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Mirrors > Home > MPE Home > Th. List > btwnleg | Structured version Visualization version GIF version |
Description: Betweenness implies less-than relation. (Contributed by Thierry Arnoux, 3-Jul-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) |
btwnleg.1 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
Ref | Expression |
---|---|
btwnleg | ⊢ (𝜑 → (𝐴 − 𝐵) ≤ (𝐴 − 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | legid.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
2 | btwnleg.1 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
3 | eqidd 2827 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − 𝐵)) | |
4 | eleq1 2895 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐴𝐼𝐶))) | |
5 | oveq2 6914 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 − 𝑥) = (𝐴 − 𝐵)) | |
6 | 5 | eqeq2d 2836 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 − 𝐵) = (𝐴 − 𝑥) ↔ (𝐴 − 𝐵) = (𝐴 − 𝐵))) |
7 | 4, 6 | anbi12d 626 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝐴 − 𝐵) = (𝐴 − 𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐶) ∧ (𝐴 − 𝐵) = (𝐴 − 𝐵)))) |
8 | 7 | rspcev 3527 | . . 3 ⊢ ((𝐵 ∈ 𝑃 ∧ (𝐵 ∈ (𝐴𝐼𝐶) ∧ (𝐴 − 𝐵) = (𝐴 − 𝐵))) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝐴 − 𝐵) = (𝐴 − 𝑥))) |
9 | 1, 2, 3, 8 | syl12anc 872 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝐴 − 𝐵) = (𝐴 − 𝑥))) |
10 | legval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
11 | legval.d | . . 3 ⊢ − = (dist‘𝐺) | |
12 | legval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
13 | legval.l | . . 3 ⊢ ≤ = (≤G‘𝐺) | |
14 | legval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
15 | legid.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
16 | legtrd.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
17 | 10, 11, 12, 13, 14, 15, 1, 15, 16 | legov 25898 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐴 − 𝐶) ↔ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝐴 − 𝐵) = (𝐴 − 𝑥)))) |
18 | 9, 17 | mpbird 249 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ≤ (𝐴 − 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∃wrex 3119 class class class wbr 4874 ‘cfv 6124 (class class class)co 6906 Basecbs 16223 distcds 16315 TarskiGcstrkg 25743 Itvcitv 25749 ≤Gcleg 25895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-pm 8126 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-card 9079 df-cda 9306 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-n0 11620 df-xnn0 11692 df-z 11706 df-uz 11970 df-fz 12621 df-fzo 12762 df-hash 13412 df-word 13576 df-concat 13632 df-s1 13657 df-s2 13970 df-s3 13971 df-trkgc 25761 df-trkgb 25762 df-trkgcb 25763 df-trkg 25766 df-cgrg 25824 df-leg 25896 |
This theorem is referenced by: legbtwn 25907 tgcgrsub2 25908 |
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