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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 485 | . 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 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG) |
9 | israg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | 9 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃) |
11 | israg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
12 | 11 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃) |
13 | israg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
14 | 13 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃) |
15 | eqid 2736 | . . . 4 ⊢ (𝑆‘𝐶) = (𝑆‘𝐶) | |
16 | 2, 3, 4, 5, 6, 8, 14, 15, 10 | mircl 27601 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐶)‘𝐴) ∈ 𝑃) |
17 | ragflat.1 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) | |
18 | 17 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
19 | 2, 3, 4, 5, 6, 8, 14, 15, 10 | mircgr 27597 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − ((𝑆‘𝐶)‘𝐴)) = (𝐶 − 𝐴)) |
20 | 2, 3, 4, 8, 14, 16, 14, 10, 19 | tgcgrcomlr 27420 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (((𝑆‘𝐶)‘𝐴) − 𝐶) = (𝐴 − 𝐶)) |
21 | 2, 3, 4, 5, 6, 8, 10, 12, 14 | israg 27637 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺) ↔ (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶)))) |
22 | 18, 21 | mpbid 231 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶))) |
23 | eqid 2736 | . . . . . . 7 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵) | |
24 | 2, 3, 4, 5, 6, 8, 12, 23, 14 | mircl 27601 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐵)‘𝐶) ∈ 𝑃) |
25 | ragflat.2 | . . . . . . . . . 10 ⊢ (𝜑 → 〈“𝐴𝐶𝐵”〉 ∈ (∟G‘𝐺)) | |
26 | 25 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“𝐴𝐶𝐵”〉 ∈ (∟G‘𝐺)) |
27 | 2, 3, 4, 5, 6, 8, 10, 14, 12, 26 | ragcom 27638 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“𝐵𝐶𝐴”〉 ∈ (∟G‘𝐺)) |
28 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶) | |
29 | 2, 3, 4, 5, 6, 8, 12, 23, 14 | mirbtwn 27598 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (((𝑆‘𝐵)‘𝐶)𝐼𝐶)) |
30 | 2, 3, 4, 8, 24, 12, 14, 29 | tgbtwncom 27428 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐵)‘𝐶))) |
31 | 2, 5, 4, 8, 14, 24, 12, 30 | btwncolg1 27495 | . . . . . . . 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 27639 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“((𝑆‘𝐵)‘𝐶)𝐶𝐴”〉 ∈ (∟G‘𝐺)) |
33 | 2, 3, 4, 5, 6, 8, 24, 14, 10 | israg 27637 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (〈“((𝑆‘𝐵)‘𝐶)𝐶𝐴”〉 ∈ (∟G‘𝐺) ↔ (((𝑆‘𝐵)‘𝐶) − 𝐴) = (((𝑆‘𝐵)‘𝐶) − ((𝑆‘𝐶)‘𝐴)))) |
34 | 32, 33 | mpbid 231 | . . . . . 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 27420 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − ((𝑆‘𝐵)‘𝐶)) = (((𝑆‘𝐶)‘𝐴) − ((𝑆‘𝐵)‘𝐶))) |
36 | 20, 22, 35 | 3eqtrd 2780 | . . . 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 27637 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (〈“((𝑆‘𝐶)‘𝐴)𝐵𝐶”〉 ∈ (∟G‘𝐺) ↔ (((𝑆‘𝐶)‘𝐴) − 𝐶) = (((𝑆‘𝐶)‘𝐴) − ((𝑆‘𝐵)‘𝐶)))) |
38 | 36, 37 | mpbird 256 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“((𝑆‘𝐶)‘𝐴)𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
39 | 2, 3, 4, 5, 6, 8, 14, 15, 10 | mirbtwn 27598 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (((𝑆‘𝐶)‘𝐴)𝐼𝐴)) |
40 | 2, 3, 4, 8, 16, 14, 10, 39 | tgbtwncom 27428 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼((𝑆‘𝐶)‘𝐴))) |
41 | 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 38, 40 | ragflat2 27643 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 = 𝐶) |
42 | 1, 41 | pm2.61dane 3032 | 1 ⊢ (𝜑 → 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ‘cfv 6496 (class class class)co 7356 〈“cs3 14730 Basecbs 17082 distcds 17141 TarskiGcstrkg 27367 Itvcitv 27373 LineGclng 27374 pInvGcmir 27592 ∟Gcrag 27633 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-oadd 8415 df-er 8647 df-map 8766 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-dju 9836 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-n0 12413 df-xnn0 12485 df-z 12499 df-uz 12763 df-fz 13424 df-fzo 13567 df-hash 14230 df-word 14402 df-concat 14458 df-s1 14483 df-s2 14736 df-s3 14737 df-trkgc 27388 df-trkgb 27389 df-trkgcb 27390 df-trkg 27393 df-cgrg 27451 df-mir 27593 df-rag 27634 |
This theorem is referenced by: ragtriva 27645 footexALT 27658 footexlem2 27660 foot 27662 |
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