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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 26445 | . . . . 5 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) |
11 | mirauto.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
12 | 10, 11 | ffvelrnd 6851 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
13 | 7, 12 | eqeltrid 2917 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
14 | eqid 2821 | . . 3 ⊢ (𝑆‘𝑋) = (𝑆‘𝑋) | |
15 | mirauto.y | . . . 4 ⊢ 𝑌 = (𝑀‘𝐵) | |
16 | mirauto.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
17 | 10, 16 | ffvelrnd 6851 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃) |
18 | 15, 17 | eqeltrid 2917 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
19 | mirauto.z | . . . 4 ⊢ 𝑍 = (𝑀‘𝐶) | |
20 | mirauto.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
21 | 10, 20 | ffvelrnd 6851 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃) |
22 | 19, 21 | eqeltrid 2917 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
23 | mirauto.4 | . . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) = 𝐶) | |
24 | 23, 20 | eqeltrd 2913 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) ∈ 𝑃) |
25 | eqid 2821 | . . . . . 6 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴) | |
26 | 1, 2, 3, 4, 5, 6, 11, 25, 16 | mircgr 26442 | . . . . 5 ⊢ (𝜑 → (𝐴 − ((𝑆‘𝐴)‘𝐵)) = (𝐴 − 𝐵)) |
27 | 1, 2, 3, 4, 5, 6, 8, 9, 11, 24, 11, 16, 26 | mircgrs 26458 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵))) = ((𝑀‘𝐴) − (𝑀‘𝐵))) |
28 | 7 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑋 = (𝑀‘𝐴)) |
29 | 23 | fveq2d 6673 | . . . . . 6 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐴)‘𝐵)) = (𝑀‘𝐶)) |
30 | 29, 19 | syl6reqr 2875 | . . . . 5 ⊢ (𝜑 → 𝑍 = (𝑀‘((𝑆‘𝐴)‘𝐵))) |
31 | 28, 30 | oveq12d 7173 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑍) = ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵)))) |
32 | 7, 15 | oveq12i 7167 | . . . . 5 ⊢ (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵)) |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵))) |
34 | 27, 31, 33 | 3eqtr4d 2866 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝑌)) |
35 | 1, 2, 3, 4, 5, 6, 11, 25, 16 | mirbtwn 26443 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (((𝑆‘𝐴)‘𝐵)𝐼𝐵)) |
36 | 23 | oveq1d 7170 | . . . . . 6 ⊢ (𝜑 → (((𝑆‘𝐴)‘𝐵)𝐼𝐵) = (𝐶𝐼𝐵)) |
37 | 35, 36 | eleqtrd 2915 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵)) |
38 | 1, 2, 3, 4, 5, 6, 8, 9, 20, 11, 16, 37 | mirbtwni 26456 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ((𝑀‘𝐶)𝐼(𝑀‘𝐵))) |
39 | 19, 15 | oveq12i 7167 | . . . 4 ⊢ (𝑍𝐼𝑌) = ((𝑀‘𝐶)𝐼(𝑀‘𝐵)) |
40 | 38, 7, 39 | 3eltr4g 2930 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌)) |
41 | 1, 2, 3, 4, 5, 6, 13, 14, 18, 22, 34, 40 | ismir 26444 | . 2 ⊢ (𝜑 → 𝑍 = ((𝑆‘𝑋)‘𝑌)) |
42 | 41 | eqcomd 2827 | 1 ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 distcds 16573 TarskiGcstrkg 26215 Itvcitv 26221 LineGclng 26222 pInvGcmir 26437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-s1 13949 df-s2 14209 df-s3 14210 df-trkgc 26233 df-trkgb 26234 df-trkgcb 26235 df-trkg 26238 df-cgrg 26296 df-mir 26438 |
This theorem is referenced by: miduniq2 26472 krippenlem 26475 |
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