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Mirrors > Home > MPE Home > Th. List > legbtwn | Structured version Visualization version GIF version |
Description: Deduce betweenness from "less than" relation. Corresponds loosely to Proposition 6.13 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 25-Aug-2019.) |
Ref | Expression |
---|---|
legval.p | ⊢ 𝑃 = (Base‘𝐺) |
legval.d | ⊢ − = (dist‘𝐺) |
legval.i | ⊢ 𝐼 = (Itv‘𝐺) |
legval.l | ⊢ ≤ = (≤G‘𝐺) |
legval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
legid.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
legid.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
legtrd.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
legtrd.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
legbtwn.1 | ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) |
legbtwn.2 | ⊢ (𝜑 → (𝐶 − 𝐴) ≤ (𝐶 − 𝐵)) |
Ref | Expression |
---|---|
legbtwn | ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐶𝐼𝐵)) | |
2 | legval.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
3 | legval.d | . . . . 5 ⊢ − = (dist‘𝐺) | |
4 | legval.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | legval.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG) |
7 | legid.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | 7 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 ∈ 𝑃) |
9 | legid.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ 𝑃) |
11 | legtrd.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
12 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐶 ∈ 𝑃) |
13 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴)) | |
14 | 2, 3, 4, 6, 12, 10, 8, 13 | tgbtwncom 28510 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶)) |
15 | 2, 3, 4, 6, 10, 12 | tgbtwntriv1 28513 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐵𝐼𝐶)) |
16 | legval.l | . . . . . . . 8 ⊢ ≤ = (≤G‘𝐺) | |
17 | legbtwn.2 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶 − 𝐴) ≤ (𝐶 − 𝐵)) | |
18 | 17 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐶 − 𝐴) ≤ (𝐶 − 𝐵)) |
19 | 2, 3, 4, 16, 6, 12, 10, 8, 13 | btwnleg 28610 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐶 − 𝐵) ≤ (𝐶 − 𝐴)) |
20 | 2, 3, 4, 16, 6, 12, 8, 12, 10, 18, 19 | legtri3 28612 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐶 − 𝐴) = (𝐶 − 𝐵)) |
21 | 2, 3, 4, 6, 12, 8, 12, 10, 20 | tgcgrcomlr 28502 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐴 − 𝐶) = (𝐵 − 𝐶)) |
22 | eqidd 2735 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐵 − 𝐶) = (𝐵 − 𝐶)) | |
23 | 2, 3, 4, 6, 8, 10, 12, 10, 10, 12, 14, 15, 21, 22 | tgcgrsub 28531 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐴 − 𝐵) = (𝐵 − 𝐵)) |
24 | 2, 3, 4, 6, 8, 10, 10, 23 | axtgcgrid 28485 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 = 𝐵) |
25 | 24, 13 | eqeltrd 2838 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 ∈ (𝐶𝐼𝐴)) |
26 | 24 | oveq2d 7446 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐶𝐼𝐴) = (𝐶𝐼𝐵)) |
27 | 25, 26 | eleqtrd 2840 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 ∈ (𝐶𝐼𝐵)) |
28 | legbtwn.1 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) | |
29 | 1, 27, 28 | mpjaodan 960 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 distcds 17306 TarskiGcstrkg 28449 Itvcitv 28455 ≤Gcleg 28604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 df-s3 14884 df-trkgc 28470 df-trkgb 28471 df-trkgcb 28472 df-trkg 28475 df-cgrg 28533 df-leg 28605 |
This theorem is referenced by: tgcgrsub2 28617 krippenlem 28712 |
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