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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 33451 | . . . . 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 5071 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 Btwn 〈𝐴, 𝐶〉 ↔ 𝐵 Btwn 〈𝐴, 𝐶〉)) | |
9 | opeq2 4806 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → 〈𝐴, 𝑥〉 = 〈𝐴, 𝐵〉) | |
10 | 9 | breq2d 5080 | . . . . . 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 3625 | . . . 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 33571 | . . . . 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 1537 ∈ wcel 2114 ∃wrex 3141 〈cop 4575 class class class wbr 5068 ‘cfv 6357 ℕcn 11640 𝔼cee 26676 Btwn cbtwn 26677 Cgrccgr 26678 Seg≤ csegle 33569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-seq 13373 df-exp 13433 df-sum 15045 df-ee 26679 df-cgr 26681 df-segle 33570 |
This theorem is referenced by: colinbtwnle 33581 outsidele 33595 |
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