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Mirrors > Home > MPE Home > Th. List > eengbas | Structured version Visualization version GIF version |
Description: The Base of the Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
Ref | Expression |
---|---|
eengbas | ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eengstr 26693 | . 2 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct 〈1, ;17〉) | |
2 | fvexd 6678 | . 2 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) ∈ V) | |
3 | opex 5347 | . . . . 5 ⊢ 〈(Base‘ndx), (𝔼‘𝑁)〉 ∈ V | |
4 | 3 | prid1 4690 | . . . 4 ⊢ 〈(Base‘ndx), (𝔼‘𝑁)〉 ∈ {〈(Base‘ndx), (𝔼‘𝑁)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} |
5 | elun1 4149 | . . . 4 ⊢ (〈(Base‘ndx), (𝔼‘𝑁)〉 ∈ {〈(Base‘ndx), (𝔼‘𝑁)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} → 〈(Base‘ndx), (𝔼‘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〉)})〉})) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ 〈(Base‘ndx), (𝔼‘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〉)})〉}) |
7 | eengv 26692 | . . 3 ⊢ (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〉)})〉})) | |
8 | 6, 7 | eleqtrrid 2917 | . 2 ⊢ (𝑁 ∈ ℕ → 〈(Base‘ndx), (𝔼‘𝑁)〉 ∈ (EEG‘𝑁)) |
9 | 1, 2, 8 | opelstrbas 16585 | 1 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1078 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ∖ cdif 3930 ∪ cun 3931 {csn 4557 {cpr 4559 〈cop 4563 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 1c1 10526 − cmin 10858 ℕcn 11626 2c2 11680 7c7 11685 ;cdc 12086 ...cfz 12880 ↑cexp 13417 Σcsu 15030 ndxcnx 16468 Basecbs 16471 distcds 16562 Itvcitv 26149 LineGclng 26150 𝔼cee 26601 Btwn cbtwn 26602 EEGceeng 26690 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-seq 13358 df-sum 15031 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-ds 16575 df-itv 26151 df-lng 26152 df-eeng 26691 |
This theorem is referenced by: ebtwntg 26695 ecgrtg 26696 elntg 26697 elntg2 26698 eengtrkg 26699 eengtrkge 26700 eenglngeehlnm 44654 |
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