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Mirrors > Home > MPE Home > Th. List > tgcolg | Structured version Visualization version GIF version |
Description: We choose the notation (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) instead of "colinear" in order to avoid defining an additional symbol for colinearity because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 25-May-2019.) |
Ref | Expression |
---|---|
tglngval.p | ⊢ 𝑃 = (Base‘𝐺) |
tglngval.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglngval.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglngval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglngval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
tglngval.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
tgcolg.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
Ref | Expression |
---|---|
tgcolg | ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | animorr 975 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) | |
2 | tglngval.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
3 | eqid 2821 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
4 | tglngval.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | tglngval.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | 5 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG) |
7 | tgcolg.z | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
8 | 7 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝑃) |
9 | tglngval.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
10 | 9 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝑃) |
11 | 2, 3, 4, 6, 8, 10 | tgbtwntriv2 26267 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑋)) |
12 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
13 | 12 | oveq2d 7166 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑍𝐼𝑋) = (𝑍𝐼𝑌)) |
14 | 11, 13 | eleqtrd 2915 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑌)) |
15 | 14 | 3mix2d 1333 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) |
16 | 1, 15 | 2thd 267 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
17 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌) | |
18 | 17 | neneqd 3021 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌) |
19 | biorf 933 | . . . . 5 ⊢ (¬ 𝑋 = 𝑌 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌 ∨ 𝑍 ∈ (𝑋𝐿𝑌)))) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌 ∨ 𝑍 ∈ (𝑋𝐿𝑌)))) |
21 | orcom 866 | . . . 4 ⊢ ((𝑋 = 𝑌 ∨ 𝑍 ∈ (𝑋𝐿𝑌)) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) | |
22 | 20, 21 | syl6bb 289 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))) |
23 | tglngval.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
24 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐺 ∈ TarskiG) |
25 | 9 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝑃) |
26 | tglngval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
27 | 26 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝑃) |
28 | 7 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑍 ∈ 𝑃) |
29 | 2, 23, 4, 24, 25, 27, 17, 28 | tgellng 26333 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
30 | 22, 29 | bitr3d 283 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
31 | 16, 30 | pm2.61dane 3104 | 1 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 distcds 16568 TarskiGcstrkg 26210 Itvcitv 26216 LineGclng 26217 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 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-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-trkgc 26228 df-trkgcb 26230 df-trkg 26233 |
This theorem is referenced by: btwncolg1 26335 btwncolg2 26336 btwncolg3 26337 colcom 26338 colrot1 26339 lnxfr 26346 lnext 26347 tgfscgr 26348 tglowdim2l 26430 outpasch 26535 |
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