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Theorem ttgelitv 28828

Description: Betweenness for a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)

**Hypotheses**
Ref	Expression
ttgval.n	⊢ 𝐺 = (toTG‘𝐻)
ttgitvval.i	⊢ 𝐼 = (Itv‘𝐺)
ttgitvval.b	⊢ 𝑃 = (Base‘𝐻)
ttgitvval.m	⊢ − = (-_g‘𝐻)
ttgitvval.s	⊢ · = ( ·_𝑠 ‘𝐻)
ttgelitv.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
ttgelitv.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
ttgelitv.h	⊢ (𝜑 → 𝐻 ∈ 𝑉)
ttgelitv.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)

**Assertion**
Ref	Expression
ttgelitv	⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ ∃𝑘 ∈ (0[,]1)(𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋))))

Distinct variable groups: − ,𝑘 𝑘,𝐻 𝑃,𝑘 𝑘,𝑉 𝑘,𝑋 𝑘,𝑌 𝑘,𝑍 Allowed substitution hints: 𝜑(𝑘) · (𝑘) 𝐺(𝑘) 𝐼(𝑘)

Proof of Theorem ttgelitv Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ttgelitv.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
2		ttgelitv.h	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝑉)
3		ttgelitv.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
4		ttgelitv.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
5		ttgval.n	. . . . . 6 ⊢ 𝐺 = (toTG‘𝐻)
6		ttgitvval.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
7		ttgitvval.b	. . . . . 6 ⊢ 𝑃 = (Base‘𝐻)
8		ttgitvval.m	. . . . . 6 ⊢ − = (-_g‘𝐻)
9		ttgitvval.s	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝐻)
10	5, 6, 7, 8, 9	ttgitvval 28827	. . . . 5 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋𝐼𝑌) = {𝑧 ∈ 𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋))})
11	2, 3, 4, 10	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝑋𝐼𝑌) = {𝑧 ∈ 𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋))})
12	11	eleq2d 2814	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ {𝑧 ∈ 𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋))}))
13		oveq1 7356	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 − 𝑋) = (𝑍 − 𝑋))
14	13	eqeq1d 2731	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋)) ↔ (𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋))))
15	14	rexbidv 3153	. . . 4 ⊢ (𝑧 = 𝑍 → (∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋)) ↔ ∃𝑘 ∈ (0[,]1)(𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋))))
16	15	elrab 3648	. . 3 ⊢ (𝑍 ∈ {𝑧 ∈ 𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋))} ↔ (𝑍 ∈ 𝑃 ∧ ∃𝑘 ∈ (0[,]1)(𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋))))
17	12, 16	bitrdi 287	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ (𝑍 ∈ 𝑃 ∧ ∃𝑘 ∈ (0[,]1)(𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋)))))
18	1, 17	mpbirand 707	1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ ∃𝑘 ∈ (0[,]1)(𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 {crab 3394 ‘cfv 6482 (class class class)co 7349 0cc0 11009 1c1 11010 [,]cicc 13251 Basecbs 17120 ·_𝑠 cvsca 17165 -_gcsg 18814 Itvcitv 28378 toTGcttg 28818

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-dec 12592 df-sets 17075 df-slot 17093 df-ndx 17105 df-itv 28380 df-lng 28381 df-ttg 28819

This theorem is referenced by: ttgbtwnid 28829 ttgcontlem1 28830

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