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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 17366 | . . 3 ⊢ dist = Slot (dist‘ndx) | |
4 | slotslnbpsd 28245 | . . . . 5 ⊢ (((LineG‘ndx) ≠ (Base‘ndx) ∧ (LineG‘ndx) ≠ (+g‘ndx)) ∧ ((LineG‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (LineG‘ndx) ≠ (dist‘ndx))) | |
5 | simprr 772 | . . . . 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 2992 | . . 3 ⊢ (dist‘ndx) ≠ (LineG‘ndx) |
8 | slotsinbpsd 28244 | . . . . 5 ⊢ (((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) ∧ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx))) | |
9 | simprr 772 | . . . . 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 2992 | . . 3 ⊢ (dist‘ndx) ≠ (Itv‘ndx) |
12 | 2, 3, 7, 11 | ttglem 28680 | . 2 ⊢ (dist‘𝐻) = (dist‘𝐺) |
13 | 1, 12 | eqtri 2756 | 1 ⊢ 𝐷 = (dist‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ≠ wne 2937 ‘cfv 6548 ndxcnx 17161 Basecbs 17179 +gcplusg 17232 ·𝑠 cvsca 17236 distcds 17241 Itvcitv 28236 LineGclng 28237 toTGcttg 28676 |
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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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-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 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-vsca 17249 df-ds 17254 df-itv 28238 df-lng 28239 df-ttg 28677 |
This theorem is referenced by: (None) |
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