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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 1213 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn ⟨𝐴, 𝐶⟩) → 𝐵 ∈ (𝔼‘𝑁)) | |
2 | simpr 483 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn ⟨𝐴, 𝐶⟩) → 𝐵 Btwn ⟨𝐴, 𝐶⟩) | |
3 | simpl 481 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ) | |
4 | simpr1 1191 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁)) | |
5 | simpr2 1192 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁)) | |
6 | 3, 4, 5 | cgrrflxd 35641 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝐵⟩) |
7 | 6 | adantr 479 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn ⟨𝐴, 𝐶⟩) → ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝐵⟩) |
8 | breq1 5146 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 Btwn ⟨𝐴, 𝐶⟩ ↔ 𝐵 Btwn ⟨𝐴, 𝐶⟩)) | |
9 | opeq2 4870 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → ⟨𝐴, 𝑥⟩ = ⟨𝐴, 𝐵⟩) | |
10 | 9 | breq2d 5155 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝑥⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝐵⟩)) |
11 | 8, 10 | anbi12d 630 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑥 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝑥⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝐵⟩))) |
12 | 11 | rspcev 3601 | . . . 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 835 | . . 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 1193 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁)) | |
15 | brsegle 35761 | . . . . 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 1376 | . . . 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 479 | . . 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 256 | . 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐵 Btwn ⟨𝐴, 𝐶⟩) → ⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐴, 𝐶⟩) |
19 | 18 | ex 411 | 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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 ⟨cop 4630 class class class wbr 5143 ‘cfv 6543 ℕcn 12242 𝔼cee 28743 Btwn cbtwn 28744 Cgrccgr 28745 Seg≤ csegle 35759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-seq 13999 df-exp 14059 df-sum 15665 df-ee 28746 df-cgr 28748 df-segle 35760 |
This theorem is referenced by: colinbtwnle 35771 outsidele 35785 |
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