-dvech 40552 df-dib 40612 df-dic 40646 df-dih 40702 df-doch 40821 df-djh 40868 df-lcdual 41060 df-mapd 41098 |
This theorem is referenced by: mapdhcl 41200 mapdheq 41201 hdmap1eq 41274 |
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