eengstr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > eengstr		Structured version Visualization version GIF version

Theorem eengstr 28809

Description: The Euclidean geometry as a structure. (Contributed by Thierry Arnoux, 15-Mar-2019.)

**Assertion**
Ref	Expression
eengstr	⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct ⟨1, ;17⟩)

Proof of Theorem eengstr Dummy variables 𝑖 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eengv 28808	. 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 12259	. . . 4 ⊢ 1 ∈ ℕ
3		basendx 17194	. . . 4 ⊢ (Base‘ndx) = 1
4		2nn0 12525	. . . . 5 ⊢ 2 ∈ ℕ₀
5		1nn0 12524	. . . . 5 ⊢ 1 ∈ ℕ₀
6		1lt10 12852	. . . . 5 ⊢ 1 < ;10
7	2, 4, 5, 6	declti 12751	. . . 4 ⊢ 1 < ;12
8		2nn 12321	. . . . 5 ⊢ 2 ∈ ℕ
9	5, 8	decnncl 12733	. . . 4 ⊢ ;12 ∈ ℕ
10		dsndx 17371	. . . 4 ⊢ (dist‘ndx) = ;12
11	2, 3, 7, 9, 10	strle2 17133	. . 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 12337	. . . . 5 ⊢ 6 ∈ ℕ
13	5, 12	decnncl 12733	. . . 4 ⊢ ;16 ∈ ℕ
14		itvndx 28259	. . . 4 ⊢ (Itv‘ndx) = ;16
15		6nn0 12529	. . . . 5 ⊢ 6 ∈ ℕ₀
16		7nn 12340	. . . . 5 ⊢ 7 ∈ ℕ
17		6lt7 12434	. . . . 5 ⊢ 6 < 7
18	5, 15, 16, 17	declt 12741	. . . 4 ⊢ ;16 < ;17
19	5, 16	decnncl 12733	. . . 4 ⊢ ;17 ∈ ℕ
20		lngndx 28260	. . . 4 ⊢ (LineG‘ndx) = ;17
21	13, 14, 18, 19, 20	strle2 17133	. . 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 12432	. . . 4 ⊢ 2 < 6
23	5, 4, 12, 22	declt 12741	. . 3 ⊢ ;12 < ;16
24	11, 21, 23	strleun 17131	. 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 5189	1 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct ⟨1, ;17⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ w3o 1083 ∈ wcel 2098 {crab 3428 ∖ cdif 3944 ∪ cun 3945 {csn 4630 {cpr 4632 ⟨cop 4636 class class class wbr 5150 ‘cfv 6551 (class class class)co 7424 ∈ cmpo 7426 1c1 11145 − cmin 11480 ℕcn 12248 2c2 12303 6c6 12307 7c7 12308 cdc 12713 ...cfz 13522 ↑cexp 14064 Σcsu 15670 Struct cstr 17120 ndxcnx 17167 Basecbs 17185 distcds 17247 Itvcitv 28255 LineGclng 28256 𝔼cee 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

W3C validator

	
		OSZAR »