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Mirrors > Home > MPE Home > Th. List > eengstr | Structured version Visualization version GIF version |
Description: The Euclidean geometry as a structure. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
Ref | Expression |
---|---|
eengstr | β’ (π β β β (EEGβπ) Struct β¨1, ;17β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eengv 27970 | . 2 β’ (π β β β (EEGβπ) = ({β¨(Baseβndx), (πΌβπ)β©, β¨(distβndx), (π₯ β (πΌβπ), π¦ β (πΌβπ) β¦ Ξ£π β (1...π)(((π₯βπ) β (π¦βπ))β2))β©} βͺ {β¨(Itvβndx), (π₯ β (πΌβπ), π¦ β (πΌβπ) β¦ {π§ β (πΌβπ) β£ π§ Btwn β¨π₯, π¦β©})β©, β¨(LineGβndx), (π₯ β (πΌβπ), π¦ β ((πΌβπ) β {π₯}) β¦ {π§ β (πΌβπ) β£ (π§ Btwn β¨π₯, π¦β© β¨ π₯ Btwn β¨π§, π¦β© β¨ π¦ Btwn β¨π₯, π§β©)})β©})) | |
2 | 1nn 12171 | . . . 4 β’ 1 β β | |
3 | basendx 17099 | . . . 4 β’ (Baseβndx) = 1 | |
4 | 2nn0 12437 | . . . . 5 β’ 2 β β0 | |
5 | 1nn0 12436 | . . . . 5 β’ 1 β β0 | |
6 | 1lt10 12764 | . . . . 5 β’ 1 < ;10 | |
7 | 2, 4, 5, 6 | declti 12663 | . . . 4 β’ 1 < ;12 |
8 | 2nn 12233 | . . . . 5 β’ 2 β β | |
9 | 5, 8 | decnncl 12645 | . . . 4 β’ ;12 β β |
10 | dsndx 17273 | . . . 4 β’ (distβndx) = ;12 | |
11 | 2, 3, 7, 9, 10 | strle2 17038 | . . 3 β’ {β¨(Baseβndx), (πΌβπ)β©, β¨(distβndx), (π₯ β (πΌβπ), π¦ β (πΌβπ) β¦ Ξ£π β (1...π)(((π₯βπ) β (π¦βπ))β2))β©} Struct β¨1, ;12β© |
12 | 6nn 12249 | . . . . 5 β’ 6 β β | |
13 | 5, 12 | decnncl 12645 | . . . 4 β’ ;16 β β |
14 | itvndx 27421 | . . . 4 β’ (Itvβndx) = ;16 | |
15 | 6nn0 12441 | . . . . 5 β’ 6 β β0 | |
16 | 7nn 12252 | . . . . 5 β’ 7 β β | |
17 | 6lt7 12346 | . . . . 5 β’ 6 < 7 | |
18 | 5, 15, 16, 17 | declt 12653 | . . . 4 β’ ;16 < ;17 |
19 | 5, 16 | decnncl 12645 | . . . 4 β’ ;17 β β |
20 | lngndx 27422 | . . . 4 β’ (LineGβndx) = ;17 | |
21 | 13, 14, 18, 19, 20 | strle2 17038 | . . 3 β’ {β¨(Itvβndx), (π₯ β (πΌβπ), π¦ β (πΌβπ) β¦ {π§ β (πΌβπ) β£ π§ Btwn β¨π₯, π¦β©})β©, β¨(LineGβndx), (π₯ β (πΌβπ), π¦ β ((πΌβπ) β {π₯}) β¦ {π§ β (πΌβπ) β£ (π§ Btwn β¨π₯, π¦β© β¨ π₯ Btwn β¨π§, π¦β© β¨ π¦ Btwn β¨π₯, π§β©)})β©} Struct β¨;16, ;17β© |
22 | 2lt6 12344 | . . . 4 β’ 2 < 6 | |
23 | 5, 4, 12, 22 | declt 12653 | . . 3 β’ ;12 < ;16 |
24 | 11, 21, 23 | strleun 17036 | . 2 β’ ({β¨(Baseβndx), (πΌβπ)β©, β¨(distβndx), (π₯ β (πΌβπ), π¦ β (πΌβπ) β¦ Ξ£π β (1...π)(((π₯βπ) β (π¦βπ))β2))β©} βͺ {β¨(Itvβndx), (π₯ β (πΌβπ), π¦ β (πΌβπ) β¦ {π§ β (πΌβπ) β£ π§ Btwn β¨π₯, π¦β©})β©, β¨(LineGβndx), (π₯ β (πΌβπ), π¦ β ((πΌβπ) β {π₯}) β¦ {π§ β (πΌβπ) β£ (π§ Btwn β¨π₯, π¦β© β¨ π₯ Btwn β¨π§, π¦β© β¨ π¦ Btwn β¨π₯, π§β©)})β©}) Struct β¨1, ;17β© |
25 | 1, 24 | eqbrtrdi 5149 | 1 β’ (π β β β (EEGβπ) Struct β¨1, ;17β©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ w3o 1087 β wcel 2107 {crab 3410 β cdif 3912 βͺ cun 3913 {csn 4591 {cpr 4593 β¨cop 4597 class class class wbr 5110 βcfv 6501 (class class class)co 7362 β cmpo 7364 1c1 11059 β cmin 11392 βcn 12160 2c2 12215 6c6 12219 7c7 12220 ;cdc 12625 ...cfz 13431 βcexp 13974 Ξ£csu 15577 Struct cstr 17025 ndxcnx 17072 Basecbs 17090 distcds 17149 Itvcitv 27417 LineGclng 27418 πΌcee 27879 Btwn cbtwn 27880 EEGceeng 27968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-seq 13914 df-sum 15578 df-struct 17026 df-slot 17061 df-ndx 17073 df-base 17091 df-ds 17162 df-itv 27419 df-lng 27420 df-eeng 27969 |
This theorem is referenced by: eengbas 27972 ebtwntg 27973 ecgrtg 27974 elntg 27975 |
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