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Mirrors > Home > MPE Home > Th. List > ragflat | Structured version Visualization version GIF version |
Description: Deduce equality from two right angles. Theorem 8.7 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 3-Sep-2019.) |
Ref | Expression |
---|---|
israg.p | ⊢ 𝑃 = (Base‘𝐺) |
israg.d | ⊢ − = (dist‘𝐺) |
israg.i | ⊢ 𝐼 = (Itv‘𝐺) |
israg.l | ⊢ 𝐿 = (LineG‘𝐺) |
israg.s | ⊢ 𝑆 = (pInvG‘𝐺) |
israg.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
israg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
israg.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
israg.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
ragflat.1 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
ragflat.2 | ⊢ (𝜑 → 〈“𝐴𝐶𝐵”〉 ∈ (∟G‘𝐺)) |
Ref | Expression |
---|---|
ragflat | ⊢ (𝜑 → 𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . 2 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶) | |
2 | israg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
3 | israg.d | . . 3 ⊢ − = (dist‘𝐺) | |
4 | israg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | israg.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
6 | israg.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
7 | israg.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG) |
9 | israg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | 9 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃) |
11 | israg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
12 | 11 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃) |
13 | israg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
14 | 13 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃) |
15 | eqid 2821 | . . . 4 ⊢ (𝑆‘𝐶) = (𝑆‘𝐶) | |
16 | 2, 3, 4, 5, 6, 8, 14, 15, 10 | mircl 26446 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐶)‘𝐴) ∈ 𝑃) |
17 | ragflat.1 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) | |
18 | 17 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
19 | 2, 3, 4, 5, 6, 8, 14, 15, 10 | mircgr 26442 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − ((𝑆‘𝐶)‘𝐴)) = (𝐶 − 𝐴)) |
20 | 2, 3, 4, 8, 14, 16, 14, 10, 19 | tgcgrcomlr 26265 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (((𝑆‘𝐶)‘𝐴) − 𝐶) = (𝐴 − 𝐶)) |
21 | 2, 3, 4, 5, 6, 8, 10, 12, 14 | israg 26482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺) ↔ (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶)))) |
22 | 18, 21 | mpbid 234 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶))) |
23 | eqid 2821 | . . . . . . 7 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵) | |
24 | 2, 3, 4, 5, 6, 8, 12, 23, 14 | mircl 26446 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐵)‘𝐶) ∈ 𝑃) |
25 | ragflat.2 | . . . . . . . . . 10 ⊢ (𝜑 → 〈“𝐴𝐶𝐵”〉 ∈ (∟G‘𝐺)) | |
26 | 25 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“𝐴𝐶𝐵”〉 ∈ (∟G‘𝐺)) |
27 | 2, 3, 4, 5, 6, 8, 10, 14, 12, 26 | ragcom 26483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“𝐵𝐶𝐴”〉 ∈ (∟G‘𝐺)) |
28 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶) | |
29 | 2, 3, 4, 5, 6, 8, 12, 23, 14 | mirbtwn 26443 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (((𝑆‘𝐵)‘𝐶)𝐼𝐶)) |
30 | 2, 3, 4, 8, 24, 12, 14, 29 | tgbtwncom 26273 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐵)‘𝐶))) |
31 | 2, 5, 4, 8, 14, 24, 12, 30 | btwncolg1 26340 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 ∈ (𝐶𝐿((𝑆‘𝐵)‘𝐶)) ∨ 𝐶 = ((𝑆‘𝐵)‘𝐶))) |
32 | 2, 3, 4, 5, 6, 8, 12, 14, 10, 24, 27, 28, 31 | ragcol 26484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“((𝑆‘𝐵)‘𝐶)𝐶𝐴”〉 ∈ (∟G‘𝐺)) |
33 | 2, 3, 4, 5, 6, 8, 24, 14, 10 | israg 26482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (〈“((𝑆‘𝐵)‘𝐶)𝐶𝐴”〉 ∈ (∟G‘𝐺) ↔ (((𝑆‘𝐵)‘𝐶) − 𝐴) = (((𝑆‘𝐵)‘𝐶) − ((𝑆‘𝐶)‘𝐴)))) |
34 | 32, 33 | mpbid 234 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (((𝑆‘𝐵)‘𝐶) − 𝐴) = (((𝑆‘𝐵)‘𝐶) − ((𝑆‘𝐶)‘𝐴))) |
35 | 2, 3, 4, 8, 24, 10, 24, 16, 34 | tgcgrcomlr 26265 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − ((𝑆‘𝐵)‘𝐶)) = (((𝑆‘𝐶)‘𝐴) − ((𝑆‘𝐵)‘𝐶))) |
36 | 20, 22, 35 | 3eqtrd 2860 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (((𝑆‘𝐶)‘𝐴) − 𝐶) = (((𝑆‘𝐶)‘𝐴) − ((𝑆‘𝐵)‘𝐶))) |
37 | 2, 3, 4, 5, 6, 8, 16, 12, 14 | israg 26482 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (〈“((𝑆‘𝐶)‘𝐴)𝐵𝐶”〉 ∈ (∟G‘𝐺) ↔ (((𝑆‘𝐶)‘𝐴) − 𝐶) = (((𝑆‘𝐶)‘𝐴) − ((𝑆‘𝐵)‘𝐶)))) |
38 | 36, 37 | mpbird 259 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“((𝑆‘𝐶)‘𝐴)𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
39 | 2, 3, 4, 5, 6, 8, 14, 15, 10 | mirbtwn 26443 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (((𝑆‘𝐶)‘𝐴)𝐼𝐴)) |
40 | 2, 3, 4, 8, 16, 14, 10, 39 | tgbtwncom 26273 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼((𝑆‘𝐶)‘𝐴))) |
41 | 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 38, 40 | ragflat2 26488 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 = 𝐶) |
42 | 1, 41 | pm2.61dane 3104 | 1 ⊢ (𝜑 → 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6354 (class class class)co 7155 〈“cs3 14203 Basecbs 16482 distcds 16573 TarskiGcstrkg 26215 Itvcitv 26221 LineGclng 26222 pInvGcmir 26437 ∟Gcrag 26478 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
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-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-s1 13949 df-s2 14209 df-s3 14210 df-trkgc 26233 df-trkgb 26234 df-trkgcb 26235 df-trkg 26238 df-cgrg 26296 df-mir 26438 df-rag 26479 |
This theorem is referenced by: ragtriva 26490 footexALT 26503 footexlem2 26505 foot 26507 |
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