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Mirrors > Home > MPE Home > Th. List > Mathboxes > btwnsegle | Structured version Visualization version GIF version |
Description: If 𝐵 falls between 𝐴 and 𝐶, then 𝐴𝐵 is no longer than 𝐴𝐶. (Contributed by Scott Fenton, 16-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
btwnsegle | ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn 〈𝐴, 𝐶〉 → 〈𝐴, 𝐵〉 Seg≤ 〈𝐴, 𝐶〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr2 1196 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 𝐵 ∈ (𝔼‘𝑁)) | |
2 | simpr 477 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 𝐵 Btwn 〈𝐴, 𝐶〉) | |
3 | simpl 475 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ) | |
4 | simpr1 1174 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁)) | |
5 | simpr2 1175 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁)) | |
6 | 3, 4, 5 | cgrrflxd 32976 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉) |
7 | 6 | adantr 473 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉) |
8 | breq1 4932 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 Btwn 〈𝐴, 𝐶〉 ↔ 𝐵 Btwn 〈𝐴, 𝐶〉)) | |
9 | opeq2 4678 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → 〈𝐴, 𝑥〉 = 〈𝐴, 𝐵〉) | |
10 | 9 | breq2d 4941 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (〈𝐴, 𝐵〉Cgr〈𝐴, 𝑥〉 ↔ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉)) |
11 | 8, 10 | anbi12d 621 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑥 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑥〉) ↔ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉))) |
12 | 11 | rspcev 3535 | . . . 4 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉)) → ∃𝑥 ∈ (𝔼‘𝑁)(𝑥 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑥〉)) |
13 | 1, 2, 7, 12 | syl12anc 824 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → ∃𝑥 ∈ (𝔼‘𝑁)(𝑥 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑥〉)) |
14 | simpr3 1176 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁)) | |
15 | brsegle 33096 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (〈𝐴, 𝐵〉 Seg≤ 〈𝐴, 𝐶〉 ↔ ∃𝑥 ∈ (𝔼‘𝑁)(𝑥 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑥〉))) | |
16 | 3, 4, 5, 4, 14, 15 | syl122anc 1359 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (〈𝐴, 𝐵〉 Seg≤ 〈𝐴, 𝐶〉 ↔ ∃𝑥 ∈ (𝔼‘𝑁)(𝑥 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑥〉))) |
17 | 16 | adantr 473 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → (〈𝐴, 𝐵〉 Seg≤ 〈𝐴, 𝐶〉 ↔ ∃𝑥 ∈ (𝔼‘𝑁)(𝑥 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑥〉))) |
18 | 13, 17 | mpbird 249 | . 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 〈𝐴, 𝐵〉 Seg≤ 〈𝐴, 𝐶〉) |
19 | 18 | ex 405 | 1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn 〈𝐴, 𝐶〉 → 〈𝐴, 𝐵〉 Seg≤ 〈𝐴, 𝐶〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∃wrex 3089 〈cop 4447 class class class wbr 4929 ‘cfv 6188 ℕcn 11439 𝔼cee 26377 Btwn cbtwn 26378 Cgrccgr 26379 Seg≤ csegle 33094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-n0 11708 df-z 11794 df-uz 12059 df-fz 12709 df-seq 13185 df-exp 13245 df-sum 14904 df-ee 26380 df-cgr 26382 df-segle 33095 |
This theorem is referenced by: colinbtwnle 33106 outsidele 33120 |
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