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Mirrors > Home > MPE Home > Th. List > mirauto | Structured version Visualization version GIF version |
Description: Point inversion preserves point inversion. (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) |
mirauto.m | ⊢ 𝑀 = (𝑆‘𝑇) |
mirauto.x | ⊢ 𝑋 = (𝑀‘𝐴) |
mirauto.y | ⊢ 𝑌 = (𝑀‘𝐵) |
mirauto.z | ⊢ 𝑍 = (𝑀‘𝐶) |
mirauto.0 | ⊢ (𝜑 → 𝑇 ∈ 𝑃) |
mirauto.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
mirauto.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
mirauto.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
mirauto.4 | ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) = 𝐶) |
Ref | Expression |
---|---|
mirauto | ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . . 3 ⊢ − = (dist‘𝐺) | |
3 | mirval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
6 | mirval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | mirauto.x | . . . 4 ⊢ 𝑋 = (𝑀‘𝐴) | |
8 | mirauto.0 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑃) | |
9 | mirauto.m | . . . . . 6 ⊢ 𝑀 = (𝑆‘𝑇) | |
10 | 1, 2, 3, 4, 5, 6, 8, 9 | mirf 27600 | . . . . 5 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) |
11 | mirauto.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
12 | 10, 11 | ffvelcdmd 7035 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
13 | 7, 12 | eqeltrid 2842 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
14 | eqid 2736 | . . 3 ⊢ (𝑆‘𝑋) = (𝑆‘𝑋) | |
15 | mirauto.y | . . . 4 ⊢ 𝑌 = (𝑀‘𝐵) | |
16 | mirauto.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
17 | 10, 16 | ffvelcdmd 7035 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃) |
18 | 15, 17 | eqeltrid 2842 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
19 | mirauto.z | . . . 4 ⊢ 𝑍 = (𝑀‘𝐶) | |
20 | mirauto.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
21 | 10, 20 | ffvelcdmd 7035 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃) |
22 | 19, 21 | eqeltrid 2842 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
23 | mirauto.4 | . . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) = 𝐶) | |
24 | 23, 20 | eqeltrd 2838 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) ∈ 𝑃) |
25 | eqid 2736 | . . . . . 6 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴) | |
26 | 1, 2, 3, 4, 5, 6, 11, 25, 16 | mircgr 27597 | . . . . 5 ⊢ (𝜑 → (𝐴 − ((𝑆‘𝐴)‘𝐵)) = (𝐴 − 𝐵)) |
27 | 1, 2, 3, 4, 5, 6, 8, 9, 11, 24, 11, 16, 26 | mircgrs 27613 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵))) = ((𝑀‘𝐴) − (𝑀‘𝐵))) |
28 | 7 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑋 = (𝑀‘𝐴)) |
29 | 23 | fveq2d 6846 | . . . . . 6 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐴)‘𝐵)) = (𝑀‘𝐶)) |
30 | 19, 29 | eqtr4id 2795 | . . . . 5 ⊢ (𝜑 → 𝑍 = (𝑀‘((𝑆‘𝐴)‘𝐵))) |
31 | 28, 30 | oveq12d 7374 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑍) = ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵)))) |
32 | 7, 15 | oveq12i 7368 | . . . . 5 ⊢ (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵)) |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵))) |
34 | 27, 31, 33 | 3eqtr4d 2786 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝑌)) |
35 | 1, 2, 3, 4, 5, 6, 11, 25, 16 | mirbtwn 27598 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (((𝑆‘𝐴)‘𝐵)𝐼𝐵)) |
36 | 23 | oveq1d 7371 | . . . . . 6 ⊢ (𝜑 → (((𝑆‘𝐴)‘𝐵)𝐼𝐵) = (𝐶𝐼𝐵)) |
37 | 35, 36 | eleqtrd 2840 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵)) |
38 | 1, 2, 3, 4, 5, 6, 8, 9, 20, 11, 16, 37 | mirbtwni 27611 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ((𝑀‘𝐶)𝐼(𝑀‘𝐵))) |
39 | 19, 15 | oveq12i 7368 | . . . 4 ⊢ (𝑍𝐼𝑌) = ((𝑀‘𝐶)𝐼(𝑀‘𝐵)) |
40 | 38, 7, 39 | 3eltr4g 2855 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌)) |
41 | 1, 2, 3, 4, 5, 6, 13, 14, 18, 22, 34, 40 | ismir 27599 | . 2 ⊢ (𝜑 → 𝑍 = ((𝑆‘𝑋)‘𝑌)) |
42 | 41 | eqcomd 2742 | 1 ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 𝑍) |
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 27367 Itvcitv 27373 LineGclng 27374 pInvGcmir 27592 |
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 27388 df-trkgb 27389 df-trkgcb 27390 df-trkg 27393 df-cgrg 27451 df-mir 27593 |
This theorem is referenced by: miduniq2 27627 krippenlem 27630 |
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