| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 28496 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶)) |
| 15 | 2, 3, 4, 6, 10, 12 | tgbtwntriv1 28499 | . . . . . 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 28596 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐶 − 𝐵) ≤ (𝐶 − 𝐴)) |
| 20 | 2, 3, 4, 16, 6, 12, 8, 12, 10, 18, 19 | legtri3 28598 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐶 − 𝐴) = (𝐶 − 𝐵)) |
| 21 | 2, 3, 4, 6, 12, 8, 12, 10, 20 | tgcgrcomlr 28488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐴 − 𝐶) = (𝐵 − 𝐶)) |
| 22 | eqidd 2738 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐵 − 𝐶) = (𝐵 − 𝐶)) | |
| 23 | 2, 3, 4, 6, 8, 10, 12, 10, 10, 12, 14, 15, 21, 22 | tgcgrsub 28517 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐴 − 𝐵) = (𝐵 − 𝐵)) |
| 24 | 2, 3, 4, 6, 8, 10, 10, 23 | axtgcgrid 28471 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 = 𝐵) |
| 25 | 24, 13 | eqeltrd 2841 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 ∈ (𝐶𝐼𝐴)) |
| 26 | 24 | oveq2d 7447 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐶𝐼𝐴) = (𝐶𝐼𝐵)) |
| 27 | 25, 26 | eleqtrd 2843 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 ∈ (𝐶𝐼𝐵)) |
| 28 | legbtwn.1 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) | |
| 29 | 1, 27, 28 | mpjaodan 961 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 ≤Gcleg 28590 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 df-cgrg 28519 df-leg 28591 |
| This theorem is referenced by: tgcgrsub2 28603 krippenlem 28698 |
| Copyright terms: Public domain | W3C validator |