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Mirrors > Home > MPE Home > Th. List > coltr3 | Structured version Visualization version GIF version |
Description: A transitivity law for colinearity. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
Ref | Expression |
---|---|
tglineintmo.p | ⊢ 𝑃 = (Base‘𝐺) |
tglineintmo.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglineintmo.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglineintmo.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
coltr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
coltr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
coltr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
coltr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
coltr.1 | ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) |
coltr3.2 | ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐶)) |
Ref | Expression |
---|---|
coltr3 | ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐿𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglineintmo.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | eqid 2819 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | tglineintmo.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tglineintmo.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG) |
6 | coltr.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃) |
8 | coltr.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ 𝑃) |
10 | coltr3.2 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐶)) | |
11 | 10 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ (𝐴𝐼𝐶)) |
12 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶) | |
13 | 12 | oveq2d 7164 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶)) |
14 | 11, 13 | eleqtrrd 2914 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ (𝐴𝐼𝐴)) |
15 | 1, 2, 3, 5, 7, 9, 14 | axtgbtwnid 26244 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐷) |
16 | coltr.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) | |
17 | 16 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ (𝐵𝐿𝐶)) |
18 | 15, 17 | eqeltrrd 2912 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ (𝐵𝐿𝐶)) |
19 | tglineintmo.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
20 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐺 ∈ TarskiG) |
21 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐴 ∈ 𝑃) |
22 | coltr.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
23 | 22 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ 𝑃) |
24 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐷 ∈ 𝑃) |
25 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐴 ≠ 𝐶) | |
26 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐷 ∈ (𝐴𝐼𝐶)) |
27 | 1, 3, 19, 20, 21, 23, 24, 25, 26 | btwnlng1 26397 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐷 ∈ (𝐴𝐿𝐶)) |
28 | 25 | necomd 3069 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐶 ≠ 𝐴) |
29 | coltr.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
30 | 29 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐵 ∈ 𝑃) |
31 | 1, 19, 3, 4, 29, 22, 16 | tglngne 26328 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
32 | 31 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
33 | 16 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐴 ∈ (𝐵𝐿𝐶)) |
34 | 1, 3, 19, 20, 23, 21, 30, 28, 33, 32 | lnrot1 26401 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐿𝐴)) |
35 | 1, 3, 19, 20, 23, 21, 28, 30, 32, 34 | tglineelsb2 26410 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → (𝐶𝐿𝐴) = (𝐶𝐿𝐵)) |
36 | 1, 3, 19, 20, 21, 23, 25 | tglinecom 26413 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → (𝐴𝐿𝐶) = (𝐶𝐿𝐴)) |
37 | 1, 3, 19, 4, 29, 22, 31 | tglinecom 26413 | . . . . 5 ⊢ (𝜑 → (𝐵𝐿𝐶) = (𝐶𝐿𝐵)) |
38 | 37 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → (𝐵𝐿𝐶) = (𝐶𝐿𝐵)) |
39 | 35, 36, 38 | 3eqtr4d 2864 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → (𝐴𝐿𝐶) = (𝐵𝐿𝐶)) |
40 | 27, 39 | eleqtrd 2913 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐷 ∈ (𝐵𝐿𝐶)) |
41 | 18, 40 | pm2.61dane 3102 | 1 ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐿𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 distcds 16566 TarskiGcstrkg 26208 Itvcitv 26214 LineGclng 26215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-dju 9322 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-xnn0 11960 df-z 11974 df-uz 12236 df-fz 12885 df-fzo 13026 df-hash 13683 df-word 13854 df-concat 13915 df-s1 13942 df-s2 14202 df-s3 14203 df-trkgc 26226 df-trkgb 26227 df-trkgcb 26228 df-trkg 26231 df-cgrg 26289 |
This theorem is referenced by: mideulem2 26512 opphllem 26513 outpasch 26533 |
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