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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 28808 | . 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 12259 | . . . 4 β’ 1 β β | |
3 | basendx 17194 | . . . 4 β’ (Baseβndx) = 1 | |
4 | 2nn0 12525 | . . . . 5 β’ 2 β β0 | |
5 | 1nn0 12524 | . . . . 5 β’ 1 β β0 | |
6 | 1lt10 12852 | . . . . 5 β’ 1 < ;10 | |
7 | 2, 4, 5, 6 | declti 12751 | . . . 4 β’ 1 < ;12 |
8 | 2nn 12321 | . . . . 5 β’ 2 β β | |
9 | 5, 8 | decnncl 12733 | . . . 4 β’ ;12 β β |
10 | dsndx 17371 | . . . 4 β’ (distβndx) = ;12 | |
11 | 2, 3, 7, 9, 10 | strle2 17133 | . . 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 12337 | . . . . 5 β’ 6 β β | |
13 | 5, 12 | decnncl 12733 | . . . 4 β’ ;16 β β |
14 | itvndx 28259 | . . . 4 β’ (Itvβndx) = ;16 | |
15 | 6nn0 12529 | . . . . 5 β’ 6 β β0 | |
16 | 7nn 12340 | . . . . 5 β’ 7 β β | |
17 | 6lt7 12434 | . . . . 5 β’ 6 < 7 | |
18 | 5, 15, 16, 17 | declt 12741 | . . . 4 β’ ;16 < ;17 |
19 | 5, 16 | decnncl 12733 | . . . 4 β’ ;17 β β |
20 | lngndx 28260 | . . . 4 β’ (LineGβndx) = ;17 | |
21 | 13, 14, 18, 19, 20 | strle2 17133 | . . 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 12432 | . . . 4 β’ 2 < 6 | |
23 | 5, 4, 12, 22 | declt 12741 | . . 3 β’ ;12 < ;16 |
24 | 11, 21, 23 | strleun 17131 | . 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 5189 | 1 β’ (π β β β (EEGβπ) Struct β¨1, ;17β©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ w3o 1083 β wcel 2098 {crab 3428 β cdif 3944 βͺ cun 3945 {csn 4630 {cpr 4632 β¨cop 4636 class class class wbr 5150 βcfv 6551 (class class class)co 7424 β cmpo 7426 1c1 11145 β cmin 11480 βcn 12248 2c2 12303 6c6 12307 7c7 12308 ;cdc 12713 ...cfz 13522 βcexp 14064 Ξ£csu 15670 Struct cstr 17120 ndxcnx 17167 Basecbs 17185 distcds 17247 Itvcitv 28255 LineGclng 28256 πΌcee 28717 Btwn cbtwn 28718 EEGceeng 28806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 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 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-fz 13523 df-seq 14005 df-sum 15671 df-struct 17121 df-slot 17156 df-ndx 17168 df-base 17186 df-ds 17260 df-itv 28257 df-lng 28258 df-eeng 28807 |
This theorem is referenced by: eengbas 28810 ebtwntg 28811 ecgrtg 28812 elntg 28813 |
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