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Mirrors > Home > MPE Home > Th. List > ttgds | Structured version Visualization version GIF version |
Description: The metric of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.) |
Ref | Expression |
---|---|
ttgval.n | ⊢ 𝐺 = (toTG‘𝐻) |
ttgds.1 | ⊢ 𝐷 = (dist‘𝐻) |
Ref | Expression |
---|---|
ttgds | ⊢ 𝐷 = (dist‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ttgds.1 | . 2 ⊢ 𝐷 = (dist‘𝐻) | |
2 | ttgval.n | . . 3 ⊢ 𝐺 = (toTG‘𝐻) | |
3 | dsid 17106 | . . 3 ⊢ dist = Slot (dist‘ndx) | |
4 | slotslnbpsd 26813 | . . . . 5 ⊢ (((LineG‘ndx) ≠ (Base‘ndx) ∧ (LineG‘ndx) ≠ (+g‘ndx)) ∧ ((LineG‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (LineG‘ndx) ≠ (dist‘ndx))) | |
5 | simprr 770 | . . . . 5 ⊢ ((((LineG‘ndx) ≠ (Base‘ndx) ∧ (LineG‘ndx) ≠ (+g‘ndx)) ∧ ((LineG‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (LineG‘ndx) ≠ (dist‘ndx))) → (LineG‘ndx) ≠ (dist‘ndx)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (LineG‘ndx) ≠ (dist‘ndx) |
7 | 6 | necomi 2998 | . . 3 ⊢ (dist‘ndx) ≠ (LineG‘ndx) |
8 | slotsinbpsd 26812 | . . . . 5 ⊢ (((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) ∧ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx))) | |
9 | simprr 770 | . . . . 5 ⊢ ((((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) ∧ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx))) → (Itv‘ndx) ≠ (dist‘ndx)) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ (Itv‘ndx) ≠ (dist‘ndx) |
11 | 10 | necomi 2998 | . . 3 ⊢ (dist‘ndx) ≠ (Itv‘ndx) |
12 | 2, 3, 7, 11 | ttglem 27248 | . 2 ⊢ (dist‘𝐻) = (dist‘𝐺) |
13 | 1, 12 | eqtri 2766 | 1 ⊢ 𝐷 = (dist‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1539 ≠ wne 2943 ‘cfv 6426 ndxcnx 16904 Basecbs 16922 +gcplusg 16972 ·𝑠 cvsca 16976 distcds 16981 Itvcitv 26804 LineGclng 26805 toTGcttg 27244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-plusg 16985 df-vsca 16989 df-ds 16994 df-itv 26806 df-lng 26807 df-ttg 27245 |
This theorem is referenced by: (None) |
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