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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 2819 | . . . 4 ⊢ (𝑆‘𝐶) = (𝑆‘𝐶) | |
16 | 2, 3, 4, 5, 6, 8, 14, 15, 10 | mircl 26439 | . . 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 26435 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − ((𝑆‘𝐶)‘𝐴)) = (𝐶 − 𝐴)) |
20 | 2, 3, 4, 8, 14, 16, 14, 10, 19 | tgcgrcomlr 26258 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (((𝑆‘𝐶)‘𝐴) − 𝐶) = (𝐴 − 𝐶)) |
21 | 2, 3, 4, 5, 6, 8, 10, 12, 14 | israg 26475 | . . . . . 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 2819 | . . . . . . 7 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵) | |
24 | 2, 3, 4, 5, 6, 8, 12, 23, 14 | mircl 26439 | . . . . . 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 26476 | . . . . . . . 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 26436 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (((𝑆‘𝐵)‘𝐶)𝐼𝐶)) |
30 | 2, 3, 4, 8, 24, 12, 14, 29 | tgbtwncom 26266 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐵)‘𝐶))) |
31 | 2, 5, 4, 8, 14, 24, 12, 30 | btwncolg1 26333 | . . . . . . . 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 26477 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 〈“((𝑆‘𝐵)‘𝐶)𝐶𝐴”〉 ∈ (∟G‘𝐺)) |
33 | 2, 3, 4, 5, 6, 8, 24, 14, 10 | israg 26475 | . . . . . . 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 26258 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − ((𝑆‘𝐵)‘𝐶)) = (((𝑆‘𝐶)‘𝐴) − ((𝑆‘𝐵)‘𝐶))) |
36 | 20, 22, 35 | 3eqtrd 2858 | . . . 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 26475 | . . . 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 26436 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (((𝑆‘𝐶)‘𝐴)𝐼𝐴)) |
40 | 2, 3, 4, 8, 16, 14, 10, 39 | tgbtwncom 26266 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼((𝑆‘𝐶)‘𝐴))) |
41 | 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 38, 40 | ragflat2 26481 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 = 𝐶) |
42 | 1, 41 | pm2.61dane 3102 | 1 ⊢ (𝜑 → 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ‘cfv 6348 (class class class)co 7148 〈“cs3 14196 Basecbs 16475 distcds 16566 TarskiGcstrkg 26208 Itvcitv 26214 LineGclng 26215 pInvGcmir 26430 ∟Gcrag 26471 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-dju 9322 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-xnn0 11960 df-z 11974 df-uz 12236 df-fz 12885 df-fzo 13026 df-hash 13683 df-word 13854 df-concat 13915 df-s1 13942 df-s2 14202 df-s3 14203 df-trkgc 26226 df-trkgb 26227 df-trkgcb 26228 df-trkg 26231 df-cgrg 26289 df-mir 26431 df-rag 26472 |
This theorem is referenced by: ragtriva 26483 footexALT 26496 footexlem2 26498 foot 26500 |
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