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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 26373 | . . . . 5 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) |
11 | mirauto.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
12 | 10, 11 | ffvelrnd 6844 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
13 | 7, 12 | eqeltrid 2914 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
14 | eqid 2818 | . . 3 ⊢ (𝑆‘𝑋) = (𝑆‘𝑋) | |
15 | mirauto.y | . . . 4 ⊢ 𝑌 = (𝑀‘𝐵) | |
16 | mirauto.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
17 | 10, 16 | ffvelrnd 6844 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃) |
18 | 15, 17 | eqeltrid 2914 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
19 | mirauto.z | . . . 4 ⊢ 𝑍 = (𝑀‘𝐶) | |
20 | mirauto.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
21 | 10, 20 | ffvelrnd 6844 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃) |
22 | 19, 21 | eqeltrid 2914 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
23 | mirauto.4 | . . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) = 𝐶) | |
24 | 23, 20 | eqeltrd 2910 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) ∈ 𝑃) |
25 | eqid 2818 | . . . . . 6 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴) | |
26 | 1, 2, 3, 4, 5, 6, 11, 25, 16 | mircgr 26370 | . . . . 5 ⊢ (𝜑 → (𝐴 − ((𝑆‘𝐴)‘𝐵)) = (𝐴 − 𝐵)) |
27 | 1, 2, 3, 4, 5, 6, 8, 9, 11, 24, 11, 16, 26 | mircgrs 26386 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵))) = ((𝑀‘𝐴) − (𝑀‘𝐵))) |
28 | 7 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑋 = (𝑀‘𝐴)) |
29 | 23 | fveq2d 6667 | . . . . . 6 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐴)‘𝐵)) = (𝑀‘𝐶)) |
30 | 29, 19 | syl6reqr 2872 | . . . . 5 ⊢ (𝜑 → 𝑍 = (𝑀‘((𝑆‘𝐴)‘𝐵))) |
31 | 28, 30 | oveq12d 7163 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑍) = ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵)))) |
32 | 7, 15 | oveq12i 7157 | . . . . 5 ⊢ (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵)) |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵))) |
34 | 27, 31, 33 | 3eqtr4d 2863 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝑌)) |
35 | 1, 2, 3, 4, 5, 6, 11, 25, 16 | mirbtwn 26371 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (((𝑆‘𝐴)‘𝐵)𝐼𝐵)) |
36 | 23 | oveq1d 7160 | . . . . . 6 ⊢ (𝜑 → (((𝑆‘𝐴)‘𝐵)𝐼𝐵) = (𝐶𝐼𝐵)) |
37 | 35, 36 | eleqtrd 2912 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵)) |
38 | 1, 2, 3, 4, 5, 6, 8, 9, 20, 11, 16, 37 | mirbtwni 26384 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ((𝑀‘𝐶)𝐼(𝑀‘𝐵))) |
39 | 19, 15 | oveq12i 7157 | . . . 4 ⊢ (𝑍𝐼𝑌) = ((𝑀‘𝐶)𝐼(𝑀‘𝐵)) |
40 | 38, 7, 39 | 3eltr4g 2927 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌)) |
41 | 1, 2, 3, 4, 5, 6, 13, 14, 18, 22, 34, 40 | ismir 26372 | . 2 ⊢ (𝜑 → 𝑍 = ((𝑆‘𝑋)‘𝑌)) |
42 | 41 | eqcomd 2824 | 1 ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 distcds 16562 TarskiGcstrkg 26143 Itvcitv 26149 LineGclng 26150 pInvGcmir 26365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-s3 14199 df-trkgc 26161 df-trkgb 26162 df-trkgcb 26163 df-trkg 26166 df-cgrg 26224 df-mir 26366 |
This theorem is referenced by: miduniq2 26400 krippenlem 26403 |
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