![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 484 | . 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 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG) |
9 | israg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃) |
11 | israg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
12 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃) |
13 | israg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
14 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃) |
15 | eqid 2734 | . . . 4 ⊢ (𝑆‘𝐶) = (𝑆‘𝐶) | |
16 | 2, 3, 4, 5, 6, 8, 14, 15, 10 | mircl 28678 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐶)‘𝐴) ∈ 𝑃) |
17 | ragflat.1 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) | |
18 | 17 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
19 | 2, 3, 4, 5, 6, 8, 14, 15, 10 | mircgr 28674 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − ((𝑆‘𝐶)‘𝐴)) = (𝐶 − 𝐴)) |
20 | 2, 3, 4, 8, 14, 16, 14, 10, 19 | tgcgrcomlr 28497 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (((𝑆‘𝐶)‘𝐴) − 𝐶) = (𝐴 − 𝐶)) |
21 | 2, 3, 4, 5, 6, 8, 10, 12, 14 | israg 28714 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺) ↔ (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶)))) |
22 | 18, 21 | mpbid 232 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶))) |
23 | eqid 2734 | . . . . . . 7 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵) | |
24 | 2, 3, 4, 5, 6, 8, 12, 23, 14 | mircl 28678 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐵)‘𝐶) ∈ 𝑃) |
25 | ragflat.2 | . . . . . . . . . 10 ⊢ (𝜑 → 〈“𝐴𝐶𝐵”〉 ∈ (∟G‘𝐺)) | |
26 | 25 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“𝐴𝐶𝐵”〉 ∈ (∟G‘𝐺)) |
27 | 2, 3, 4, 5, 6, 8, 10, 14, 12, 26 | ragcom 28715 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“𝐵𝐶𝐴”〉 ∈ (∟G‘𝐺)) |
28 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶) | |
29 | 2, 3, 4, 5, 6, 8, 12, 23, 14 | mirbtwn 28675 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (((𝑆‘𝐵)‘𝐶)𝐼𝐶)) |
30 | 2, 3, 4, 8, 24, 12, 14, 29 | tgbtwncom 28505 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐵)‘𝐶))) |
31 | 2, 5, 4, 8, 14, 24, 12, 30 | btwncolg1 28572 | . . . . . . . 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 28716 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“((𝑆‘𝐵)‘𝐶)𝐶𝐴”〉 ∈ (∟G‘𝐺)) |
33 | 2, 3, 4, 5, 6, 8, 24, 14, 10 | israg 28714 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (〈“((𝑆‘𝐵)‘𝐶)𝐶𝐴”〉 ∈ (∟G‘𝐺) ↔ (((𝑆‘𝐵)‘𝐶) − 𝐴) = (((𝑆‘𝐵)‘𝐶) − ((𝑆‘𝐶)‘𝐴)))) |
34 | 32, 33 | mpbid 232 | . . . . . 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 28497 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − ((𝑆‘𝐵)‘𝐶)) = (((𝑆‘𝐶)‘𝐴) − ((𝑆‘𝐵)‘𝐶))) |
36 | 20, 22, 35 | 3eqtrd 2778 | . . . 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 28714 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (〈“((𝑆‘𝐶)‘𝐴)𝐵𝐶”〉 ∈ (∟G‘𝐺) ↔ (((𝑆‘𝐶)‘𝐴) − 𝐶) = (((𝑆‘𝐶)‘𝐴) − ((𝑆‘𝐵)‘𝐶)))) |
38 | 36, 37 | mpbird 257 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“((𝑆‘𝐶)‘𝐴)𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
39 | 2, 3, 4, 5, 6, 8, 14, 15, 10 | mirbtwn 28675 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (((𝑆‘𝐶)‘𝐴)𝐼𝐴)) |
40 | 2, 3, 4, 8, 16, 14, 10, 39 | tgbtwncom 28505 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼((𝑆‘𝐶)‘𝐴))) |
41 | 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 38, 40 | ragflat2 28720 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 = 𝐶) |
42 | 1, 41 | pm2.61dane 3031 | 1 ⊢ (𝜑 → 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 ‘cfv 6572 (class class class)co 7445 〈“cs3 14887 Basecbs 17253 distcds 17315 TarskiGcstrkg 28444 Itvcitv 28450 LineGclng 28451 pInvGcmir 28669 ∟Gcrag 28710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-oadd 8522 df-er 8759 df-map 8882 df-pm 8883 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-dju 9966 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-n0 12550 df-xnn0 12622 df-z 12636 df-uz 12900 df-fz 13564 df-fzo 13708 df-hash 14376 df-word 14559 df-concat 14615 df-s1 14640 df-s2 14893 df-s3 14894 df-trkgc 28465 df-trkgb 28466 df-trkgcb 28467 df-trkg 28470 df-cgrg 28528 df-mir 28670 df-rag 28711 |
This theorem is referenced by: ragtriva 28722 footexALT 28735 footexlem2 28737 foot 28739 |
Copyright terms: Public domain | W3C validator |