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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 | baseid 15919 | . 2 ⊢ Base = Slot (Base‘ndx) | |
2 | fvexd 6203 | . 2 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ V) | |
3 | eengstr 25860 | . . . 4 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct 〈1, ;17〉) | |
4 | isstruct 15870 | . . . . 5 ⊢ ((EEG‘𝑁) Struct 〈1, ;17〉 ↔ ((1 ∈ ℕ ∧ ;17 ∈ ℕ ∧ 1 ≤ ;17) ∧ Fun ((EEG‘𝑁) ∖ {∅}) ∧ dom (EEG‘𝑁) ⊆ (1...;17))) | |
5 | 4 | simp2bi 1077 | . . . 4 ⊢ ((EEG‘𝑁) Struct 〈1, ;17〉 → Fun ((EEG‘𝑁) ∖ {∅})) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ → Fun ((EEG‘𝑁) ∖ {∅})) |
7 | structcnvcnv 15871 | . . . . 5 ⊢ ((EEG‘𝑁) Struct 〈1, ;17〉 → ◡◡(EEG‘𝑁) = ((EEG‘𝑁) ∖ {∅})) | |
8 | 3, 7 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → ◡◡(EEG‘𝑁) = ((EEG‘𝑁) ∖ {∅})) |
9 | 8 | funeqd 5910 | . . 3 ⊢ (𝑁 ∈ ℕ → (Fun ◡◡(EEG‘𝑁) ↔ Fun ((EEG‘𝑁) ∖ {∅}))) |
10 | 6, 9 | mpbird 247 | . 2 ⊢ (𝑁 ∈ ℕ → Fun ◡◡(EEG‘𝑁)) |
11 | opex 4932 | . . . . 5 ⊢ 〈(Base‘ndx), (𝔼‘𝑁)〉 ∈ V | |
12 | 11 | prid1 4297 | . . . 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))〉} |
13 | elun1 3780 | . . . 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〉)})〉})) | |
14 | 12, 13 | 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〉)})〉}) |
15 | eengv 25859 | . . 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〉)})〉})) | |
16 | 14, 15 | syl5eleqr 2708 | . 2 ⊢ (𝑁 ∈ ℕ → 〈(Base‘ndx), (𝔼‘𝑁)〉 ∈ (EEG‘𝑁)) |
17 | fvexd 6203 | . 2 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) ∈ V) | |
18 | 1, 2, 10, 16, 17 | strfv2d 15905 | 1 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 {csn 4177 {cpr 4179 〈cop 4183 class class class wbr 4653 ◡ccnv 5113 dom cdm 5114 Fun wfun 5882 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1c1 9937 ≤ cle 10075 − cmin 10266 ℕcn 11020 2c2 11070 7c7 11075 ;cdc 11493 ...cfz 12326 ↑cexp 12860 Σcsu 14416 Struct cstr 15853 ndxcnx 15854 Basecbs 15857 distcds 15950 Itvcitv 25335 LineGclng 25336 𝔼cee 25768 Btwn cbtwn 25769 EEGceeng 25857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-seq 12802 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-ds 15964 df-itv 25337 df-lng 25338 df-eeng 25858 |
This theorem is referenced by: ebtwntg 25862 ecgrtg 25863 elntg 25864 eengtrkg 25865 eengtrkge 25866 |
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