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Mirrors > Home > MPE Home > Th. List > ttgplusg | Structured version Visualization version GIF version |
Description: The addition operation 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) |
ttgplusg.1 | ⊢ + = (+g‘𝐻) |
Ref | Expression |
---|---|
ttgplusg | ⊢ + = (+g‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ttgplusg.1 | . 2 ⊢ + = (+g‘𝐻) | |
2 | ttgval.n | . . 3 ⊢ 𝐺 = (toTG‘𝐻) | |
3 | plusgid 17268 | . . 3 ⊢ +g = Slot (+g‘ndx) | |
4 | slotslnbpsd 28323 | . . . . 5 ⊢ (((LineG‘ndx) ≠ (Base‘ndx) ∧ (LineG‘ndx) ≠ (+g‘ndx)) ∧ ((LineG‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (LineG‘ndx) ≠ (dist‘ndx))) | |
5 | simplr 767 | . . . . 5 ⊢ ((((LineG‘ndx) ≠ (Base‘ndx) ∧ (LineG‘ndx) ≠ (+g‘ndx)) ∧ ((LineG‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (LineG‘ndx) ≠ (dist‘ndx))) → (LineG‘ndx) ≠ (+g‘ndx)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (LineG‘ndx) ≠ (+g‘ndx) |
7 | 6 | necomi 2984 | . . 3 ⊢ (+g‘ndx) ≠ (LineG‘ndx) |
8 | slotsinbpsd 28322 | . . . . 5 ⊢ (((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) ∧ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx))) | |
9 | simplr 767 | . . . . 5 ⊢ ((((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) ∧ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx))) → (Itv‘ndx) ≠ (+g‘ndx)) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ (Itv‘ndx) ≠ (+g‘ndx) |
11 | 10 | necomi 2984 | . . 3 ⊢ (+g‘ndx) ≠ (Itv‘ndx) |
12 | 2, 3, 7, 11 | ttglem 28758 | . 2 ⊢ (+g‘𝐻) = (+g‘𝐺) |
13 | 1, 12 | eqtri 2753 | 1 ⊢ + = (+g‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ≠ wne 2929 ‘cfv 6549 ndxcnx 17170 Basecbs 17188 +gcplusg 17241 ·𝑠 cvsca 17245 distcds 17250 Itvcitv 28314 LineGclng 28315 toTGcttg 28754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-plusg 17254 df-vsca 17258 df-ds 17263 df-itv 28316 df-lng 28317 df-ttg 28755 |
This theorem is referenced by: ttgsub 28764 |
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