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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 1234 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 𝐵 ∈ (𝔼‘𝑁)) | |
2 | simpr 479 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 𝐵 Btwn 〈𝐴, 𝐶〉) | |
3 | simpl 476 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ) | |
4 | simpr1 1205 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁)) | |
5 | simpr2 1207 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁)) | |
6 | 3, 4, 5 | cgrrflxd 32692 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉) |
7 | 6 | adantr 474 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉) |
8 | breq1 4891 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 Btwn 〈𝐴, 𝐶〉 ↔ 𝐵 Btwn 〈𝐴, 𝐶〉)) | |
9 | opeq2 4639 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → 〈𝐴, 𝑥〉 = 〈𝐴, 𝐵〉) | |
10 | 9 | breq2d 4900 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (〈𝐴, 𝐵〉Cgr〈𝐴, 𝑥〉 ↔ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉)) |
11 | 8, 10 | anbi12d 624 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑥 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑥〉) ↔ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉))) |
12 | 11 | rspcev 3511 | . . . 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 827 | . . 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 1209 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁)) | |
15 | brsegle 32812 | . . . . 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 1447 | . . . 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 474 | . . 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 403 | 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 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 〈cop 4404 class class class wbr 4888 ‘cfv 6137 ℕcn 11379 𝔼cee 26254 Btwn cbtwn 26255 Cgrccgr 26256 Seg≤ csegle 32810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-seq 13125 df-exp 13184 df-sum 14834 df-ee 26257 df-cgr 26259 df-segle 32811 |
This theorem is referenced by: colinbtwnle 32822 outsidele 32836 |
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