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Mirrors > Home > MPE Home > Th. List > ttgelitv | Structured version Visualization version GIF version |
Description: Betweenness for a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) |
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) |
Ref | Expression |
---|---|
ttgelitv | ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ ∃𝑘 ∈ (0[,]1)(𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋)))) |
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 26668 | . . . . 5 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋𝐼𝑌) = {𝑧 ∈ 𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋))}) |
11 | 2, 3, 4, 10 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝑋𝐼𝑌) = {𝑧 ∈ 𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋))}) |
12 | 11 | eleq2d 2898 | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ {𝑧 ∈ 𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋))})) |
13 | oveq1 7163 | . . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 − 𝑋) = (𝑍 − 𝑋)) | |
14 | 13 | eqeq1d 2823 | . . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋)) ↔ (𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋)))) |
15 | 14 | rexbidv 3297 | . . . 4 ⊢ (𝑧 = 𝑍 → (∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋)) ↔ ∃𝑘 ∈ (0[,]1)(𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋)))) |
16 | 15 | elrab 3680 | . . 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 | syl6bb 289 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ (𝑍 ∈ 𝑃 ∧ ∃𝑘 ∈ (0[,]1)(𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋))))) |
18 | 1, 17 | mpbirand 705 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ ∃𝑘 ∈ (0[,]1)(𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 {crab 3142 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 [,]cicc 12742 Basecbs 16483 ·𝑠 cvsca 16569 -gcsg 18105 Itvcitv 26222 toTGcttg 26659 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-dec 12100 df-ndx 16486 df-slot 16487 df-sets 16490 df-itv 26224 df-lng 26225 df-ttg 26660 |
This theorem is referenced by: ttgbtwnid 26670 ttgcontlem1 26671 |
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