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| 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 480 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐺 ∈ TarskiG) |
| 6 | lnxfr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 7 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐴 ∈ 𝑃) |
| 8 | lnxfr.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
| 9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐶 ∈ 𝑃) |
| 10 | lnxfr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐵 ∈ 𝑃) |
| 12 | eqid 2737 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
| 13 | lnxfr.r | . . . 4 ⊢ ∼ = (cgrG‘𝐺) | |
| 14 | tglngval.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
| 15 | 14 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑋 ∈ 𝑃) |
| 16 | tglngval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
| 17 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ 𝑃) |
| 18 | tgcolg.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
| 19 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑍 ∈ 𝑃) |
| 20 | lnxfr.2 | . . . . 5 ⊢ (𝜑 → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝐶”〉) | |
| 21 | 20 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝐶”〉) |
| 22 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ (𝑋𝐼𝑍)) | |
| 23 | 1, 12, 3, 13, 5, 15, 17, 19, 7, 11, 9, 21, 22 | tgbtwnxfr 28538 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐵 ∈ (𝐴𝐼𝐶)) |
| 24 | 1, 2, 3, 5, 7, 9, 11, 23 | btwncolg1 28563 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) |
| 25 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐺 ∈ TarskiG) |
| 26 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐴 ∈ 𝑃) |
| 27 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐶 ∈ 𝑃) |
| 28 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐵 ∈ 𝑃) |
| 29 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑌 ∈ 𝑃) |
| 30 | 14 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑋 ∈ 𝑃) |
| 31 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑍 ∈ 𝑃) |
| 32 | 20 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝐶”〉) |
| 33 | 1, 12, 3, 13, 25, 30, 29, 31, 26, 28, 27, 32 | cgr3swap12 28531 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 〈“𝑌𝑋𝑍”〉 ∼ 〈“𝐵𝐴𝐶”〉) |
| 34 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑋 ∈ (𝑌𝐼𝑍)) | |
| 35 | 1, 12, 3, 13, 25, 29, 30, 31, 28, 26, 27, 33, 34 | tgbtwnxfr 28538 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐴 ∈ (𝐵𝐼𝐶)) |
| 36 | 1, 2, 3, 25, 26, 27, 28, 35 | btwncolg2 28564 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) |
| 37 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐺 ∈ TarskiG) |
| 38 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐴 ∈ 𝑃) |
| 39 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ 𝑃) |
| 40 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐵 ∈ 𝑃) |
| 41 | 14 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝑃) |
| 42 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ 𝑃) |
| 43 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝑃) |
| 44 | 20 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝐶”〉) |
| 45 | 1, 12, 3, 13, 37, 41, 43, 42, 38, 40, 39, 44 | cgr3swap23 28532 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 〈“𝑋𝑍𝑌”〉 ∼ 〈“𝐴𝐶𝐵”〉) |
| 46 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ (𝑋𝐼𝑌)) | |
| 47 | 1, 12, 3, 13, 37, 41, 42, 43, 38, 39, 40, 45, 46 | tgbtwnxfr 28538 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ (𝐴𝐼𝐵)) |
| 48 | 1, 2, 3, 37, 38, 39, 40, 47 | btwncolg3 28565 | . 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 28562 | . . 3 ⊢ (𝜑 → ((𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍) ↔ (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))) |
| 51 | 49, 50 | mpbid 232 | . 2 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌))) |
| 52 | 24, 36, 48, 51 | mpjao3dan 1434 | 1 ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 〈“cs3 14881 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 LineGclng 28442 cgrGccgrg 28518 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 df-cgrg 28519 |
| This theorem is referenced by: symquadlem 28697 midexlem 28700 trgcopy 28812 |
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