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Theorem trkgstr 28261

Description: Functionality of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)

**Hypothesis**
Ref	Expression
trkgstr.w	⊢ 𝑊 = {⟨(Base‘ndx), 𝑈⟩, ⟨(dist‘ndx), 𝐷⟩, ⟨(Itv‘ndx), 𝐼⟩}

**Assertion**
Ref	Expression
trkgstr	⊢ 𝑊 Struct ⟨1, ;16⟩

**Proof of Theorem trkgstr**
Step	Hyp	Ref	Expression
1		trkgstr.w	. 2 ⊢ 𝑊 = {⟨(Base‘ndx), 𝑈⟩, ⟨(dist‘ndx), 𝐷⟩, ⟨(Itv‘ndx), 𝐼⟩}
2		1nn 12254	. . 3 ⊢ 1 ∈ ℕ
3		basendx 17189	. . 3 ⊢ (Base‘ndx) = 1
4		2nn0 12520	. . . 4 ⊢ 2 ∈ ℕ₀
5		1nn0 12519	. . . 4 ⊢ 1 ∈ ℕ₀
6		1lt10 12847	. . . 4 ⊢ 1 < ;10
7	2, 4, 5, 6	declti 12746	. . 3 ⊢ 1 < ;12
8		2nn 12316	. . . 4 ⊢ 2 ∈ ℕ
9	5, 8	decnncl 12728	. . 3 ⊢ ;12 ∈ ℕ
10		dsndx 17366	. . 3 ⊢ (dist‘ndx) = ;12
11		6nn 12332	. . . 4 ⊢ 6 ∈ ℕ
12		2lt6 12427	. . . 4 ⊢ 2 < 6
13	5, 4, 11, 12	declt 12736	. . 3 ⊢ ;12 < ;16
14	5, 11	decnncl 12728	. . 3 ⊢ ;16 ∈ ℕ
15		itvndx 28254	. . 3 ⊢ (Itv‘ndx) = ;16
16	2, 3, 7, 9, 10, 13, 14, 15	strle3 17129	. 2 ⊢ {⟨(Base‘ndx), 𝑈⟩, ⟨(dist‘ndx), 𝐷⟩, ⟨(Itv‘ndx), 𝐼⟩} Struct ⟨1, ;16⟩
17	1, 16	eqbrtri 5169	1 ⊢ 𝑊 Struct ⟨1, ;16⟩

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 {ctp 4633 ⟨cop 4635 class class class wbr 5148 ‘cfv 6548 1c1 11140 2c2 12298 6c6 12302 cdc 12708 Struct cstr 17115 ndxcnx 17162 Basecbs 17180 distcds 17242 Itvcitv 28250

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-ds 17255 df-itv 28252

This theorem is referenced by: trkgbas 28262 trkgdist 28263 trkgitv 28264

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