![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lnxfr | Structured version Visualization version GIF version |
Description: Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-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) |
lnxfr.r | ⊢ ∼ = (cgrG‘𝐺) |
lnxfr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
lnxfr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
lnxfr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
lnxfr.1 | ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) |
lnxfr.2 | ⊢ (𝜑 → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝐶”〉) |
Ref | Expression |
---|---|
lnxfr | ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglngval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tglngval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
3 | tglngval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tglngval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐺 ∈ TarskiG) |
6 | lnxfr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐴 ∈ 𝑃) |
8 | lnxfr.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
9 | 8 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐶 ∈ 𝑃) |
10 | lnxfr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐵 ∈ 𝑃) |
12 | eqid 2726 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
13 | lnxfr.r | . . . 4 ⊢ ∼ = (cgrG‘𝐺) | |
14 | tglngval.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
15 | 14 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑋 ∈ 𝑃) |
16 | tglngval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
17 | 16 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ 𝑃) |
18 | tgcolg.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
19 | 18 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑍 ∈ 𝑃) |
20 | lnxfr.2 | . . . . 5 ⊢ (𝜑 → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝐶”〉) | |
21 | 20 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝐶”〉) |
22 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ (𝑋𝐼𝑍)) | |
23 | 1, 12, 3, 13, 5, 15, 17, 19, 7, 11, 9, 21, 22 | tgbtwnxfr 28454 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐵 ∈ (𝐴𝐼𝐶)) |
24 | 1, 2, 3, 5, 7, 9, 11, 23 | btwncolg1 28479 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) |
25 | 4 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐺 ∈ TarskiG) |
26 | 6 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐴 ∈ 𝑃) |
27 | 8 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐶 ∈ 𝑃) |
28 | 10 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐵 ∈ 𝑃) |
29 | 16 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑌 ∈ 𝑃) |
30 | 14 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑋 ∈ 𝑃) |
31 | 18 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑍 ∈ 𝑃) |
32 | 20 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝐶”〉) |
33 | 1, 12, 3, 13, 25, 30, 29, 31, 26, 28, 27, 32 | cgr3swap12 28447 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 〈“𝑌𝑋𝑍”〉 ∼ 〈“𝐵𝐴𝐶”〉) |
34 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑋 ∈ (𝑌𝐼𝑍)) | |
35 | 1, 12, 3, 13, 25, 29, 30, 31, 28, 26, 27, 33, 34 | tgbtwnxfr 28454 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐴 ∈ (𝐵𝐼𝐶)) |
36 | 1, 2, 3, 25, 26, 27, 28, 35 | btwncolg2 28480 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) |
37 | 4 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐺 ∈ TarskiG) |
38 | 6 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐴 ∈ 𝑃) |
39 | 8 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ 𝑃) |
40 | 10 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐵 ∈ 𝑃) |
41 | 14 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝑃) |
42 | 18 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ 𝑃) |
43 | 16 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝑃) |
44 | 20 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝐶”〉) |
45 | 1, 12, 3, 13, 37, 41, 43, 42, 38, 40, 39, 44 | cgr3swap23 28448 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 〈“𝑋𝑍𝑌”〉 ∼ 〈“𝐴𝐶𝐵”〉) |
46 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ (𝑋𝐼𝑌)) | |
47 | 1, 12, 3, 13, 37, 41, 42, 43, 38, 39, 40, 45, 46 | tgbtwnxfr 28454 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ (𝐴𝐼𝐵)) |
48 | 1, 2, 3, 37, 38, 39, 40, 47 | btwncolg3 28481 | . 2 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) |
49 | lnxfr.1 | . . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) | |
50 | 1, 2, 3, 4, 14, 18, 16 | tgcolg 28478 | . . 3 ⊢ (𝜑 → ((𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍) ↔ (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))) |
51 | 49, 50 | mpbid 231 | . 2 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌))) |
52 | 24, 36, 48, 51 | mpjao3dan 1429 | 1 ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 ∨ w3o 1083 = wceq 1534 ∈ wcel 2099 class class class wbr 5145 ‘cfv 6546 (class class class)co 7416 〈“cs3 14846 Basecbs 17208 distcds 17270 TarskiGcstrkg 28351 Itvcitv 28357 LineGclng 28358 cgrGccgrg 28434 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-oadd 8492 df-er 8726 df-pm 8850 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-dju 9937 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-xnn0 12591 df-z 12605 df-uz 12869 df-fz 13533 df-fzo 13676 df-hash 14343 df-word 14518 df-concat 14574 df-s1 14599 df-s2 14852 df-s3 14853 df-trkgc 28372 df-trkgb 28373 df-trkgcb 28374 df-trkg 28377 df-cgrg 28435 |
This theorem is referenced by: symquadlem 28613 midexlem 28616 trgcopy 28728 |
Copyright terms: Public domain | W3C validator |