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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 29009 | . 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 12275 | . . . 4 ⊢ 1 ∈ ℕ | |
3 | basendx 17254 | . . . 4 ⊢ (Base‘ndx) = 1 | |
4 | 2nn0 12541 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
5 | 1nn0 12540 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 12870 | . . . . 5 ⊢ 1 < ;10 | |
7 | 2, 4, 5, 6 | declti 12769 | . . . 4 ⊢ 1 < ;12 |
8 | 2nn 12337 | . . . . 5 ⊢ 2 ∈ ℕ | |
9 | 5, 8 | decnncl 12751 | . . . 4 ⊢ ;12 ∈ ℕ |
10 | dsndx 17431 | . . . 4 ⊢ (dist‘ndx) = ;12 | |
11 | 2, 3, 7, 9, 10 | strle2 17193 | . . 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 12353 | . . . . 5 ⊢ 6 ∈ ℕ | |
13 | 5, 12 | decnncl 12751 | . . . 4 ⊢ ;16 ∈ ℕ |
14 | itvndx 28460 | . . . 4 ⊢ (Itv‘ndx) = ;16 | |
15 | 6nn0 12545 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
16 | 7nn 12356 | . . . . 5 ⊢ 7 ∈ ℕ | |
17 | 6lt7 12450 | . . . . 5 ⊢ 6 < 7 | |
18 | 5, 15, 16, 17 | declt 12759 | . . . 4 ⊢ ;16 < ;17 |
19 | 5, 16 | decnncl 12751 | . . . 4 ⊢ ;17 ∈ ℕ |
20 | lngndx 28461 | . . . 4 ⊢ (LineG‘ndx) = ;17 | |
21 | 13, 14, 18, 19, 20 | strle2 17193 | . . 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 12448 | . . . 4 ⊢ 2 < 6 | |
23 | 5, 4, 12, 22 | declt 12759 | . . 3 ⊢ ;12 < ;16 |
24 | 11, 21, 23 | strleun 17191 | . 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 1085 ∈ wcel 2106 {crab 3433 ∖ cdif 3960 ∪ cun 3961 {csn 4631 {cpr 4633 〈cop 4637 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1c1 11154 − cmin 11490 ℕcn 12264 2c2 12319 6c6 12323 7c7 12324 ;cdc 12731 ...cfz 13544 ↑cexp 14099 Σcsu 15719 Struct cstr 17180 ndxcnx 17227 Basecbs 17245 distcds 17307 Itvcitv 28456 LineGclng 28457 𝔼cee 28918 Btwn cbtwn 28919 EEGceeng 29007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-seq 14040 df-sum 15720 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-ds 17320 df-itv 28458 df-lng 28459 df-eeng 29008 |
This theorem is referenced by: eengbas 29011 ebtwntg 29012 ecgrtg 29013 elntg 29014 |
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