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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 26693 | . 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 11638 | . . . 4 ⊢ 1 ∈ ℕ | |
3 | basendx 16537 | . . . 4 ⊢ (Base‘ndx) = 1 | |
4 | 2nn0 11903 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
5 | 1nn0 11902 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 12226 | . . . . 5 ⊢ 1 < ;10 | |
7 | 2, 4, 5, 6 | declti 12125 | . . . 4 ⊢ 1 < ;12 |
8 | 2nn 11699 | . . . . 5 ⊢ 2 ∈ ℕ | |
9 | 5, 8 | decnncl 12107 | . . . 4 ⊢ ;12 ∈ ℕ |
10 | dsndx 16665 | . . . 4 ⊢ (dist‘ndx) = ;12 | |
11 | 2, 3, 7, 9, 10 | strle2 16583 | . . 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 11715 | . . . . 5 ⊢ 6 ∈ ℕ | |
13 | 5, 12 | decnncl 12107 | . . . 4 ⊢ ;16 ∈ ℕ |
14 | itvndx 26154 | . . . 4 ⊢ (Itv‘ndx) = ;16 | |
15 | 6nn0 11907 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
16 | 7nn 11718 | . . . . 5 ⊢ 7 ∈ ℕ | |
17 | 6lt7 11812 | . . . . 5 ⊢ 6 < 7 | |
18 | 5, 15, 16, 17 | declt 12115 | . . . 4 ⊢ ;16 < ;17 |
19 | 5, 16 | decnncl 12107 | . . . 4 ⊢ ;17 ∈ ℕ |
20 | lngndx 26155 | . . . 4 ⊢ (LineG‘ndx) = ;17 | |
21 | 13, 14, 18, 19, 20 | strle2 16583 | . . 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 11810 | . . . 4 ⊢ 2 < 6 | |
23 | 5, 4, 12, 22 | declt 12115 | . . 3 ⊢ ;12 < ;16 |
24 | 11, 21, 23 | strleun 16581 | . 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 5097 | 1 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct 〈1, ;17〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1078 ∈ wcel 2105 {crab 3142 ∖ cdif 3932 ∪ cun 3933 {csn 4559 {cpr 4561 〈cop 4565 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 ∈ cmpo 7147 1c1 10527 − cmin 10859 ℕcn 11627 2c2 11681 6c6 11685 7c7 11686 ;cdc 12087 ...cfz 12882 ↑cexp 13419 Σcsu 15032 Struct cstr 16469 ndxcnx 16470 Basecbs 16473 distcds 16564 Itvcitv 26150 LineGclng 26151 𝔼cee 26602 Btwn cbtwn 26603 EEGceeng 26691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-fz 12883 df-seq 13360 df-sum 15033 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-ds 16577 df-itv 26152 df-lng 26153 df-eeng 26692 |
This theorem is referenced by: eengbas 26695 ebtwntg 26696 ecgrtg 26697 elntg 26698 |
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