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Mirrors > Home > MPE Home > Th. List > trkgstr | Structured version Visualization version GIF version |
Description: Functionality of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.) |
Ref | Expression |
---|---|
trkgstr.w | ⊢ 𝑊 = {〈(Base‘ndx), 𝑈〉, 〈(dist‘ndx), 𝐷〉, 〈(Itv‘ndx), 𝐼〉} |
Ref | Expression |
---|---|
trkgstr | ⊢ 𝑊 Struct 〈1, ;16〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trkgstr.w | . 2 ⊢ 𝑊 = {〈(Base‘ndx), 𝑈〉, 〈(dist‘ndx), 𝐷〉, 〈(Itv‘ndx), 𝐼〉} | |
2 | 1nn 11392 | . . 3 ⊢ 1 ∈ ℕ | |
3 | basendx 16330 | . . 3 ⊢ (Base‘ndx) = 1 | |
4 | 2nn0 11666 | . . . 4 ⊢ 2 ∈ ℕ0 | |
5 | 1nn0 11665 | . . . 4 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 11991 | . . . 4 ⊢ 1 < ;10 | |
7 | 2, 4, 5, 6 | declti 11889 | . . 3 ⊢ 1 < ;12 |
8 | 2nn 11453 | . . . 4 ⊢ 2 ∈ ℕ | |
9 | 5, 8 | decnncl 11871 | . . 3 ⊢ ;12 ∈ ℕ |
10 | dsndx 16459 | . . 3 ⊢ (dist‘ndx) = ;12 | |
11 | 6nn 11472 | . . . 4 ⊢ 6 ∈ ℕ | |
12 | 2lt6 11571 | . . . 4 ⊢ 2 < 6 | |
13 | 5, 4, 11, 12 | declt 11879 | . . 3 ⊢ ;12 < ;16 |
14 | 5, 11 | decnncl 11871 | . . 3 ⊢ ;16 ∈ ℕ |
15 | itvndx 25808 | . . 3 ⊢ (Itv‘ndx) = ;16 | |
16 | 2, 3, 7, 9, 10, 13, 14, 15 | strle3 16378 | . 2 ⊢ {〈(Base‘ndx), 𝑈〉, 〈(dist‘ndx), 𝐷〉, 〈(Itv‘ndx), 𝐼〉} Struct 〈1, ;16〉 |
17 | 1, 16 | eqbrtri 4909 | 1 ⊢ 𝑊 Struct 〈1, ;16〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 {ctp 4402 〈cop 4404 class class class wbr 4888 ‘cfv 6137 1c1 10275 2c2 11435 6c6 11439 ;cdc 11850 Struct cstr 16262 ndxcnx 16263 Basecbs 16266 distcds 16358 Itvcitv 25804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-fz 12649 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-ds 16371 df-itv 25806 |
This theorem is referenced by: trkgbas 25813 trkgdist 25814 trkgitv 25815 |
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