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Mirrors > Home > MPE Home > Th. List > Mathboxes > colinearperm4 | Structured version Visualization version GIF version |
Description: Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.) |
Ref | Expression |
---|---|
colinearperm4 | ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear 〈𝐵, 𝐶〉 ↔ 𝐶 Colinear 〈𝐴, 𝐵〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | colinearperm3 33526 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear 〈𝐵, 𝐶〉 ↔ 𝐵 Colinear 〈𝐶, 𝐴〉)) | |
2 | 3anrot 1096 | . . 3 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))) | |
3 | colinearperm3 33526 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))) → (𝐵 Colinear 〈𝐶, 𝐴〉 ↔ 𝐶 Colinear 〈𝐴, 𝐵〉)) | |
4 | 2, 3 | sylan2b 595 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Colinear 〈𝐶, 𝐴〉 ↔ 𝐶 Colinear 〈𝐴, 𝐵〉)) |
5 | 1, 4 | bitrd 281 | 1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear 〈𝐵, 𝐶〉 ↔ 𝐶 Colinear 〈𝐴, 𝐵〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 〈cop 4575 class class class wbr 5068 ‘cfv 6357 ℕcn 11640 𝔼cee 26676 Colinear ccolin 33500 |
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-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 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-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-iota 6316 df-fv 6365 df-oprab 7162 df-colinear 33502 |
This theorem is referenced by: colinearperm5 33529 btwncolinear5 33536 |
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