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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 2798 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | tglineintmo.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tglineintmo.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG) |
6 | coltr.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃) |
8 | coltr.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | 8 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ 𝑃) |
10 | coltr3.2 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐶)) | |
11 | 10 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ (𝐴𝐼𝐶)) |
12 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶) | |
13 | 12 | oveq2d 7151 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶)) |
14 | 11, 13 | eleqtrrd 2893 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ (𝐴𝐼𝐴)) |
15 | 1, 2, 3, 5, 7, 9, 14 | axtgbtwnid 26260 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐷) |
16 | coltr.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) | |
17 | 16 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ (𝐵𝐿𝐶)) |
18 | 15, 17 | eqeltrrd 2891 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ (𝐵𝐿𝐶)) |
19 | tglineintmo.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
20 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐺 ∈ TarskiG) |
21 | 6 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐴 ∈ 𝑃) |
22 | coltr.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
23 | 22 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ 𝑃) |
24 | 8 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐷 ∈ 𝑃) |
25 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐴 ≠ 𝐶) | |
26 | 10 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐷 ∈ (𝐴𝐼𝐶)) |
27 | 1, 3, 19, 20, 21, 23, 24, 25, 26 | btwnlng1 26413 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐷 ∈ (𝐴𝐿𝐶)) |
28 | 25 | necomd 3042 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐶 ≠ 𝐴) |
29 | coltr.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
30 | 29 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐵 ∈ 𝑃) |
31 | 1, 19, 3, 4, 29, 22, 16 | tglngne 26344 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
32 | 31 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
33 | 16 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐴 ∈ (𝐵𝐿𝐶)) |
34 | 1, 3, 19, 20, 23, 21, 30, 28, 33, 32 | lnrot1 26417 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐿𝐴)) |
35 | 1, 3, 19, 20, 23, 21, 28, 30, 32, 34 | tglineelsb2 26426 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → (𝐶𝐿𝐴) = (𝐶𝐿𝐵)) |
36 | 1, 3, 19, 20, 21, 23, 25 | tglinecom 26429 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → (𝐴𝐿𝐶) = (𝐶𝐿𝐴)) |
37 | 1, 3, 19, 4, 29, 22, 31 | tglinecom 26429 | . . . . 5 ⊢ (𝜑 → (𝐵𝐿𝐶) = (𝐶𝐿𝐵)) |
38 | 37 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → (𝐵𝐿𝐶) = (𝐶𝐿𝐵)) |
39 | 35, 36, 38 | 3eqtr4d 2843 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → (𝐴𝐿𝐶) = (𝐵𝐿𝐶)) |
40 | 27, 39 | eleqtrd 2892 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → 𝐷 ∈ (𝐵𝐿𝐶)) |
41 | 18, 40 | pm2.61dane 3074 | 1 ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐿𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 LineGclng 26231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 df-cgrg 26305 |
This theorem is referenced by: mideulem2 26528 opphllem 26529 outpasch 26549 |
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