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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 28897 | . . . . 5 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋𝐼𝑌) = {𝑧 ∈ 𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋))}) | 
| 11 | 2, 3, 4, 10 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → (𝑋𝐼𝑌) = {𝑧 ∈ 𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋))}) | 
| 12 | 11 | eleq2d 2826 | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ {𝑧 ∈ 𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋))})) | 
| 13 | oveq1 7439 | . . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 − 𝑋) = (𝑍 − 𝑋)) | |
| 14 | 13 | eqeq1d 2738 | . . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋)) ↔ (𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋)))) | 
| 15 | 14 | rexbidv 3178 | . . . 4 ⊢ (𝑧 = 𝑍 → (∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋)) ↔ ∃𝑘 ∈ (0[,]1)(𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋)))) | 
| 16 | 15 | elrab 3691 | . . 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 1539 ∈ wcel 2107 ∃wrex 3069 {crab 3435 ‘cfv 6560 (class class class)co 7432 0cc0 11156 1c1 11157 [,]cicc 13391 Basecbs 17248 ·𝑠 cvsca 17302 -gcsg 18954 Itvcitv 28442 toTGcttg 28882 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-dec 12736 df-sets 17202 df-slot 17220 df-ndx 17232 df-itv 28444 df-lng 28445 df-ttg 28883 | 
| This theorem is referenced by: ttgbtwnid 28899 ttgcontlem1 28900 | 
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