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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 | ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct 〈1, ;17〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eengv 28803 | . 2 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) = ({〈(Base‘ndx), (𝔼‘𝑁)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} ∪ {〈(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ 𝑧 Btwn 〈𝑥, 𝑦〉})〉, 〈(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn 〈𝑥, 𝑦〉 ∨ 𝑥 Btwn 〈𝑧, 𝑦〉 ∨ 𝑦 Btwn 〈𝑥, 𝑧〉)})〉})) | |
2 | 1nn 12254 | . . . 4 ⊢ 1 ∈ ℕ | |
3 | basendx 17189 | . . . 4 ⊢ (Base‘ndx) = 1 | |
4 | 2nn0 12520 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
5 | 1nn0 12519 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 12847 | . . . . 5 ⊢ 1 < ;10 | |
7 | 2, 4, 5, 6 | declti 12746 | . . . 4 ⊢ 1 < ;12 |
8 | 2nn 12316 | . . . . 5 ⊢ 2 ∈ ℕ | |
9 | 5, 8 | decnncl 12728 | . . . 4 ⊢ ;12 ∈ ℕ |
10 | dsndx 17366 | . . . 4 ⊢ (dist‘ndx) = ;12 | |
11 | 2, 3, 7, 9, 10 | strle2 17128 | . . 3 ⊢ {〈(Base‘ndx), (𝔼‘𝑁)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} Struct 〈1, ;12〉 |
12 | 6nn 12332 | . . . . 5 ⊢ 6 ∈ ℕ | |
13 | 5, 12 | decnncl 12728 | . . . 4 ⊢ ;16 ∈ ℕ |
14 | itvndx 28254 | . . . 4 ⊢ (Itv‘ndx) = ;16 | |
15 | 6nn0 12524 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
16 | 7nn 12335 | . . . . 5 ⊢ 7 ∈ ℕ | |
17 | 6lt7 12429 | . . . . 5 ⊢ 6 < 7 | |
18 | 5, 15, 16, 17 | declt 12736 | . . . 4 ⊢ ;16 < ;17 |
19 | 5, 16 | decnncl 12728 | . . . 4 ⊢ ;17 ∈ ℕ |
20 | lngndx 28255 | . . . 4 ⊢ (LineG‘ndx) = ;17 | |
21 | 13, 14, 18, 19, 20 | strle2 17128 | . . 3 ⊢ {〈(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ 𝑧 Btwn 〈𝑥, 𝑦〉})〉, 〈(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn 〈𝑥, 𝑦〉 ∨ 𝑥 Btwn 〈𝑧, 𝑦〉 ∨ 𝑦 Btwn 〈𝑥, 𝑧〉)})〉} Struct 〈;16, ;17〉 |
22 | 2lt6 12427 | . . . 4 ⊢ 2 < 6 | |
23 | 5, 4, 12, 22 | declt 12736 | . . 3 ⊢ ;12 < ;16 |
24 | 11, 21, 23 | strleun 17126 | . 2 ⊢ ({〈(Base‘ndx), (𝔼‘𝑁)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} ∪ {〈(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ 𝑧 Btwn 〈𝑥, 𝑦〉})〉, 〈(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn 〈𝑥, 𝑦〉 ∨ 𝑥 Btwn 〈𝑧, 𝑦〉 ∨ 𝑦 Btwn 〈𝑥, 𝑧〉)})〉}) Struct 〈1, ;17〉 |
25 | 1, 24 | eqbrtrdi 5187 | 1 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct 〈1, ;17〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1084 ∈ wcel 2099 {crab 3429 ∖ cdif 3944 ∪ cun 3945 {csn 4629 {cpr 4631 〈cop 4635 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 1c1 11140 − cmin 11475 ℕcn 12243 2c2 12298 6c6 12302 7c7 12303 ;cdc 12708 ...cfz 13517 ↑cexp 14059 Σcsu 15665 Struct cstr 17115 ndxcnx 17162 Basecbs 17180 distcds 17242 Itvcitv 28250 LineGclng 28251 𝔼cee 28712 Btwn cbtwn 28713 EEGceeng 28801 |
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-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 |
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-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-op 4636 df-uni 4909 df-iun 4998 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-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 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-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-seq 14000 df-sum 15666 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-ds 17255 df-itv 28252 df-lng 28253 df-eeng 28802 |
This theorem is referenced by: eengbas 28805 ebtwntg 28806 ecgrtg 28807 elntg 28808 |
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