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Mirrors > Home > MPE Home > Th. List > miduniq | Structured version Visualization version GIF version |
Description: Uniqueness of the middle point, expressed with point inversion. Theorem 7.17 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 30-Jul-2019.) |
Ref | Expression |
---|---|
mirval.p | ⊢ 𝑃 = (Base‘𝐺) |
mirval.d | ⊢ − = (dist‘𝐺) |
mirval.i | ⊢ 𝐼 = (Itv‘𝐺) |
mirval.l | ⊢ 𝐿 = (LineG‘𝐺) |
mirval.s | ⊢ 𝑆 = (pInvG‘𝐺) |
mirval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
miduniq.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
miduniq.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
miduniq.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
miduniq.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
miduniq.e | ⊢ (𝜑 → ((𝑆‘𝐴)‘𝑋) = 𝑌) |
miduniq.f | ⊢ (𝜑 → ((𝑆‘𝐵)‘𝑋) = 𝑌) |
Ref | Expression |
---|---|
miduniq | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
3 | mirval.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | mirval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | miduniq.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
6 | miduniq.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
7 | miduniq.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | eqid 2736 | . . . 4 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺) | |
9 | mirval.d | . . . . 5 ⊢ − = (dist‘𝐺) | |
10 | mirval.s | . . . . 5 ⊢ 𝑆 = (pInvG‘𝐺) | |
11 | miduniq.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
12 | eqid 2736 | . . . . 5 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴) | |
13 | 1, 9, 3, 2, 10, 4, 11, 12, 7 | mircl 27550 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) ∈ 𝑃) |
14 | eqid 2736 | . . . . . . 7 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵) | |
15 | 1, 9, 3, 2, 10, 4, 7, 14, 5 | mirbtwn 27547 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (((𝑆‘𝐵)‘𝑋)𝐼𝑋)) |
16 | miduniq.f | . . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐵)‘𝑋) = 𝑌) | |
17 | 16 | oveq1d 7371 | . . . . . 6 ⊢ (𝜑 → (((𝑆‘𝐵)‘𝑋)𝐼𝑋) = (𝑌𝐼𝑋)) |
18 | 15, 17 | eleqtrd 2840 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝑌𝐼𝑋)) |
19 | 1, 9, 3, 4, 6, 7, 5, 18 | tgbtwncom 27377 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑋𝐼𝑌)) |
20 | 1, 9, 3, 2, 10, 4, 11, 12, 6, 7 | miriso 27559 | . . . . 5 ⊢ (𝜑 → (((𝑆‘𝐴)‘𝑌) − ((𝑆‘𝐴)‘𝐵)) = (𝑌 − 𝐵)) |
21 | miduniq.e | . . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝑋) = 𝑌) | |
22 | 1, 9, 3, 2, 10, 4, 11, 12, 5, 21 | mircom 27552 | . . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝑌) = 𝑋) |
23 | 22 | oveq1d 7371 | . . . . 5 ⊢ (𝜑 → (((𝑆‘𝐴)‘𝑌) − ((𝑆‘𝐴)‘𝐵)) = (𝑋 − ((𝑆‘𝐴)‘𝐵))) |
24 | 1, 9, 3, 2, 10, 4, 7, 14, 5 | mircgr 27546 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 − ((𝑆‘𝐵)‘𝑋)) = (𝐵 − 𝑋)) |
25 | 16 | oveq2d 7372 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 − ((𝑆‘𝐵)‘𝑋)) = (𝐵 − 𝑌)) |
26 | 24, 25 | eqtr3d 2778 | . . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝑋) = (𝐵 − 𝑌)) |
27 | 26 | eqcomd 2742 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝑌) = (𝐵 − 𝑋)) |
28 | 1, 9, 3, 4, 7, 6, 7, 5, 27 | tgcgrcomlr 27369 | . . . . 5 ⊢ (𝜑 → (𝑌 − 𝐵) = (𝑋 − 𝐵)) |
29 | 20, 23, 28 | 3eqtr3rd 2785 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝐵) = (𝑋 − ((𝑆‘𝐴)‘𝐵))) |
30 | 1, 9, 3, 2, 10, 4, 11, 12, 5, 7 | miriso 27559 | . . . . 5 ⊢ (𝜑 → (((𝑆‘𝐴)‘𝑋) − ((𝑆‘𝐴)‘𝐵)) = (𝑋 − 𝐵)) |
31 | 21 | oveq1d 7371 | . . . . 5 ⊢ (𝜑 → (((𝑆‘𝐴)‘𝑋) − ((𝑆‘𝐴)‘𝐵)) = (𝑌 − ((𝑆‘𝐴)‘𝐵))) |
32 | 1, 9, 3, 4, 7, 5, 7, 6, 26 | tgcgrcomlr 27369 | . . . . 5 ⊢ (𝜑 → (𝑋 − 𝐵) = (𝑌 − 𝐵)) |
33 | 30, 31, 32 | 3eqtr3rd 2785 | . . . 4 ⊢ (𝜑 → (𝑌 − 𝐵) = (𝑌 − ((𝑆‘𝐴)‘𝐵))) |
34 | 1, 2, 3, 4, 5, 6, 7, 8, 13, 11, 9, 19, 29, 33 | tgidinside 27460 | . . 3 ⊢ (𝜑 → 𝐵 = ((𝑆‘𝐴)‘𝐵)) |
35 | 34 | eqcomd 2742 | . 2 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) = 𝐵) |
36 | 1, 9, 3, 2, 10, 4, 11, 12, 7 | mirinv 27555 | . 2 ⊢ (𝜑 → (((𝑆‘𝐴)‘𝐵) = 𝐵 ↔ 𝐴 = 𝐵)) |
37 | 35, 36 | mpbid 231 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 distcds 17141 TarskiGcstrkg 27316 Itvcitv 27322 LineGclng 27323 cgrGccgrg 27399 pInvGcmir 27541 |
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-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 27337 df-trkgb 27338 df-trkgcb 27339 df-trkg 27342 df-cgrg 27400 df-mir 27542 |
This theorem is referenced by: miduniq1 27575 krippenlem 27579 mideu 27627 opphllem3 27638 |
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