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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 1212 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 𝐵 ∈ (𝔼‘𝑁)) | |
2 | simpr 487 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 𝐵 Btwn 〈𝐴, 𝐶〉) | |
3 | simpl 485 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ) | |
4 | simpr1 1190 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁)) | |
5 | simpr2 1191 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁)) | |
6 | 3, 4, 5 | cgrrflxd 33444 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉) |
7 | 6 | adantr 483 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉) |
8 | breq1 5061 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 Btwn 〈𝐴, 𝐶〉 ↔ 𝐵 Btwn 〈𝐴, 𝐶〉)) | |
9 | opeq2 4797 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → 〈𝐴, 𝑥〉 = 〈𝐴, 𝐵〉) | |
10 | 9 | breq2d 5070 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (〈𝐴, 𝐵〉Cgr〈𝐴, 𝑥〉 ↔ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉)) |
11 | 8, 10 | anbi12d 632 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑥 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑥〉) ↔ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉))) |
12 | 11 | rspcev 3622 | . . . 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 834 | . . 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 1192 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁)) | |
15 | brsegle 33564 | . . . . 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 1375 | . . . 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 483 | . . 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 259 | . 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 〈𝐴, 𝐵〉 Seg≤ 〈𝐴, 𝐶〉) |
19 | 18 | ex 415 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 〈cop 4566 class class class wbr 5058 ‘cfv 6349 ℕcn 11632 𝔼cee 26668 Btwn cbtwn 26669 Cgrccgr 26670 Seg≤ csegle 33562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-seq 13364 df-exp 13424 df-sum 15037 df-ee 26671 df-cgr 26673 df-segle 33563 |
This theorem is referenced by: colinbtwnle 33574 outsidele 33588 |
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