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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 26768 | . 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 11652 | . . . 4 ⊢ 1 ∈ ℕ | |
3 | basendx 16550 | . . . 4 ⊢ (Base‘ndx) = 1 | |
4 | 2nn0 11917 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
5 | 1nn0 11916 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 12240 | . . . . 5 ⊢ 1 < ;10 | |
7 | 2, 4, 5, 6 | declti 12139 | . . . 4 ⊢ 1 < ;12 |
8 | 2nn 11713 | . . . . 5 ⊢ 2 ∈ ℕ | |
9 | 5, 8 | decnncl 12121 | . . . 4 ⊢ ;12 ∈ ℕ |
10 | dsndx 16678 | . . . 4 ⊢ (dist‘ndx) = ;12 | |
11 | 2, 3, 7, 9, 10 | strle2 16596 | . . 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 11729 | . . . . 5 ⊢ 6 ∈ ℕ | |
13 | 5, 12 | decnncl 12121 | . . . 4 ⊢ ;16 ∈ ℕ |
14 | itvndx 26229 | . . . 4 ⊢ (Itv‘ndx) = ;16 | |
15 | 6nn0 11921 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
16 | 7nn 11732 | . . . . 5 ⊢ 7 ∈ ℕ | |
17 | 6lt7 11826 | . . . . 5 ⊢ 6 < 7 | |
18 | 5, 15, 16, 17 | declt 12129 | . . . 4 ⊢ ;16 < ;17 |
19 | 5, 16 | decnncl 12121 | . . . 4 ⊢ ;17 ∈ ℕ |
20 | lngndx 26230 | . . . 4 ⊢ (LineG‘ndx) = ;17 | |
21 | 13, 14, 18, 19, 20 | strle2 16596 | . . 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 11824 | . . . 4 ⊢ 2 < 6 | |
23 | 5, 4, 12, 22 | declt 12129 | . . 3 ⊢ ;12 < ;16 |
24 | 11, 21, 23 | strleun 16594 | . 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 5108 | 1 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct 〈1, ;17〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1082 ∈ wcel 2113 {crab 3145 ∖ cdif 3936 ∪ cun 3937 {csn 4570 {cpr 4572 〈cop 4576 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 1c1 10541 − cmin 10873 ℕcn 11641 2c2 11695 6c6 11699 7c7 11700 ;cdc 12101 ...cfz 12895 ↑cexp 13432 Σcsu 15045 Struct cstr 16482 ndxcnx 16483 Basecbs 16486 distcds 16577 Itvcitv 26225 LineGclng 26226 𝔼cee 26677 Btwn cbtwn 26678 EEGceeng 26766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-seq 13373 df-sum 15046 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-ds 16590 df-itv 26227 df-lng 26228 df-eeng 26767 |
This theorem is referenced by: eengbas 26770 ebtwntg 26771 ecgrtg 26772 elntg 26773 |
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